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Convergence analysis for gradient descent optimization ...
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Convergence analysis for gradient descent optimization methods in the training of ReLU neural networks Arnulf Jentzen (CUHK Shenzhen, China & University of Münster, Germany) Funded by German Research Foundation through Cluster of Excellence Mathematics Münster Joint works with Patrick Cheridito (ETH Zurich, Switzerland) Benjamin Fehrman (University of Oxford, UK) Benjamin Gess (University of Bielefeld, Germany) Martin Hutzenthaler (University of Duisburg-Essen, Germany) Benno Kuckuck (University of Münster, Germany) Thomas Kruse (University of Giessen, Germany) Tuan Anh Nguyen (University of Duisburg-Essen, Germany) Joshua Lee Padgett (University of Arkansas, USA) Florian Rossmannek (ETH Zurich, Switzerland) Adrian Riekert (University of Münster, Germany) Wednesday, September 27nd, 2021
Transcript of Convergence analysis for gradient descent optimization ...
Convergence analysis for gradient descent optimizationmethods in the training of ReLU neural networks
Arnulf Jentzen (CUHK Shenzhen, China & University of Münster, Germany)Funded by German Research Foundation through Cluster of Excellence Mathematics Münster
Joint works with
Patrick Cheridito (ETH Zurich, Switzerland)Benjamin Fehrman (University of Oxford, UK)
Benjamin Gess (University of Bielefeld, Germany)Martin Hutzenthaler (University of Duisburg-Essen, Germany)
Benno Kuckuck (University of Münster, Germany)Thomas Kruse (University of Giessen, Germany)
Tuan Anh Nguyen (University of Duisburg-Essen, Germany)Joshua Lee Padgett (University of Arkansas, USA)
Florian Rossmannek (ETH Zurich, Switzerland)Adrian Riekert (University of Münster, Germany)
Wednesday, September 27nd, 2021
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Theorem (Hutzenthaler, J, Kruse, Nguyen 2019 SN PDE; Kuckuck, J, Padgett 2021)
Let T , p, κ > 0, let f : R→ R be Lipschitz, ∀ d ∈ N let gd ∈ C1(Rd ,R) andud : [0, T ]× Rd → R be an at most poly. grow. solution of
∂ud∂t = ∆x ud + f(ud ) with ud (0, ·) = gd ,
assume |gd (x)|+ ‖(∇gd )(x)‖ ≤ κdκ(1 + ‖x‖κ), letAl : Rl → Rl , l ∈ N, satisfyAl(x1, . . . , xl) = (maxx1, 0, . . . ,maxxl , 0), let
N = ∪L∈N ∪l0,...,lL∈N (×Ln=1(Rln×ln−1 × Rln )),
letR : N→ ∪∞a,b=1C(Ra,Rb) satisfy for all L ∈ N, l0, . . . , lL ∈ N,Φ = ((W1, B1), . . . , (WL, BL)) ∈ ×L
n=1(Rln×ln−1 × Rln ), x0 ∈ Rl0 , . . ., xL ∈ RlL with∀ n ∈ 1, . . . , L : xn = Aln (Wnxn−1 + Bn) that
(RΦ)(x0) = WLxL−1 + BL,let P : N→ N be the number of parameters, and let (Gd,ε)d∈N, ε∈(0,1] ⊆ N satisfyP(Gd,ε)≤κdκε−κ and |gd (x)− (RGd,ε)(x)| ≤ εκdκ(1 + ‖x‖κ). Then∃ (Ud,ε)d∈N,ε∈(0,1] ⊆ N, c > 0 : ∀ d ∈ N, ε ∈ (0, 1] :[ ∫
[0,T ]×[0,1]d
|ud (y)− (RUd,ε)(y)|p dy
]1/p
≤ ε and P(Ud,ε) ≤ c dcε−c.
Linear PDEs: Grohs, Hornung, J, von Wurstemberger 2018 Mem. AMS; Berner,Grohs, J 2018 SIAM J. Math. Data Sci.; Elbrächter, Grohs, J, Schwab 2018; . . .
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Let d,H,P ∈ N, a ∈ R, b > a, u ∈ C([a,b]d ,R) satisfy P = dH + 2H + 1, letµ : B([a,b]d )→ [0,∞) be measure, let LH : RP → R satisfy ∀ θ ∈ RP :
LH(θ)=∫
[a,b]d
[u(x)−θP−
∑Hi=1θH(d+1)+i maxθHd+i +
∑dj=1 θ(i−1)d+jxj , 0
]2µ(dx),
let GH : RP → RP be an appropriately generalized gradient of LH , and letΘ ∈ C([0,∞),RP) satisfy for all t ∈ [0,∞) that Θt = Θ0 −
∫ t0 GH(Θs) ds.
Theorem (Cheridito, J, Riekert, Rossmannek 2021; J, Riekert 2021)
Assume for all x, y ∈ [a,b]d that u(x) = u(y). Then there exist no non-global localminima and no saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Cheridito, J, Rossmannek 2021; J, Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R satisfy u(x) = αx + β, and assume
lim inft→∞‖Θt‖ <∞ and LH(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then there exist∞-many
non-global local minima and∞-many saddle points of LH and limt→∞ LH(Θt) = 0.
Theorem (Eberle, J, Riekert, Weiss 2021)
Assume µ λ[a,b]d with a piecewise polynomial density, assume that u is piecewisepolynomial, and assume lim inft→∞‖Θt‖ <∞. Then there exist ϑ ∈ (GH)−1(0),c, β > 0 such that ‖Θt −ϑ‖ ≤ c(1 + t)−β and 0 ≤ LH(Θt)−L(ϑ) ≤ c(1 + t)−1.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Theorem (J, Riekert 2021)
Assume µ λ[a,b]d and lim inft→∞‖Θt‖ <∞. Then there existsϑ ∈ (GH)−1(0) such that limt→∞ LH(Θt) = LH(ϑ).
Theorem (J, Riekert 2021)
Assume µ ∼ λ[a,b] with a Lipschitz continuous density, assume that u is piecewiseaffine linear, let (Ω,F ,P) be a probability space, for every H, k ∈ N, γ ∈ R letΘH,k,γ
n : Ω→ R3H+1, n ∈ N0, and kH,k,γn : Ω→ N, n ∈ N0, be random variables
satisfying for all n ∈ N0 that
ΘH,k,γn+1 = ΘH,k,γ
n − γGH(ΘH,k,γn ) and kH,k,γ
n ∈ arg min`∈1,...,k LH(ΘH,`,γn ),
and assume for all H ∈ N, γ ∈ R that ΘH,k,γ0 , k ∈ N, are i.i.d. standard normal. Then
∃ g > 0 : ∀ γ ∈ (0, g] :
lim infH→∞ lim infK→∞ P(
lim supn→∞ LH
(ΘH,kH,K ,γ
n ,γn
)= 0)
= 1.
Proof based on Fehrman, Gess, J 2020 JMLR.
Appendix
Let d,H,P ∈ N, a ∈ R, b ∈ (a,∞), f ∈ C([a,b]d ,R) satisfyP = dH + 2H + 1, let Rr ∈ C(R,R), r ∈ N ∪ ∞, satisfy for all x ∈ R that(⋃
r∈NRr) ⊆ C1(R,R), R∞(x) = maxx, 0,supr∈N supy∈[−|x|,|x|]|(Rr )
′(y)| <∞, and
lim supr→∞(|Rr (x)−R∞(x)|+ |(Rr )
′(x)− 1(0,∞)(x)|)
= 0,
let µ : B([a,b]d )→ [0,∞] be a measure, let Lr : RP → R, r ∈ N ∪ ∞, satisfyfor all r ∈ N ∪ ∞, θ = (θ1, . . . , θP) ∈ RP that
Lr (θ) =∫
[a,b]d
(f(x1, . . . , xd )
−θP −∑H
i=1 θH(d+1)+i
[Rr (θHd+i +
∑dj=1 θ(i−1)d+jxj)
])2µ(d(x1, . . . , xd )),
let G : RP → RP satisfy for all θ ∈ ϑ ∈ RP : ((∇Lr )(ϑ))r∈N is convergent thatG(θ) = limr→∞(∇Lr )(θ), and let Θ ∈ C([0,∞),RP) satisfy
∀ t ∈ [0,∞) : Θt = Θ0 −∫ t
0 G(Θs) ds.
Lemma
There exists an open U ⊆ RP such that∫RP\U 1 dx = 0, (L∞)|U ∈ C1(U,R), and
∇((L∞)|U) = G|U .
Let d,H,P ∈ N, a ∈ R, b ∈ (a,∞), f ∈ C([a,b]d ,R) satisfyP = dH + 2H + 1, let Rr ∈ C(R,R), r ∈ N ∪ ∞, satisfy for all x ∈ R that(⋃
r∈NRr) ⊆ C1(R,R), R∞(x) = maxx, 0,supr∈N supy∈[−|x|,|x|]|(Rr )
′(y)| <∞, and
lim supr→∞(|Rr (x)−R∞(x)|+ |(Rr )
′(x)− 1(0,∞)(x)|)
= 0,
let µ : B([a,b]d )→ [0,∞] be a measure, let Lr : RP → R, r ∈ N ∪ ∞, satisfyfor all r ∈ N ∪ ∞, θ = (θ1, . . . , θP) ∈ RP that
Lr (θ) =∫
[a,b]d
(f(x1, . . . , xd )
−θP −∑H
i=1 θH(d+1)+i
[Rr (θHd+i +
∑dj=1 θ(i−1)d+jxj)
])2µ(d(x1, . . . , xd )),
let G : RP → RP satisfy for all θ ∈ ϑ ∈ RP : ((∇Lr )(ϑ))r∈N is convergent thatG(θ) = limr→∞(∇Lr )(θ), and let Θ ∈ C([0,∞),RP) satisfy
∀ t ∈ [0,∞) : Θt = Θ0 −∫ t
0 G(Θs) ds.
Lemma
There exists an open U ⊆ RP such that∫RP\U 1 dx = 0, (L∞)|U ∈ C1(U,R), and
∇((L∞)|U) = G|U .
Let d,H,P ∈ N, a ∈ R, b ∈ (a,∞), f ∈ C([a,b]d ,R) satisfyP = dH + 2H + 1, let Rr ∈ C(R,R), r ∈ N ∪ ∞, satisfy for all x ∈ R that(⋃
r∈NRr) ⊆ C1(R,R), R∞(x) = maxx, 0,supr∈N supy∈[−|x|,|x|]|(Rr )
′(y)| <∞, and
lim supr→∞(|Rr (x)−R∞(x)|+ |(Rr )
′(x)− 1(0,∞)(x)|)
= 0,
let µ : B([a,b]d )→ [0,∞] be a measure, let Lr : RP → R, r ∈ N ∪ ∞, satisfyfor all r ∈ N ∪ ∞, θ = (θ1, . . . , θP) ∈ RP that
Lr (θ) =∫
[a,b]d
(f(x1, . . . , xd )
−θP −∑H
i=1 θH(d+1)+i
[Rr (θHd+i +
∑dj=1 θ(i−1)d+jxj)
])2µ(d(x1, . . . , xd )),
let G : RP → RP satisfy for all θ ∈ ϑ ∈ RP : ((∇Lr )(ϑ))r∈N is convergent thatG(θ) = limr→∞(∇Lr )(θ), and let Θ ∈ C([0,∞),RP) satisfy
∀ t ∈ [0,∞) : Θt = Θ0 −∫ t
0 G(Θs) ds.
Lemma
There exists an open U ⊆ RP such that∫RP\U 1 dx = 0, (L∞)|U ∈ C1(U,R), and
∇((L∞)|U) = G|U .
Let d,H,P ∈ N, a ∈ R, b ∈ (a,∞), f ∈ C([a,b]d ,R) satisfyP = dH + 2H + 1, let Rr ∈ C(R,R), r ∈ N ∪ ∞, satisfy for all x ∈ R that(⋃
r∈NRr) ⊆ C1(R,R), R∞(x) = maxx, 0,supr∈N supy∈[−|x|,|x|]|(Rr )
′(y)| <∞, and
lim supr→∞(|Rr (x)−R∞(x)|+ |(Rr )
′(x)− 1(0,∞)(x)|)
= 0,
let µ : B([a,b]d )→ [0,∞] be a measure, let Lr : RP → R, r ∈ N ∪ ∞, satisfyfor all r ∈ N ∪ ∞, θ = (θ1, . . . , θP) ∈ RP that
Lr (θ) =∫
[a,b]d
(f(x1, . . . , xd )
−θP −∑H
i=1 θH(d+1)+i
[Rr (θHd+i +
∑dj=1 θ(i−1)d+jxj)
])2µ(d(x1, . . . , xd )),
let G : RP → RP satisfy for all θ ∈ ϑ ∈ RP : ((∇Lr )(ϑ))r∈N is convergent thatG(θ) = limr→∞(∇Lr )(θ), and let Θ ∈ C([0,∞),RP) satisfy
∀ t ∈ [0,∞) : Θt = Θ0 −∫ t
0 G(Θs) ds.
Lemma
There exists an open U ⊆ RP such that∫RP\U 1 dx = 0, (L∞)|U ∈ C1(U,R), and
∇((L∞)|U) = G|U .
Let d,H,P ∈ N, a ∈ R, b ∈ (a,∞), f ∈ C([a,b]d ,R) satisfyP = dH + 2H + 1, let Rr ∈ C(R,R), r ∈ N ∪ ∞, satisfy for all x ∈ R that(⋃
r∈NRr) ⊆ C1(R,R), R∞(x) = maxx, 0,supr∈N supy∈[−|x|,|x|]|(Rr )
′(y)| <∞, and
lim supr→∞(|Rr (x)−R∞(x)|+ |(Rr )
′(x)− 1(0,∞)(x)|)
= 0,
let µ : B([a,b]d )→ [0,∞] be a measure, let Lr : RP → R, r ∈ N ∪ ∞, satisfyfor all r ∈ N ∪ ∞, θ = (θ1, . . . , θP) ∈ RP that
Lr (θ) =∫
[a,b]d
(f(x1, . . . , xd )
−θP −∑H
i=1 θH(d+1)+i
[Rr (θHd+i +
∑dj=1 θ(i−1)d+jxj)
])2µ(d(x1, . . . , xd )),
let G : RP → RP satisfy for all θ ∈ ϑ ∈ RP : ((∇Lr )(ϑ))r∈N is convergent thatG(θ) = limr→∞(∇Lr )(θ), and let Θ ∈ C([0,∞),RP) satisfy
∀ t ∈ [0,∞) : Θt = Θ0 −∫ t
0 G(Θs) ds.
Lemma
There exists an open U ⊆ RP such that∫RP\U 1 dx = 0, (L∞)|U ∈ C1(U,R), and
∇((L∞)|U) = G|U .
Let d,H,P ∈ N, a ∈ R, b ∈ (a,∞), f ∈ C([a,b]d ,R) satisfyP = dH + 2H + 1, let Rr ∈ C(R,R), r ∈ N ∪ ∞, satisfy for all x ∈ R that(⋃
r∈NRr) ⊆ C1(R,R), R∞(x) = maxx, 0,supr∈N supy∈[−|x|,|x|]|(Rr )
′(y)| <∞, and
lim supr→∞(|Rr (x)−R∞(x)|+ |(Rr )
′(x)− 1(0,∞)(x)|)
= 0,
let µ : B([a,b]d )→ [0,∞] be a measure, let Lr : RP → R, r ∈ N ∪ ∞, satisfyfor all r ∈ N ∪ ∞, θ = (θ1, . . . , θP) ∈ RP that
Lr (θ) =∫
[a,b]d
(f(x1, . . . , xd )
−θP −∑H
i=1 θH(d+1)+i
[Rr (θHd+i +
∑dj=1 θ(i−1)d+jxj)
])2µ(d(x1, . . . , xd )),
let G : RP → RP satisfy for all θ ∈ ϑ ∈ RP : ((∇Lr )(ϑ))r∈N is convergent thatG(θ) = limr→∞(∇Lr )(θ), and let Θ ∈ C([0,∞),RP) satisfy
∀ t ∈ [0,∞) : Θt = Θ0 −∫ t
0 G(Θs) ds.
Lemma
There exists an open U ⊆ RP such that∫RP\U 1 dx = 0, (L∞)|U ∈ C1(U,R), and
∇((L∞)|U) = G|U .
Let d,H,P ∈ N, a ∈ R, b ∈ (a,∞), f ∈ C([a,b]d ,R) satisfyP = dH + 2H + 1, let Rr ∈ C(R,R), r ∈ N ∪ ∞, satisfy for all x ∈ R that(⋃
r∈NRr) ⊆ C1(R,R), R∞(x) = maxx, 0,supr∈N supy∈[−|x|,|x|]|(Rr )
′(y)| <∞, and
lim supr→∞(|Rr (x)−R∞(x)|+ |(Rr )
′(x)− 1(0,∞)(x)|)
= 0,
let µ : B([a,b]d )→ [0,∞] be a measure, let Lr : RP → R, r ∈ N ∪ ∞, satisfyfor all r ∈ N ∪ ∞, θ = (θ1, . . . , θP) ∈ RP that
Lr (θ) =∫
[a,b]d
(f(x1, . . . , xd )
−θP −∑H
i=1 θH(d+1)+i
[Rr (θHd+i +
∑dj=1 θ(i−1)d+jxj)
])2µ(d(x1, . . . , xd )),
let G : RP → RP satisfy for all θ ∈ ϑ ∈ RP : ((∇Lr )(ϑ))r∈N is convergent thatG(θ) = limr→∞(∇Lr )(θ), and let Θ ∈ C([0,∞),RP) satisfy
∀ t ∈ [0,∞) : Θt = Θ0 −∫ t
0 G(Θs) ds.
Lemma
There exists an open U ⊆ RP such that∫RP\U 1 dx = 0, (L∞)|U ∈ C1(U,R), and
∇((L∞)|U) = G|U .
Let d,H,P ∈ N, a ∈ R, b ∈ (a,∞), f ∈ C([a,b]d ,R) satisfyP = dH + 2H + 1, let Rr ∈ C(R,R), r ∈ N ∪ ∞, satisfy for all x ∈ R that(⋃
r∈NRr) ⊆ C1(R,R), R∞(x) = maxx, 0,supr∈N supy∈[−|x|,|x|]|(Rr )
′(y)| <∞, and
lim supr→∞(|Rr (x)−R∞(x)|+ |(Rr )
′(x)− 1(0,∞)(x)|)
= 0,
let µ : B([a,b]d )→ [0,∞] be a measure, let Lr : RP → R, r ∈ N ∪ ∞, satisfyfor all r ∈ N ∪ ∞, θ = (θ1, . . . , θP) ∈ RP that
Lr (θ) =∫
[a,b]d
(f(x1, . . . , xd )
−θP −∑H
i=1 θH(d+1)+i
[Rr (θHd+i +
∑dj=1 θ(i−1)d+jxj)
])2µ(d(x1, . . . , xd )),
let G : RP → RP satisfy for all θ ∈ ϑ ∈ RP : ((∇Lr )(ϑ))r∈N is convergent thatG(θ) = limr→∞(∇Lr )(θ), and let Θ ∈ C([0,∞),RP) satisfy
∀ t ∈ [0,∞) : Θt = Θ0 −∫ t
0 G(Θs) ds.
Lemma
There exists an open U ⊆ RP such that∫RP\U 1 dx = 0, (L∞)|U ∈ C1(U,R), and
∇((L∞)|U) = G|U .
Let d,H,P ∈ N, a ∈ R, b ∈ (a,∞), f ∈ C([a,b]d ,R) satisfyP = dH + 2H + 1, let Rr ∈ C(R,R), r ∈ N ∪ ∞, satisfy for all x ∈ R that(⋃
r∈NRr) ⊆ C1(R,R), R∞(x) = maxx, 0,supr∈N supy∈[−|x|,|x|]|(Rr )
′(y)| <∞, and
lim supr→∞(|Rr (x)−R∞(x)|+ |(Rr )
′(x)− 1(0,∞)(x)|)
= 0,
let µ : B([a,b]d )→ [0,∞] be a measure, let Lr : RP → R, r ∈ N ∪ ∞, satisfyfor all r ∈ N ∪ ∞, θ = (θ1, . . . , θP) ∈ RP that
Lr (θ) =∫
[a,b]d
(f(x1, . . . , xd )
−θP −∑H
i=1 θH(d+1)+i
[Rr (θHd+i +
∑dj=1 θ(i−1)d+jxj)
])2µ(d(x1, . . . , xd )),
let G : RP → RP satisfy for all θ ∈ ϑ ∈ RP : ((∇Lr )(ϑ))r∈N is convergent thatG(θ) = limr→∞(∇Lr )(θ), and let Θ ∈ C([0,∞),RP) satisfy
∀ t ∈ [0,∞) : Θt = Θ0 −∫ t
0 G(Θs) ds.
Lemma
There exists an open U ⊆ RP such that∫RP\U 1 dx = 0, (L∞)|U ∈ C1(U,R), and
∇((L∞)|U) = G|U .
Let d,H,P ∈ N, a ∈ R, b ∈ (a,∞), f ∈ C([a,b]d ,R) satisfyP = dH + 2H + 1, let Rr ∈ C(R,R), r ∈ N ∪ ∞, satisfy for all x ∈ R that(⋃
r∈NRr) ⊆ C1(R,R), R∞(x) = maxx, 0,supr∈N supy∈[−|x|,|x|]|(Rr )
′(y)| <∞, and
lim supr→∞(|Rr (x)−R∞(x)|+ |(Rr )
′(x)− 1(0,∞)(x)|)
= 0,
let µ : B([a,b]d )→ [0,∞] be a measure, let Lr : RP → R, r ∈ N ∪ ∞, satisfyfor all r ∈ N ∪ ∞, θ = (θ1, . . . , θP) ∈ RP that
Lr (θ) =∫
[a,b]d
(f(x1, . . . , xd )
−θP −∑H
i=1 θH(d+1)+i
[Rr (θHd+i +
∑dj=1 θ(i−1)d+jxj)
])2µ(d(x1, . . . , xd )),
let G : RP → RP satisfy for all θ ∈ ϑ ∈ RP : ((∇Lr )(ϑ))r∈N is convergent thatG(θ) = limr→∞(∇Lr )(θ), and let Θ ∈ C([0,∞),RP) satisfy
∀ t ∈ [0,∞) : Θt = Θ0 −∫ t
0 G(Θs) ds.
Lemma
There exists an open U ⊆ RP such that∫RP\U 1 dx = 0, (L∞)|U ∈ C1(U,R), and
∇((L∞)|U) = G|U .
Let d,H,P ∈ N, a ∈ R, b ∈ (a,∞), f ∈ C([a,b]d ,R) satisfyP = dH + 2H + 1, let Rr ∈ C(R,R), r ∈ N ∪ ∞, satisfy for all x ∈ R that(⋃
r∈NRr) ⊆ C1(R,R), R∞(x) = maxx, 0,supr∈N supy∈[−|x|,|x|]|(Rr )
′(y)| <∞, and
lim supr→∞(|Rr (x)−R∞(x)|+ |(Rr )
′(x)− 1(0,∞)(x)|)
= 0,
let µ : B([a,b]d )→ [0,∞] be a measure, let Lr : RP → R, r ∈ N ∪ ∞, satisfyfor all r ∈ N ∪ ∞, θ = (θ1, . . . , θP) ∈ RP that
Lr (θ) =∫
[a,b]d
(f(x1, . . . , xd )
−θP −∑H
i=1 θH(d+1)+i
[Rr (θHd+i +
∑dj=1 θ(i−1)d+jxj)
])2µ(d(x1, . . . , xd )),
let G : RP → RP satisfy for all θ ∈ ϑ ∈ RP : ((∇Lr )(ϑ))r∈N is convergent thatG(θ) = limr→∞(∇Lr )(θ), and let Θ ∈ C([0,∞),RP) satisfy
∀ t ∈ [0,∞) : Θt = Θ0 −∫ t
0 G(Θs) ds.
Lemma
There exists an open U ⊆ RP such that∫RP\U 1 dx = 0, (L∞)|U ∈ C1(U,R), and
∇((L∞)|U) = G|U .
Let d,H,P ∈ N, a ∈ R, b ∈ (a,∞), f ∈ C([a,b]d ,R) satisfyP = dH + 2H + 1, let Rr ∈ C(R,R), r ∈ N ∪ ∞, satisfy for all x ∈ R that(⋃
r∈NRr) ⊆ C1(R,R), R∞(x) = maxx, 0,supr∈N supy∈[−|x|,|x|]|(Rr )
′(y)| <∞, and
lim supr→∞(|Rr (x)−R∞(x)|+ |(Rr )
′(x)− 1(0,∞)(x)|)
= 0,
let µ : B([a,b]d )→ [0,∞] be a measure, let Lr : RP → R, r ∈ N ∪ ∞, satisfyfor all r ∈ N ∪ ∞, θ = (θ1, . . . , θP) ∈ RP that
Lr (θ) =∫
[a,b]d
(f(x1, . . . , xd )
−θP −∑H
i=1 θH(d+1)+i
[Rr (θHd+i +
∑dj=1 θ(i−1)d+jxj)
])2µ(d(x1, . . . , xd )),
let G : RP → RP satisfy for all θ ∈ ϑ ∈ RP : ((∇Lr )(ϑ))r∈N is convergent thatG(θ) = limr→∞(∇Lr )(θ), and let Θ ∈ C([0,∞),RP) satisfy
∀ t ∈ [0,∞) : Θt = Θ0 −∫ t
0 G(Θs) ds.
Lemma
There exists an open U ⊆ RP such that∫RP\U 1 dx = 0, (L∞)|U ∈ C1(U,R), and
∇((L∞)|U) = G|U .
Let d,H,P ∈ N, a ∈ R, b ∈ (a,∞), f ∈ C([a,b]d ,R) satisfyP = dH + 2H + 1, let Rr ∈ C(R,R), r ∈ N ∪ ∞, satisfy for all x ∈ R that(⋃
r∈NRr) ⊆ C1(R,R), R∞(x) = maxx, 0,supr∈N supy∈[−|x|,|x|]|(Rr )
′(y)| <∞, and
lim supr→∞(|Rr (x)−R∞(x)|+ |(Rr )
′(x)− 1(0,∞)(x)|)
= 0,
let µ : B([a,b]d )→ [0,∞] be a measure, let Lr : RP → R, r ∈ N ∪ ∞, satisfyfor all r ∈ N ∪ ∞, θ = (θ1, . . . , θP) ∈ RP that
Lr (θ) =∫
[a,b]d
(f(x1, . . . , xd )
−θP −∑H
i=1 θH(d+1)+i
[Rr (θHd+i +
∑dj=1 θ(i−1)d+jxj)
])2µ(d(x1, . . . , xd )),
let G : RP → RP satisfy for all θ ∈ ϑ ∈ RP : ((∇Lr )(ϑ))r∈N is convergent thatG(θ) = limr→∞(∇Lr )(θ), and let Θ ∈ C([0,∞),RP) satisfy
∀ t ∈ [0,∞) : Θt = Θ0 −∫ t
0 G(Θs) ds.
Lemma
There exists an open U ⊆ RP such that∫RP\U 1 dx = 0, (L∞)|U ∈ C1(U,R), and
∇((L∞)|U) = G|U .
Let d,H,P ∈ N, a ∈ R, b ∈ (a,∞), f ∈ C([a,b]d ,R) satisfyP = dH + 2H + 1, let Rr ∈ C(R,R), r ∈ N ∪ ∞, satisfy for all x ∈ R that(⋃
r∈NRr) ⊆ C1(R,R), R∞(x) = maxx, 0,supr∈N supy∈[−|x|,|x|]|(Rr )
′(y)| <∞, and
lim supr→∞(|Rr (x)−R∞(x)|+ |(Rr )
′(x)− 1(0,∞)(x)|)
= 0,
let µ : B([a,b]d )→ [0,∞] be a measure, let Lr : RP → R, r ∈ N ∪ ∞, satisfyfor all r ∈ N ∪ ∞, θ = (θ1, . . . , θP) ∈ RP that
Lr (θ) =∫
[a,b]d
(f(x1, . . . , xd )
−θP −∑H
i=1 θH(d+1)+i
[Rr (θHd+i +
∑dj=1 θ(i−1)d+jxj)
])2µ(d(x1, . . . , xd )),
let G : RP → RP satisfy for all θ ∈ ϑ ∈ RP : ((∇Lr )(ϑ))r∈N is convergent thatG(θ) = limr→∞(∇Lr )(θ), and let Θ ∈ C([0,∞),RP) satisfy
∀ t ∈ [0,∞) : Θt = Θ0 −∫ t
0 G(Θs) ds.
Lemma
There exists an open U ⊆ RP such that∫RP\U 1 dx = 0, (L∞)|U ∈ C1(U,R), and
∇((L∞)|U) = G|U .
Let d,H,P ∈ N, a ∈ R, b ∈ (a,∞), f ∈ C([a,b]d ,R) satisfyP = dH + 2H + 1, let Rr ∈ C(R,R), r ∈ N ∪ ∞, satisfy for all x ∈ R that(⋃
r∈NRr) ⊆ C1(R,R), R∞(x) = maxx, 0,supr∈N supy∈[−|x|,|x|]|(Rr )
′(y)| <∞, and
lim supr→∞(|Rr (x)−R∞(x)|+ |(Rr )
′(x)− 1(0,∞)(x)|)
= 0,
let µ : B([a,b]d )→ [0,∞] be a measure, let Lr : RP → R, r ∈ N ∪ ∞, satisfyfor all r ∈ N ∪ ∞, θ = (θ1, . . . , θP) ∈ RP that
Lr (θ) =∫
[a,b]d
(f(x1, . . . , xd )
−θP −∑H
i=1 θH(d+1)+i
[Rr (θHd+i +
∑dj=1 θ(i−1)d+jxj)
])2µ(d(x1, . . . , xd )),
let G : RP → RP satisfy for all θ ∈ ϑ ∈ RP : ((∇Lr )(ϑ))r∈N is convergent thatG(θ) = limr→∞(∇Lr )(θ), and let Θ ∈ C([0,∞),RP) satisfy
∀ t ∈ [0,∞) : Θt = Θ0 −∫ t
0 G(Θs) ds.
Lemma
There exists an open U ⊆ RP such that∫RP\U 1 dx = 0, (L∞)|U ∈ C1(U,R), and
∇((L∞)|U) = G|U .
Let d,H,P ∈ N, a ∈ R, b ∈ (a,∞), f ∈ C([a,b]d ,R) satisfyP = dH + 2H + 1, let Rr ∈ C(R,R), r ∈ N ∪ ∞, satisfy for all x ∈ R that(⋃
r∈NRr) ⊆ C1(R,R), R∞(x) = maxx, 0,supr∈N supy∈[−|x|,|x|]|(Rr )
′(y)| <∞, and
lim supr→∞(|Rr (x)−R∞(x)|+ |(Rr )
′(x)− 1(0,∞)(x)|)
= 0,
let µ : B([a,b]d )→ [0,∞] be a measure, let Lr : RP → R, r ∈ N ∪ ∞, satisfyfor all r ∈ N ∪ ∞, θ = (θ1, . . . , θP) ∈ RP that
Lr (θ) =∫
[a,b]d
(f(x1, . . . , xd )
−θP −∑H
i=1 θH(d+1)+i
[Rr (θHd+i +
∑dj=1 θ(i−1)d+jxj)
])2µ(d(x1, . . . , xd )),
let G : RP → RP satisfy for all θ ∈ ϑ ∈ RP : ((∇Lr )(ϑ))r∈N is convergent thatG(θ) = limr→∞(∇Lr )(θ), and let Θ ∈ C([0,∞),RP) satisfy
∀ t ∈ [0,∞) : Θt = Θ0 −∫ t
0 G(Θs) ds.
Lemma
There exists an open U ⊆ RP such that∫RP\U 1 dx = 0, (L∞)|U ∈ C1(U,R), and
∇((L∞)|U) = G|U .
Theorem (Cheridito J Riekert Rossmannek 2021, J Riekert 2021)
Assume for all x, y ∈ [a,b]d that f(x) = f(y). Then
1 there exist no non-global local minima of L∞,
2 there exist no saddle points of L∞,
3 it holds for all θ ∈ G−1(0) that θ is a global minima of L∞, and
4 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (Cheridito J Rossmannek 2021, J Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R, satisfy for all x ∈ [a,b] that f(x) = αx + β, and
assume supt∈[0,∞)(H − 1)‖Θt‖ <∞, and L∞(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then
1 there exist infinitely many non-global local minima of L∞,
2 there exist infinitely many saddle points of L∞, and
3 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (J Riekert 2021)
Assume supt∈[0,∞)‖Θt‖ <∞ and µ λ[a,b]d . Then there exists ϑ ∈ G−1(0)such that lim supt→∞ L∞(Θt) = L∞(ϑ).
Theorem (Cheridito J Riekert Rossmannek 2021, J Riekert 2021)
Assume for all x, y ∈ [a,b]d that f(x) = f(y). Then
1 there exist no non-global local minima of L∞,
2 there exist no saddle points of L∞,
3 it holds for all θ ∈ G−1(0) that θ is a global minima of L∞, and
4 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (Cheridito J Rossmannek 2021, J Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R, satisfy for all x ∈ [a,b] that f(x) = αx + β, and
assume supt∈[0,∞)(H − 1)‖Θt‖ <∞, and L∞(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then
1 there exist infinitely many non-global local minima of L∞,
2 there exist infinitely many saddle points of L∞, and
3 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (J Riekert 2021)
Assume supt∈[0,∞)‖Θt‖ <∞ and µ λ[a,b]d . Then there exists ϑ ∈ G−1(0)such that lim supt→∞ L∞(Θt) = L∞(ϑ).
Theorem (Cheridito J Riekert Rossmannek 2021, J Riekert 2021)
Assume for all x, y ∈ [a,b]d that f(x) = f(y). Then
1 there exist no non-global local minima of L∞,
2 there exist no saddle points of L∞,
3 it holds for all θ ∈ G−1(0) that θ is a global minima of L∞, and
4 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (Cheridito J Rossmannek 2021, J Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R, satisfy for all x ∈ [a,b] that f(x) = αx + β, and
assume supt∈[0,∞)(H − 1)‖Θt‖ <∞, and L∞(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then
1 there exist infinitely many non-global local minima of L∞,
2 there exist infinitely many saddle points of L∞, and
3 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (J Riekert 2021)
Assume supt∈[0,∞)‖Θt‖ <∞ and µ λ[a,b]d . Then there exists ϑ ∈ G−1(0)such that lim supt→∞ L∞(Θt) = L∞(ϑ).
Theorem (Cheridito J Riekert Rossmannek 2021, J Riekert 2021)
Assume for all x, y ∈ [a,b]d that f(x) = f(y). Then
1 there exist no non-global local minima of L∞,
2 there exist no saddle points of L∞,
3 it holds for all θ ∈ G−1(0) that θ is a global minima of L∞, and
4 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (Cheridito J Rossmannek 2021, J Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R, satisfy for all x ∈ [a,b] that f(x) = αx + β, and
assume supt∈[0,∞)(H − 1)‖Θt‖ <∞, and L∞(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then
1 there exist infinitely many non-global local minima of L∞,
2 there exist infinitely many saddle points of L∞, and
3 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (J Riekert 2021)
Assume supt∈[0,∞)‖Θt‖ <∞ and µ λ[a,b]d . Then there exists ϑ ∈ G−1(0)such that lim supt→∞ L∞(Θt) = L∞(ϑ).
Theorem (Cheridito J Riekert Rossmannek 2021, J Riekert 2021)
Assume for all x, y ∈ [a,b]d that f(x) = f(y). Then
1 there exist no non-global local minima of L∞,
2 there exist no saddle points of L∞,
3 it holds for all θ ∈ G−1(0) that θ is a global minima of L∞, and
4 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (Cheridito J Rossmannek 2021, J Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R, satisfy for all x ∈ [a,b] that f(x) = αx + β, and
assume supt∈[0,∞)(H − 1)‖Θt‖ <∞, and L∞(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then
1 there exist infinitely many non-global local minima of L∞,
2 there exist infinitely many saddle points of L∞, and
3 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (J Riekert 2021)
Assume supt∈[0,∞)‖Θt‖ <∞ and µ λ[a,b]d . Then there exists ϑ ∈ G−1(0)such that lim supt→∞ L∞(Θt) = L∞(ϑ).
Theorem (Cheridito J Riekert Rossmannek 2021, J Riekert 2021)
Assume for all x, y ∈ [a,b]d that f(x) = f(y). Then
1 there exist no non-global local minima of L∞,
2 there exist no saddle points of L∞,
3 it holds for all θ ∈ G−1(0) that θ is a global minima of L∞, and
4 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (Cheridito J Rossmannek 2021, J Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R, satisfy for all x ∈ [a,b] that f(x) = αx + β, and
assume supt∈[0,∞)(H − 1)‖Θt‖ <∞, and L∞(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then
1 there exist infinitely many non-global local minima of L∞,
2 there exist infinitely many saddle points of L∞, and
3 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (J Riekert 2021)
Assume supt∈[0,∞)‖Θt‖ <∞ and µ λ[a,b]d . Then there exists ϑ ∈ G−1(0)such that lim supt→∞ L∞(Θt) = L∞(ϑ).
Theorem (Cheridito J Riekert Rossmannek 2021, J Riekert 2021)
Assume for all x, y ∈ [a,b]d that f(x) = f(y). Then
1 there exist no non-global local minima of L∞,
2 there exist no saddle points of L∞,
3 it holds for all θ ∈ G−1(0) that θ is a global minima of L∞, and
4 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (Cheridito J Rossmannek 2021, J Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R, satisfy for all x ∈ [a,b] that f(x) = αx + β, and
assume supt∈[0,∞)(H − 1)‖Θt‖ <∞, and L∞(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then
1 there exist infinitely many non-global local minima of L∞,
2 there exist infinitely many saddle points of L∞, and
3 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (J Riekert 2021)
Assume supt∈[0,∞)‖Θt‖ <∞ and µ λ[a,b]d . Then there exists ϑ ∈ G−1(0)such that lim supt→∞ L∞(Θt) = L∞(ϑ).
Theorem (Cheridito J Riekert Rossmannek 2021, J Riekert 2021)
Assume for all x, y ∈ [a,b]d that f(x) = f(y). Then
1 there exist no non-global local minima of L∞,
2 there exist no saddle points of L∞,
3 it holds for all θ ∈ G−1(0) that θ is a global minima of L∞, and
4 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (Cheridito J Rossmannek 2021, J Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R, satisfy for all x ∈ [a,b] that f(x) = αx + β, and
assume supt∈[0,∞)(H − 1)‖Θt‖ <∞, and L∞(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then
1 there exist infinitely many non-global local minima of L∞,
2 there exist infinitely many saddle points of L∞, and
3 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (J Riekert 2021)
Assume supt∈[0,∞)‖Θt‖ <∞ and µ λ[a,b]d . Then there exists ϑ ∈ G−1(0)such that lim supt→∞ L∞(Θt) = L∞(ϑ).
Theorem (Cheridito J Riekert Rossmannek 2021, J Riekert 2021)
Assume for all x, y ∈ [a,b]d that f(x) = f(y). Then
1 there exist no non-global local minima of L∞,
2 there exist no saddle points of L∞,
3 it holds for all θ ∈ G−1(0) that θ is a global minima of L∞, and
4 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (Cheridito J Rossmannek 2021, J Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R, satisfy for all x ∈ [a,b] that f(x) = αx + β, and
assume supt∈[0,∞)(H − 1)‖Θt‖ <∞, and L∞(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then
1 there exist infinitely many non-global local minima of L∞,
2 there exist infinitely many saddle points of L∞, and
3 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (J Riekert 2021)
Assume supt∈[0,∞)‖Θt‖ <∞ and µ λ[a,b]d . Then there exists ϑ ∈ G−1(0)such that lim supt→∞ L∞(Θt) = L∞(ϑ).
Theorem (Cheridito J Riekert Rossmannek 2021, J Riekert 2021)
Assume for all x, y ∈ [a,b]d that f(x) = f(y). Then
1 there exist no non-global local minima of L∞,
2 there exist no saddle points of L∞,
3 it holds for all θ ∈ G−1(0) that θ is a global minima of L∞, and
4 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (Cheridito J Rossmannek 2021, J Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R, satisfy for all x ∈ [a,b] that f(x) = αx + β, and
assume supt∈[0,∞)(H − 1)‖Θt‖ <∞, and L∞(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then
1 there exist infinitely many non-global local minima of L∞,
2 there exist infinitely many saddle points of L∞, and
3 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (J Riekert 2021)
Assume supt∈[0,∞)‖Θt‖ <∞ and µ λ[a,b]d . Then there exists ϑ ∈ G−1(0)such that lim supt→∞ L∞(Θt) = L∞(ϑ).
Theorem (Cheridito J Riekert Rossmannek 2021, J Riekert 2021)
Assume for all x, y ∈ [a,b]d that f(x) = f(y). Then
1 there exist no non-global local minima of L∞,
2 there exist no saddle points of L∞,
3 it holds for all θ ∈ G−1(0) that θ is a global minima of L∞, and
4 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (Cheridito J Rossmannek 2021, J Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R, satisfy for all x ∈ [a,b] that f(x) = αx + β, and
assume supt∈[0,∞)(H − 1)‖Θt‖ <∞, and L∞(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then
1 there exist infinitely many non-global local minima of L∞,
2 there exist infinitely many saddle points of L∞, and
3 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (J Riekert 2021)
Assume supt∈[0,∞)‖Θt‖ <∞ and µ λ[a,b]d . Then there exists ϑ ∈ G−1(0)such that lim supt→∞ L∞(Θt) = L∞(ϑ).
Theorem (Cheridito J Riekert Rossmannek 2021, J Riekert 2021)
Assume for all x, y ∈ [a,b]d that f(x) = f(y). Then
1 there exist no non-global local minima of L∞,
2 there exist no saddle points of L∞,
3 it holds for all θ ∈ G−1(0) that θ is a global minima of L∞, and
4 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (Cheridito J Rossmannek 2021, J Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R, satisfy for all x ∈ [a,b] that f(x) = αx + β, and
assume supt∈[0,∞)(H − 1)‖Θt‖ <∞, and L∞(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then
1 there exist infinitely many non-global local minima of L∞,
2 there exist infinitely many saddle points of L∞, and
3 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (J Riekert 2021)
Assume supt∈[0,∞)‖Θt‖ <∞ and µ λ[a,b]d . Then there exists ϑ ∈ G−1(0)such that lim supt→∞ L∞(Θt) = L∞(ϑ).
Theorem (Cheridito J Riekert Rossmannek 2021, J Riekert 2021)
Assume for all x, y ∈ [a,b]d that f(x) = f(y). Then
1 there exist no non-global local minima of L∞,
2 there exist no saddle points of L∞,
3 it holds for all θ ∈ G−1(0) that θ is a global minima of L∞, and
4 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (Cheridito J Rossmannek 2021, J Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R, satisfy for all x ∈ [a,b] that f(x) = αx + β, and
assume supt∈[0,∞)(H − 1)‖Θt‖ <∞, and L∞(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then
1 there exist infinitely many non-global local minima of L∞,
2 there exist infinitely many saddle points of L∞, and
3 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (J Riekert 2021)
Assume supt∈[0,∞)‖Θt‖ <∞ and µ λ[a,b]d . Then there exists ϑ ∈ G−1(0)such that lim supt→∞ L∞(Θt) = L∞(ϑ).
Theorem (Cheridito J Riekert Rossmannek 2021, J Riekert 2021)
Assume for all x, y ∈ [a,b]d that f(x) = f(y). Then
1 there exist no non-global local minima of L∞,
2 there exist no saddle points of L∞,
3 it holds for all θ ∈ G−1(0) that θ is a global minima of L∞, and
4 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (Cheridito J Rossmannek 2021, J Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R, satisfy for all x ∈ [a,b] that f(x) = αx + β, and
assume supt∈[0,∞)(H − 1)‖Θt‖ <∞, and L∞(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then
1 there exist infinitely many non-global local minima of L∞,
2 there exist infinitely many saddle points of L∞, and
3 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (J Riekert 2021)
Assume supt∈[0,∞)‖Θt‖ <∞ and µ λ[a,b]d . Then there exists ϑ ∈ G−1(0)such that lim supt→∞ L∞(Θt) = L∞(ϑ).
Theorem (Cheridito J Riekert Rossmannek 2021, J Riekert 2021)
Assume for all x, y ∈ [a,b]d that f(x) = f(y). Then
1 there exist no non-global local minima of L∞,
2 there exist no saddle points of L∞,
3 it holds for all θ ∈ G−1(0) that θ is a global minima of L∞, and
4 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (Cheridito J Rossmannek 2021, J Riekert 2021)
Assume µ = λ[a,b], let α, β ∈ R, satisfy for all x ∈ [a,b] that f(x) = αx + β, and
assume supt∈[0,∞)(H − 1)‖Θt‖ <∞, and L∞(Θ0) < α2(b−a)3
12(2bH/2c+1)4 . Then
1 there exist infinitely many non-global local minima of L∞,
2 there exist infinitely many saddle points of L∞, and
3 it holds that lim supt→∞ L∞(Θt) = 0.
Theorem (J Riekert 2021)
Assume supt∈[0,∞)‖Θt‖ <∞ and µ λ[a,b]d . Then there exists ϑ ∈ G−1(0)such that lim supt→∞ L∞(Θt) = L∞(ϑ).
Hamiltonian-Jacobi-Bellman equations
Consider∂u∂t = ∆x u − ‖∇x u‖2
with u(0, x) =√‖x‖ for t ∈ [0, 1], x ∈ Rd .
d Mean Std. dev. Ref. value rel. L1-error Std. dev. rel. error avg. runtime
10 2.07017 0.00634850 2.04629 0.01167 0.00310245 58.200
50 3.15098 0.00275839 3.13788 0.00417 0.00087906 58.359
100 3.75329 0.00136920 3.74471 0.00229 0.00036564 58.329
200 4.46734 0.00079688 4.46172 0.00126 0.00017860 58.159
300 4.94586 0.00087736 4.94105 0.00097 0.00017756 58.819
500 5.62126 0.00045092 5.61735 0.00070 0.00008027 57.670
1000 6.68594 0.00040334 6.68335 0.00039 0.00006035 66.546
5000 9.97266 0.00047098 9.99835 0.00257 0.00004711 393.894
10000 11.87860 0.00022705 11.89099 0.00104 0.00001909 1687.680
Approximations for u(1, 0); Time steps: 24;SGD steps: 500 (d < 104), 600 (d = 104); Learning rates: 1
10 , 1100 , 1
1000Beck, Becker, Cheridito, J, Neufeld 2019 SISC (to appear)
Hamiltonian-Jacobi-Bellman equations
Consider∂u∂t = ∆x u − ‖∇x u‖2
with u(0, x) =√‖x‖ for t ∈ [0, 1], x ∈ Rd .
d Mean Std. dev. Ref. value rel. L1-error Std. dev. rel. error avg. runtime
10 2.07017 0.00634850 2.04629 0.01167 0.00310245 58.200
50 3.15098 0.00275839 3.13788 0.00417 0.00087906 58.359
100 3.75329 0.00136920 3.74471 0.00229 0.00036564 58.329
200 4.46734 0.00079688 4.46172 0.00126 0.00017860 58.159
300 4.94586 0.00087736 4.94105 0.00097 0.00017756 58.819
500 5.62126 0.00045092 5.61735 0.00070 0.00008027 57.670
1000 6.68594 0.00040334 6.68335 0.00039 0.00006035 66.546
5000 9.97266 0.00047098 9.99835 0.00257 0.00004711 393.894
10000 11.87860 0.00022705 11.89099 0.00104 0.00001909 1687.680
Approximations for u(1, 0); Time steps: 24;SGD steps: 500 (d < 104), 600 (d = 104); Learning rates: 1
10 , 1100 , 1
1000Beck, Becker, Cheridito, J, Neufeld 2019 SISC (to appear)
Hamiltonian-Jacobi-Bellman equations
Consider∂u∂t = ∆x u − ‖∇x u‖2
with u(0, x) =√‖x‖ for t ∈ [0, 1], x ∈ Rd .
d Mean Std. dev. Ref. value rel. L1-error Std. dev. rel. error avg. runtime
10 2.07017 0.00634850 2.04629 0.01167 0.00310245 58.200
50 3.15098 0.00275839 3.13788 0.00417 0.00087906 58.359
100 3.75329 0.00136920 3.74471 0.00229 0.00036564 58.329
200 4.46734 0.00079688 4.46172 0.00126 0.00017860 58.159
300 4.94586 0.00087736 4.94105 0.00097 0.00017756 58.819
500 5.62126 0.00045092 5.61735 0.00070 0.00008027 57.670
1000 6.68594 0.00040334 6.68335 0.00039 0.00006035 66.546
5000 9.97266 0.00047098 9.99835 0.00257 0.00004711 393.894
10000 11.87860 0.00022705 11.89099 0.00104 0.00001909 1687.680
Approximations for u(1, 0); Time steps: 24;SGD steps: 500 (d < 104), 600 (d = 104); Learning rates: 1
10 , 1100 , 1
1000Beck, Becker, Cheridito, J, Neufeld 2019 SISC (to appear)
Hamiltonian-Jacobi-Bellman equations
Consider∂u∂t = ∆x u − ‖∇x u‖2
with u(0, x) =√‖x‖ for t ∈ [0, 1], x ∈ Rd .
d Mean Std. dev. Ref. value rel. L1-error Std. dev. rel. error avg. runtime
10 2.07017 0.00634850 2.04629 0.01167 0.00310245 58.200
50 3.15098 0.00275839 3.13788 0.00417 0.00087906 58.359
100 3.75329 0.00136920 3.74471 0.00229 0.00036564 58.329
200 4.46734 0.00079688 4.46172 0.00126 0.00017860 58.159
300 4.94586 0.00087736 4.94105 0.00097 0.00017756 58.819
500 5.62126 0.00045092 5.61735 0.00070 0.00008027 57.670
1000 6.68594 0.00040334 6.68335 0.00039 0.00006035 66.546
5000 9.97266 0.00047098 9.99835 0.00257 0.00004711 393.894
10000 11.87860 0.00022705 11.89099 0.00104 0.00001909 1687.680
Approximations for u(1, 0); Time steps: 24;SGD steps: 500 (d < 104), 600 (d = 104); Learning rates: 1
10 , 1100 , 1
1000Beck, Becker, Cheridito, J, Neufeld 2019 SISC (to appear)
Hamiltonian-Jacobi-Bellman equations
Consider∂u∂t = ∆x u − ‖∇x u‖2
with u(0, x) =√‖x‖ for t ∈ [0, 1], x ∈ Rd .
d Mean Std. dev. Ref. value rel. L1-error Std. dev. rel. error avg. runtime
10 2.07017 0.00634850 2.04629 0.01167 0.00310245 58.200
50 3.15098 0.00275839 3.13788 0.00417 0.00087906 58.359
100 3.75329 0.00136920 3.74471 0.00229 0.00036564 58.329
200 4.46734 0.00079688 4.46172 0.00126 0.00017860 58.159
300 4.94586 0.00087736 4.94105 0.00097 0.00017756 58.819
500 5.62126 0.00045092 5.61735 0.00070 0.00008027 57.670
1000 6.68594 0.00040334 6.68335 0.00039 0.00006035 66.546
5000 9.97266 0.00047098 9.99835 0.00257 0.00004711 393.894
10000 11.87860 0.00022705 11.89099 0.00104 0.00001909 1687.680
Approximations for u(1, 0); Time steps: 24;SGD steps: 500 (d < 104), 600 (d = 104); Learning rates: 1
10 , 1100 , 1
1000Beck, Becker, Cheridito, J, Neufeld 2019 SISC (to appear)
Hamiltonian-Jacobi-Bellman equations
Consider∂u∂t = ∆x u − ‖∇x u‖2
with u(0, x) =√‖x‖ for t ∈ [0, 1], x ∈ Rd .
d Mean Std. dev. Ref. value rel. L1-error Std. dev. rel. error avg. runtime
10 2.07017 0.00634850 2.04629 0.01167 0.00310245 58.200
50 3.15098 0.00275839 3.13788 0.00417 0.00087906 58.359
100 3.75329 0.00136920 3.74471 0.00229 0.00036564 58.329
200 4.46734 0.00079688 4.46172 0.00126 0.00017860 58.159
300 4.94586 0.00087736 4.94105 0.00097 0.00017756 58.819
500 5.62126 0.00045092 5.61735 0.00070 0.00008027 57.670
1000 6.68594 0.00040334 6.68335 0.00039 0.00006035 66.546
5000 9.97266 0.00047098 9.99835 0.00257 0.00004711 393.894
10000 11.87860 0.00022705 11.89099 0.00104 0.00001909 1687.680
Approximations for u(1, 0); Time steps: 24;SGD steps: 500 (d < 104), 600 (d = 104); Learning rates: 1
10 , 1100 , 1
1000Beck, Becker, Cheridito, J, Neufeld 2019 SISC (to appear)
Hamiltonian-Jacobi-Bellman equations
Consider∂u∂t = ∆x u − ‖∇x u‖2
with u(0, x) =√‖x‖ for t ∈ [0, 1], x ∈ Rd .
d Mean Std. dev. Ref. value rel. L1-error Std. dev. rel. error avg. runtime
10 2.07017 0.00634850 2.04629 0.01167 0.00310245 58.200
50 3.15098 0.00275839 3.13788 0.00417 0.00087906 58.359
100 3.75329 0.00136920 3.74471 0.00229 0.00036564 58.329
200 4.46734 0.00079688 4.46172 0.00126 0.00017860 58.159
300 4.94586 0.00087736 4.94105 0.00097 0.00017756 58.819
500 5.62126 0.00045092 5.61735 0.00070 0.00008027 57.670
1000 6.68594 0.00040334 6.68335 0.00039 0.00006035 66.546
5000 9.97266 0.00047098 9.99835 0.00257 0.00004711 393.894
10000 11.87860 0.00022705 11.89099 0.00104 0.00001909 1687.680
Approximations for u(1, 0); Time steps: 24;SGD steps: 500 (d < 104), 600 (d = 104); Learning rates: 1
10 , 1100 , 1
1000Beck, Becker, Cheridito, J, Neufeld 2019 SISC (to appear)
Hamiltonian-Jacobi-Bellman equations
Consider∂u∂t = ∆x u − ‖∇x u‖2
with u(0, x) =√‖x‖ for t ∈ [0, 1], x ∈ Rd .
d Mean Std. dev. Ref. value rel. L1-error Std. dev. rel. error avg. runtime
10 2.07017 0.00634850 2.04629 0.01167 0.00310245 58.200
50 3.15098 0.00275839 3.13788 0.00417 0.00087906 58.359
100 3.75329 0.00136920 3.74471 0.00229 0.00036564 58.329
200 4.46734 0.00079688 4.46172 0.00126 0.00017860 58.159
300 4.94586 0.00087736 4.94105 0.00097 0.00017756 58.819
500 5.62126 0.00045092 5.61735 0.00070 0.00008027 57.670
1000 6.68594 0.00040334 6.68335 0.00039 0.00006035 66.546
5000 9.97266 0.00047098 9.99835 0.00257 0.00004711 393.894
10000 11.87860 0.00022705 11.89099 0.00104 0.00001909 1687.680
Approximations for u(1, 0); Time steps: 24;SGD steps: 500 (d < 104), 600 (d = 104); Learning rates: 1
10 , 1100 , 1
1000Beck, Becker, Cheridito, J, Neufeld 2019 SISC (to appear)
Hamiltonian-Jacobi-Bellman equations
Consider∂u∂t = ∆x u − ‖∇x u‖2
with u(0, x) =√‖x‖ for t ∈ [0, 1], x ∈ Rd .
d Mean Std. dev. Ref. value rel. L1-error Std. dev. rel. error avg. runtime
10 2.07017 0.00634850 2.04629 0.01167 0.00310245 58.200
50 3.15098 0.00275839 3.13788 0.00417 0.00087906 58.359
100 3.75329 0.00136920 3.74471 0.00229 0.00036564 58.329
200 4.46734 0.00079688 4.46172 0.00126 0.00017860 58.159
300 4.94586 0.00087736 4.94105 0.00097 0.00017756 58.819
500 5.62126 0.00045092 5.61735 0.00070 0.00008027 57.670
1000 6.68594 0.00040334 6.68335 0.00039 0.00006035 66.546
5000 9.97266 0.00047098 9.99835 0.00257 0.00004711 393.894
10000 11.87860 0.00022705 11.89099 0.00104 0.00001909 1687.680
Approximations for u(1, 0); Time steps: 24;SGD steps: 500 (d < 104), 600 (d = 104); Learning rates: 1
10 , 1100 , 1
1000Beck, Becker, Cheridito, J, Neufeld 2019 SISC (to appear)
Hamiltonian-Jacobi-Bellman equations
Consider∂u∂t = ∆x u − ‖∇x u‖2
with u(0, x) =√‖x‖ for t ∈ [0, 1], x ∈ Rd .
d Mean Std. dev. Ref. value rel. L1-error Std. dev. rel. error avg. runtime
10 2.07017 0.00634850 2.04629 0.01167 0.00310245 58.200
50 3.15098 0.00275839 3.13788 0.00417 0.00087906 58.359
100 3.75329 0.00136920 3.74471 0.00229 0.00036564 58.329
200 4.46734 0.00079688 4.46172 0.00126 0.00017860 58.159
300 4.94586 0.00087736 4.94105 0.00097 0.00017756 58.819
500 5.62126 0.00045092 5.61735 0.00070 0.00008027 57.670
1000 6.68594 0.00040334 6.68335 0.00039 0.00006035 66.546
5000 9.97266 0.00047098 9.99835 0.00257 0.00004711 393.894
10000 11.87860 0.00022705 11.89099 0.00104 0.00001909 1687.680
Approximations for u(1, 0); Time steps: 24;SGD steps: 500 (d < 104), 600 (d = 104); Learning rates: 1
10 , 1100 , 1
1000Beck, Becker, Cheridito, J, Neufeld 2019 SISC (to appear)
Hamiltonian-Jacobi-Bellman equations
Consider∂u∂t = ∆x u − ‖∇x u‖2
with u(0, x) =√‖x‖ for t ∈ [0, 1], x ∈ Rd .
d Mean Std. dev. Ref. value rel. L1-error Std. dev. rel. error avg. runtime
10 2.07017 0.00634850 2.04629 0.01167 0.00310245 58.200
50 3.15098 0.00275839 3.13788 0.00417 0.00087906 58.359
100 3.75329 0.00136920 3.74471 0.00229 0.00036564 58.329
200 4.46734 0.00079688 4.46172 0.00126 0.00017860 58.159
300 4.94586 0.00087736 4.94105 0.00097 0.00017756 58.819
500 5.62126 0.00045092 5.61735 0.00070 0.00008027 57.670
1000 6.68594 0.00040334 6.68335 0.00039 0.00006035 66.546
5000 9.97266 0.00047098 9.99835 0.00257 0.00004711 393.894
10000 11.87860 0.00022705 11.89099 0.00104 0.00001909 1687.680
Approximations for u(1, 0); Time steps: 24;SGD steps: 500 (d < 104), 600 (d = 104); Learning rates: 1
10 , 1100 , 1
1000Beck, Becker, Cheridito, J, Neufeld 2019 SISC (to appear)
Hamiltonian-Jacobi-Bellman equations
Consider∂u∂t = ∆x u − ‖∇x u‖2
with u(0, x) =√‖x‖ for t ∈ [0, 1], x ∈ Rd .
d Mean Std. dev. Ref. value rel. L1-error Std. dev. rel. error avg. runtime
10 2.07017 0.00634850 2.04629 0.01167 0.00310245 58.200
50 3.15098 0.00275839 3.13788 0.00417 0.00087906 58.359
100 3.75329 0.00136920 3.74471 0.00229 0.00036564 58.329
200 4.46734 0.00079688 4.46172 0.00126 0.00017860 58.159
300 4.94586 0.00087736 4.94105 0.00097 0.00017756 58.819
500 5.62126 0.00045092 5.61735 0.00070 0.00008027 57.670
1000 6.68594 0.00040334 6.68335 0.00039 0.00006035 66.546
5000 9.97266 0.00047098 9.99835 0.00257 0.00004711 393.894
10000 11.87860 0.00022705 11.89099 0.00104 0.00001909 1687.680
Approximations for u(1, 0); Time steps: 24;SGD steps: 500 (d < 104), 600 (d = 104); Learning rates: 1
10 , 1100 , 1
1000Beck, Becker, Cheridito, J, Neufeld 2019 SISC (to appear)
Hamiltonian-Jacobi-Bellman equations
Consider∂u∂t = ∆x u − ‖∇x u‖2
with u(0, x) =√‖x‖ for t ∈ [0, 1], x ∈ Rd .
d Mean Std. dev. Ref. value rel. L1-error Std. dev. rel. error avg. runtime
10 2.07017 0.00634850 2.04629 0.01167 0.00310245 58.200
50 3.15098 0.00275839 3.13788 0.00417 0.00087906 58.359
100 3.75329 0.00136920 3.74471 0.00229 0.00036564 58.329
200 4.46734 0.00079688 4.46172 0.00126 0.00017860 58.159
300 4.94586 0.00087736 4.94105 0.00097 0.00017756 58.819
500 5.62126 0.00045092 5.61735 0.00070 0.00008027 57.670
1000 6.68594 0.00040334 6.68335 0.00039 0.00006035 66.546
5000 9.97266 0.00047098 9.99835 0.00257 0.00004711 393.894
10000 11.87860 0.00022705 11.89099 0.00104 0.00001909 1687.680
Approximations for u(1, 0); Time steps: 24;SGD steps: 500 (d < 104), 600 (d = 104); Learning rates: 1
10 , 1100 , 1
1000Beck, Becker, Cheridito, J, Neufeld 2019 SISC (to appear)
Hamiltonian-Jacobi-Bellman equations
Consider∂u∂t = ∆x u − ‖∇x u‖2
with u(0, x) =√‖x‖ for t ∈ [0, 1], x ∈ Rd .
d Mean Std. dev. Ref. value rel. L1-error Std. dev. rel. error avg. runtime
10 2.07017 0.00634850 2.04629 0.01167 0.00310245 58.200
50 3.15098 0.00275839 3.13788 0.00417 0.00087906 58.359
100 3.75329 0.00136920 3.74471 0.00229 0.00036564 58.329
200 4.46734 0.00079688 4.46172 0.00126 0.00017860 58.159
300 4.94586 0.00087736 4.94105 0.00097 0.00017756 58.819
500 5.62126 0.00045092 5.61735 0.00070 0.00008027 57.670
1000 6.68594 0.00040334 6.68335 0.00039 0.00006035 66.546
5000 9.97266 0.00047098 9.99835 0.00257 0.00004711 393.894
10000 11.87860 0.00022705 11.89099 0.00104 0.00001909 1687.680
Approximations for u(1, 0); Time steps: 24;SGD steps: 500 (d < 104), 600 (d = 104); Learning rates: 1
10 , 1100 , 1
1000Beck, Becker, Cheridito, J, Neufeld 2019 SISC (to appear)
Hamiltonian-Jacobi-Bellman equations
Consider∂u∂t = ∆x u − ‖∇x u‖2
with u(0, x) =√‖x‖ for t ∈ [0, 1], x ∈ Rd .
d Mean Std. dev. Ref. value rel. L1-error Std. dev. rel. error avg. runtime
10 2.07017 0.00634850 2.04629 0.01167 0.00310245 58.200
50 3.15098 0.00275839 3.13788 0.00417 0.00087906 58.359
100 3.75329 0.00136920 3.74471 0.00229 0.00036564 58.329
200 4.46734 0.00079688 4.46172 0.00126 0.00017860 58.159
300 4.94586 0.00087736 4.94105 0.00097 0.00017756 58.819
500 5.62126 0.00045092 5.61735 0.00070 0.00008027 57.670
1000 6.68594 0.00040334 6.68335 0.00039 0.00006035 66.546
5000 9.97266 0.00047098 9.99835 0.00257 0.00004711 393.894
10000 11.87860 0.00022705 11.89099 0.00104 0.00001909 1687.680
Approximations for u(1, 0); Time steps: 24;SGD steps: 500 (d < 104), 600 (d = 104); Learning rates: 1
10 , 1100 , 1
1000Beck, Becker, Cheridito, J, Neufeld 2019 SISC (to appear)
Hamiltonian-Jacobi-Bellman equations
Consider∂u∂t = ∆x u − ‖∇x u‖2
with u(0, x) =√‖x‖ for t ∈ [0, 1], x ∈ Rd .
d Mean Std. dev. Ref. value rel. L1-error Std. dev. rel. error avg. runtime
10 2.07017 0.00634850 2.04629 0.01167 0.00310245 58.200
50 3.15098 0.00275839 3.13788 0.00417 0.00087906 58.359
100 3.75329 0.00136920 3.74471 0.00229 0.00036564 58.329
200 4.46734 0.00079688 4.46172 0.00126 0.00017860 58.159
300 4.94586 0.00087736 4.94105 0.00097 0.00017756 58.819
500 5.62126 0.00045092 5.61735 0.00070 0.00008027 57.670
1000 6.68594 0.00040334 6.68335 0.00039 0.00006035 66.546
5000 9.97266 0.00047098 9.99835 0.00257 0.00004711 393.894
10000 11.87860 0.00022705 11.89099 0.00104 0.00001909 1687.680
Approximations for u(1, 0); Time steps: 24;SGD steps: 500 (d < 104), 600 (d = 104); Learning rates: 1
10 , 1100 , 1
1000Beck, Becker, Cheridito, J, Neufeld 2019 SISC (to appear)
Hamiltonian-Jacobi-Bellman equations
Consider∂u∂t = ∆x u − ‖∇x u‖2
with u(0, x) =√‖x‖ for t ∈ [0, 1], x ∈ Rd .
d Mean Std. dev. Ref. value rel. L1-error Std. dev. rel. error avg. runtime
10 2.07017 0.00634850 2.04629 0.01167 0.00310245 58.200
50 3.15098 0.00275839 3.13788 0.00417 0.00087906 58.359
100 3.75329 0.00136920 3.74471 0.00229 0.00036564 58.329
200 4.46734 0.00079688 4.46172 0.00126 0.00017860 58.159
300 4.94586 0.00087736 4.94105 0.00097 0.00017756 58.819
500 5.62126 0.00045092 5.61735 0.00070 0.00008027 57.670
1000 6.68594 0.00040334 6.68335 0.00039 0.00006035 66.546
5000 9.97266 0.00047098 9.99835 0.00257 0.00004711 393.894
10000 11.87860 0.00022705 11.89099 0.00104 0.00001909 1687.680
Approximations for u(1, 0); Time steps: 24;SGD steps: 500 (d < 104), 600 (d = 104); Learning rates: 1
10 , 1100 , 1
1000Beck, Becker, Cheridito, J, Neufeld 2019 SISC (to appear)
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Full history recursive Multilevel-Picard method: LetT > 0,L, p ≥ 0,Θ = ∪∞n=1Zn,∀ d ∈ N let gd ∈ C(Rd ,R) satisfy ∀ x ∈ Rd : |gd (x)| ≤ L(1 + ‖x‖p), letf : R→ R be Lipschitz, let (Ω,F ,P) probab. sp., let W d,θ : [0, T ]× Ω→ Rd ,d ∈ N, θ ∈ Θ, be i.i.d. Brownian motions, let Sθ : [0, T ]× Ω→ R, θ ∈ Θ, i.i.d.continuous satisfying ∀ t ∈ [0, T ], θ ∈ Θ that Sθt is U[t,T ]-distributed, assume that
(Sθ)θ∈Θ and (W d,θ)θ∈Θ,d∈N are independent, let Ud,θn,M : [0, T ]× Rd × Ω→ R,
d, n,M ∈ Z, θ ∈ Θ, satisfy ∀ d,M ∈ N, n ∈ N0, θ ∈ Θ, t ∈ [0, T ], x ∈ Rd :
Ud,θn,M(t, x) =
Mn∑m=1
gd (x + Wd,(θ,0,−m)T−t )1N(n)
Mn
+
[n−1∑l=0
(T−t)Mn−l
Mn−l∑m=1
f(
Ud,(θ,l,m)l,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))−1N(l) f
(U
d,(θ,l,m)l−1,M
(S
(θ,l,m)t , x + W
d,(θ,l,m)
S(θ,l,m)t −t
))]
and ∀ d, n ∈ N let Costd,n ∈ N be the comp. cost of Ud,0n,n (0, 0).
Then
(i) ∀ d ∈ N : there exists ud : [0, T ]× Rd → R at most polyn. grow. solution of
∂ud∂t + 1
2 ∆x ud + f(ud ) = 0 with ud (T , ·) = gd
and
(ii) ∀ δ > 0 : there exist n : N× (0,∞)→ N and C > 0 : ∀ d ∈ N, ε > 0 :(E[|ud (0, 0)− Ud,0
nd,ε,nd,ε(0, 0)|2
])1/2 ≤ ε
andCostd,nd,ε ≤ Cd1+p(1+δ)ε−(2+δ).
Extensions: Algorithms/Simulations/Proofs: Fully nonlinear PDEs (Beck, E, J 2018JNS), Optimal stopping (Becker, Cheridito, J 2018 JMLR), Uniform errors (Beck,Becker, Grohs, Jaafari, J 2018), Semilinear PDEs/CVA (Hutzenthaler, J, vonWurstemberger 2019 EJP, Hutzenthaler, J, Kruse, Nguyen 2020), Nonlipschitznonlinearities (Beck, Hornung, Hutzenthaler, Jentzen, Kruse 2019 J. Numer. Math.),Gradient dependent nonlinearities (Hutzenthaler, J, Kruse 2019), Elliptic PDEs (Beck,Gonon, J 2020), Discrete problems (Beck, J, Kruse 2020), . . .
Then
(i) ∀ d ∈ N : there exists ud : [0, T ]× Rd → R at most polyn. grow. solution of
∂ud∂t + 1
2 ∆x ud + f(ud ) = 0 with ud (T , ·) = gd
and
(ii) ∀ δ > 0 : there exist n : N× (0,∞)→ N and C > 0 : ∀ d ∈ N, ε > 0 :(E[|ud (0, 0)− Ud,0
nd,ε,nd,ε(0, 0)|2
])1/2 ≤ ε
andCostd,nd,ε ≤ Cd1+p(1+δ)ε−(2+δ).
Extensions: Algorithms/Simulations/Proofs: Fully nonlinear PDEs (Beck, E, J 2018JNS), Optimal stopping (Becker, Cheridito, J 2018 JMLR), Uniform errors (Beck,Becker, Grohs, Jaafari, J 2018), Semilinear PDEs/CVA (Hutzenthaler, J, vonWurstemberger 2019 EJP, Hutzenthaler, J, Kruse, Nguyen 2020), Nonlipschitznonlinearities (Beck, Hornung, Hutzenthaler, Jentzen, Kruse 2019 J. Numer. Math.),Gradient dependent nonlinearities (Hutzenthaler, J, Kruse 2019), Elliptic PDEs (Beck,Gonon, J 2020), Discrete problems (Beck, J, Kruse 2020), . . .
Then
(i) ∀ d ∈ N : there exists ud : [0, T ]× Rd → R at most polyn. grow. solution of
∂ud∂t + 1
2 ∆x ud + f(ud ) = 0 with ud (T , ·) = gd
and
(ii) ∀ δ > 0 : there exist n : N× (0,∞)→ N and C > 0 : ∀ d ∈ N, ε > 0 :(E[|ud (0, 0)− Ud,0
nd,ε,nd,ε(0, 0)|2
])1/2 ≤ ε
andCostd,nd,ε ≤ Cd1+p(1+δ)ε−(2+δ).
Extensions: Algorithms/Simulations/Proofs: Fully nonlinear PDEs (Beck, E, J 2018JNS), Optimal stopping (Becker, Cheridito, J 2018 JMLR), Uniform errors (Beck,Becker, Grohs, Jaafari, J 2018), Semilinear PDEs/CVA (Hutzenthaler, J, vonWurstemberger 2019 EJP, Hutzenthaler, J, Kruse, Nguyen 2020), Nonlipschitznonlinearities (Beck, Hornung, Hutzenthaler, Jentzen, Kruse 2019 J. Numer. Math.),Gradient dependent nonlinearities (Hutzenthaler, J, Kruse 2019), Elliptic PDEs (Beck,Gonon, J 2020), Discrete problems (Beck, J, Kruse 2020), . . .
Then
(i) ∀ d ∈ N : there exists ud : [0, T ]× Rd → R at most polyn. grow. solution of
∂ud∂t + 1
2 ∆x ud + f(ud ) = 0 with ud (T , ·) = gd
and
(ii) ∀ δ > 0 : there exist n : N× (0,∞)→ N and C > 0 : ∀ d ∈ N, ε > 0 :(E[|ud (0, 0)− Ud,0
nd,ε,nd,ε(0, 0)|2
])1/2 ≤ ε
andCostd,nd,ε ≤ Cd1+p(1+δ)ε−(2+δ).
Extensions: Algorithms/Simulations/Proofs: Fully nonlinear PDEs (Beck, E, J 2018JNS), Optimal stopping (Becker, Cheridito, J 2018 JMLR), Uniform errors (Beck,Becker, Grohs, Jaafari, J 2018), Semilinear PDEs/CVA (Hutzenthaler, J, vonWurstemberger 2019 EJP, Hutzenthaler, J, Kruse, Nguyen 2020), Nonlipschitznonlinearities (Beck, Hornung, Hutzenthaler, Jentzen, Kruse 2019 J. Numer. Math.),Gradient dependent nonlinearities (Hutzenthaler, J, Kruse 2019), Elliptic PDEs (Beck,Gonon, J 2020), Discrete problems (Beck, J, Kruse 2020), . . .
Then
(i) ∀ d ∈ N : there exists ud : [0, T ]× Rd → R at most polyn. grow. solution of
∂ud∂t + 1
2 ∆x ud + f(ud ) = 0 with ud (T , ·) = gd
and
(ii) ∀ δ > 0 : there exist n : N× (0,∞)→ N and C > 0 : ∀ d ∈ N, ε > 0 :(E[|ud (0, 0)− Ud,0
nd,ε,nd,ε(0, 0)|2
])1/2 ≤ ε
andCostd,nd,ε ≤ Cd1+p(1+δ)ε−(2+δ).
Extensions: Algorithms/Simulations/Proofs: Fully nonlinear PDEs (Beck, E, J 2018JNS), Optimal stopping (Becker, Cheridito, J 2018 JMLR), Uniform errors (Beck,Becker, Grohs, Jaafari, J 2018), Semilinear PDEs/CVA (Hutzenthaler, J, vonWurstemberger 2019 EJP, Hutzenthaler, J, Kruse, Nguyen 2020), Nonlipschitznonlinearities (Beck, Hornung, Hutzenthaler, Jentzen, Kruse 2019 J. Numer. Math.),Gradient dependent nonlinearities (Hutzenthaler, J, Kruse 2019), Elliptic PDEs (Beck,Gonon, J 2020), Discrete problems (Beck, J, Kruse 2020), . . .
Then
(i) ∀ d ∈ N : there exists ud : [0, T ]× Rd → R at most polyn. grow. solution of
∂ud∂t + 1
2 ∆x ud + f(ud ) = 0 with ud (T , ·) = gd
and
(ii) ∀ δ > 0 : there exist n : N× (0,∞)→ N and C > 0 : ∀ d ∈ N, ε > 0 :(E[|ud (0, 0)− Ud,0
nd,ε,nd,ε(0, 0)|2
])1/2 ≤ ε
andCostd,nd,ε ≤ Cd1+p(1+δ)ε−(2+δ).
Extensions: Algorithms/Simulations/Proofs: Fully nonlinear PDEs (Beck, E, J 2018JNS), Optimal stopping (Becker, Cheridito, J 2018 JMLR), Uniform errors (Beck,Becker, Grohs, Jaafari, J 2018), Semilinear PDEs/CVA (Hutzenthaler, J, vonWurstemberger 2019 EJP, Hutzenthaler, J, Kruse, Nguyen 2020), Nonlipschitznonlinearities (Beck, Hornung, Hutzenthaler, Jentzen, Kruse 2019 J. Numer. Math.),Gradient dependent nonlinearities (Hutzenthaler, J, Kruse 2019), Elliptic PDEs (Beck,Gonon, J 2020), Discrete problems (Beck, J, Kruse 2020), . . .
Then
(i) ∀ d ∈ N : there exists ud : [0, T ]× Rd → R at most polyn. grow. solution of
∂ud∂t + 1
2 ∆x ud + f(ud ) = 0 with ud (T , ·) = gd
and
(ii) ∀ δ > 0 : there exist n : N× (0,∞)→ N and C > 0 : ∀ d ∈ N, ε > 0 :(E[|ud (0, 0)− Ud,0
nd,ε,nd,ε(0, 0)|2
])1/2 ≤ ε
andCostd,nd,ε ≤ Cd1+p(1+δ)ε−(2+δ).
Extensions: Algorithms/Simulations/Proofs: Fully nonlinear PDEs (Beck, E, J 2018JNS), Optimal stopping (Becker, Cheridito, J 2018 JMLR), Uniform errors (Beck,Becker, Grohs, Jaafari, J 2018), Semilinear PDEs/CVA (Hutzenthaler, J, vonWurstemberger 2019 EJP, Hutzenthaler, J, Kruse, Nguyen 2020), Nonlipschitznonlinearities (Beck, Hornung, Hutzenthaler, Jentzen, Kruse 2019 J. Numer. Math.),Gradient dependent nonlinearities (Hutzenthaler, J, Kruse 2019), Elliptic PDEs (Beck,Gonon, J 2020), Discrete problems (Beck, J, Kruse 2020), . . .
Then
(i) ∀ d ∈ N : there exists ud : [0, T ]× Rd → R at most polyn. grow. solution of
∂ud∂t + 1
2 ∆x ud + f(ud ) = 0 with ud (T , ·) = gd
and
(ii) ∀ δ > 0 : there exist n : N× (0,∞)→ N and C > 0 : ∀ d ∈ N, ε > 0 :(E[|ud (0, 0)− Ud,0
nd,ε,nd,ε(0, 0)|2
])1/2 ≤ ε
andCostd,nd,ε ≤ Cd1+p(1+δ)ε−(2+δ).
Extensions: Algorithms/Simulations/Proofs: Fully nonlinear PDEs (Beck, E, J 2018JNS), Optimal stopping (Becker, Cheridito, J 2018 JMLR), Uniform errors (Beck,Becker, Grohs, Jaafari, J 2018), Semilinear PDEs/CVA (Hutzenthaler, J, vonWurstemberger 2019 EJP, Hutzenthaler, J, Kruse, Nguyen 2020), Nonlipschitznonlinearities (Beck, Hornung, Hutzenthaler, Jentzen, Kruse 2019 J. Numer. Math.),Gradient dependent nonlinearities (Hutzenthaler, J, Kruse 2019), Elliptic PDEs (Beck,Gonon, J 2020), Discrete problems (Beck, J, Kruse 2020), . . .
Then
(i) ∀ d ∈ N : there exists ud : [0, T ]× Rd → R at most polyn. grow. solution of
∂ud∂t + 1
2 ∆x ud + f(ud ) = 0 with ud (T , ·) = gd
and
(ii) ∀ δ > 0 : there exist n : N× (0,∞)→ N and C > 0 : ∀ d ∈ N, ε > 0 :(E[|ud (0, 0)− Ud,0
nd,ε,nd,ε(0, 0)|2
])1/2 ≤ ε
andCostd,nd,ε ≤ Cd1+p(1+δ)ε−(2+δ).
Extensions: Algorithms/Simulations/Proofs: Fully nonlinear PDEs (Beck, E, J 2018JNS), Optimal stopping (Becker, Cheridito, J 2018 JMLR), Uniform errors (Beck,Becker, Grohs, Jaafari, J 2018), Semilinear PDEs/CVA (Hutzenthaler, J, vonWurstemberger 2019 EJP, Hutzenthaler, J, Kruse, Nguyen 2020), Nonlipschitznonlinearities (Beck, Hornung, Hutzenthaler, Jentzen, Kruse 2019 J. Numer. Math.),Gradient dependent nonlinearities (Hutzenthaler, J, Kruse 2019), Elliptic PDEs (Beck,Gonon, J 2020), Discrete problems (Beck, J, Kruse 2020), . . .
Then
(i) ∀ d ∈ N : there exists ud : [0, T ]× Rd → R at most polyn. grow. solution of
∂ud∂t + 1
2 ∆x ud + f(ud ) = 0 with ud (T , ·) = gd
and
(ii) ∀ δ > 0 : there exist n : N× (0,∞)→ N and C > 0 : ∀ d ∈ N, ε > 0 :(E[|ud (0, 0)− Ud,0
nd,ε,nd,ε(0, 0)|2
])1/2 ≤ ε
andCostd,nd,ε ≤ Cd1+p(1+δ)ε−(2+δ).
Extensions: Algorithms/Simulations/Proofs: Fully nonlinear PDEs (Beck, E, J 2018JNS), Optimal stopping (Becker, Cheridito, J 2018 JMLR), Uniform errors (Beck,Becker, Grohs, Jaafari, J 2018), Semilinear PDEs/CVA (Hutzenthaler, J, vonWurstemberger 2019 EJP, Hutzenthaler, J, Kruse, Nguyen 2020), Nonlipschitznonlinearities (Beck, Hornung, Hutzenthaler, Jentzen, Kruse 2019 J. Numer. Math.),Gradient dependent nonlinearities (Hutzenthaler, J, Kruse 2019), Elliptic PDEs (Beck,Gonon, J 2020), Discrete problems (Beck, J, Kruse 2020), . . .
Then
(i) ∀ d ∈ N : there exists ud : [0, T ]× Rd → R at most polyn. grow. solution of
∂ud∂t + 1
2 ∆x ud + f(ud ) = 0 with ud (T , ·) = gd
and
(ii) ∀ δ > 0 : there exist n : N× (0,∞)→ N and C > 0 : ∀ d ∈ N, ε > 0 :(E[|ud (0, 0)− Ud,0
nd,ε,nd,ε(0, 0)|2
])1/2 ≤ ε
andCostd,nd,ε ≤ Cd1+p(1+δ)ε−(2+δ).
Extensions: Algorithms/Simulations/Proofs: Fully nonlinear PDEs (Beck, E, J 2018JNS), Optimal stopping (Becker, Cheridito, J 2018 JMLR), Uniform errors (Beck,Becker, Grohs, Jaafari, J 2018), Semilinear PDEs/CVA (Hutzenthaler, J, vonWurstemberger 2019 EJP, Hutzenthaler, J, Kruse, Nguyen 2020), Nonlipschitznonlinearities (Beck, Hornung, Hutzenthaler, Jentzen, Kruse 2019 J. Numer. Math.),Gradient dependent nonlinearities (Hutzenthaler, J, Kruse 2019), Elliptic PDEs (Beck,Gonon, J 2020), Discrete problems (Beck, J, Kruse 2020), . . .
Then
(i) ∀ d ∈ N : there exists ud : [0, T ]× Rd → R at most polyn. grow. solution of
∂ud∂t + 1
2 ∆x ud + f(ud ) = 0 with ud (T , ·) = gd
and
(ii) ∀ δ > 0 : there exist n : N× (0,∞)→ N and C > 0 : ∀ d ∈ N, ε > 0 :(E[|ud (0, 0)− Ud,0
nd,ε,nd,ε(0, 0)|2
])1/2 ≤ ε
andCostd,nd,ε ≤ Cd1+p(1+δ)ε−(2+δ).
Extensions: Algorithms/Simulations/Proofs: Fully nonlinear PDEs (Beck, E, J 2018JNS), Optimal stopping (Becker, Cheridito, J 2018 JMLR), Uniform errors (Beck,Becker, Grohs, Jaafari, J 2018), Semilinear PDEs/CVA (Hutzenthaler, J, vonWurstemberger 2019 EJP, Hutzenthaler, J, Kruse, Nguyen 2020), Nonlipschitznonlinearities (Beck, Hornung, Hutzenthaler, Jentzen, Kruse 2019 J. Numer. Math.),Gradient dependent nonlinearities (Hutzenthaler, J, Kruse 2019), Elliptic PDEs (Beck,Gonon, J 2020), Discrete problems (Beck, J, Kruse 2020), . . .
Then
(i) ∀ d ∈ N : there exists ud : [0, T ]× Rd → R at most polyn. grow. solution of
∂ud∂t + 1
2 ∆x ud + f(ud ) = 0 with ud (T , ·) = gd
and
(ii) ∀ δ > 0 : there exist n : N× (0,∞)→ N and C > 0 : ∀ d ∈ N, ε > 0 :(E[|ud (0, 0)− Ud,0
nd,ε,nd,ε(0, 0)|2
])1/2 ≤ ε
andCostd,nd,ε ≤ Cd1+p(1+δ)ε−(2+δ).
Extensions: Algorithms/Simulations/Proofs: Fully nonlinear PDEs (Beck, E, J 2018JNS), Optimal stopping (Becker, Cheridito, J 2018 JMLR), Uniform errors (Beck,Becker, Grohs, Jaafari, J 2018), Semilinear PDEs/CVA (Hutzenthaler, J, vonWurstemberger 2019 EJP, Hutzenthaler, J, Kruse, Nguyen 2020), Nonlipschitznonlinearities (Beck, Hornung, Hutzenthaler, Jentzen, Kruse 2019 J. Numer. Math.),Gradient dependent nonlinearities (Hutzenthaler, J, Kruse 2019), Elliptic PDEs (Beck,Gonon, J 2020), Discrete problems (Beck, J, Kruse 2020), . . .
