Data-Driven FilteredReduced Order Modelsof Nonlinear Systems
T. Iliescu
Department of MathematicsICAM
Virginia Tech
MoRePaS 2018
NantesApril 11, 2018
National Science Foundation DMS-1522656 2015-2018
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 1 / 17
Outline
1 Problem
2 Solution
3 Connections
4 Conclusions
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 2 / 17
Outline
Collaborators
Muhammad Mohebujjaman (postdoc, Virginia Tech)
Xuping Xie (postdoc, Oak Ridge National Laboratory)
Leo Rebholz (professor, Clemson University)
Birgul Koc (student, Virginia Tech)
Changhong Mou (student, Virginia Tech)
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 3 / 17
Outline
Collaborators
Muhammad Mohebujjaman (postdoc, Virginia Tech)
Xuping Xie (postdoc, Oak Ridge National Laboratory)
Leo Rebholz (professor, Clemson University)
Birgul Koc (student, Virginia Tech)
Changhong Mou (student, Virginia Tech)
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 3 / 17
Outline
Collaborators
Muhammad Mohebujjaman (postdoc, Virginia Tech)
Xuping Xie (postdoc, Oak Ridge National Laboratory)
Leo Rebholz (professor, Clemson University)
Birgul Koc (student, Virginia Tech)
Changhong Mou (student, Virginia Tech)
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 3 / 17
Outline
Collaborators
Muhammad Mohebujjaman (postdoc, Virginia Tech)
Xuping Xie (postdoc, Oak Ridge National Laboratory)
Leo Rebholz (professor, Clemson University)
Birgul Koc (student, Virginia Tech)
Changhong Mou (student, Virginia Tech)
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 3 / 17
Outline
Collaborators
Muhammad Mohebujjaman (postdoc, Virginia Tech)
Xuping Xie (postdoc, Oak Ridge National Laboratory)
Leo Rebholz (professor, Clemson University)
Birgul Koc (student, Virginia Tech)
Changhong Mou (student, Virginia Tech)
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 3 / 17
Problem
General Problem
POD
RBM
DMD
systems theoretic ROMs (Serkan ,)
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 4 / 17
Problem
General Problem
POD
RBM
DMD
systems theoretic ROMs (Serkan ,)
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 4 / 17
Problem
General Problem
POD
RBM
DMD
systems theoretic ROMs (Serkan ,)
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 4 / 17
Problem
General Problem
POD
RBM
DMD
systems theoretic ROMs (Serkan ,)
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 4 / 17
Problem
What is the Problem?
(I) PDE•u = f(u) given
(II) FEM(
•uh ,vh
)=
(f(uh) ,vh
)given
(III) ROM
r given
{ϕ1, . . . ,ϕr} given
model ???
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 5 / 17
Problem
What is the Problem?
(I) PDE•u = f(u) given
(II) FEM(
•uh ,vh
)=
(f(uh) ,vh
)given
(III) ROM
r given
{ϕ1, . . . ,ϕr} given
model ???
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 5 / 17
Problem
What is the Problem?
(I) PDE•u = f(u) given
(II) FEM(
•uh ,vh
)=
(f(uh) ,vh
)given
(III) ROM
r given
{ϕ1, . . . ,ϕr} given
model ???
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 5 / 17
Problem
What is the Problem?
(I) PDE•u = f(u) given
(II) FEM(
•uh ,vh
)=
(f(uh) ,vh
)given
(III) ROM
r given
{ϕ1, . . . ,ϕr} given
model ???
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 5 / 17
Problem
What is the Problem?
(I) PDE•u = f(u) given
(II) FEM(
•uh ,vh
)=
(f(uh) ,vh
)given
(III) ROM
r given
{ϕ1, . . . ,ϕr} given
model ???
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 5 / 17
Problem
What is the Problem?
(I) PDE•u = f(u) given
(II) FEM(
•uh ,vh
)=
(f(uh) ,vh
)given
(III) ROM
r given
{ϕ1, . . . ,ϕr} given
model ???
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 5 / 17
Solution
Standard Solution Discretization Levels
(I) PDE dim =∞
(II) FEM dim = N = O(106)
(III) ROM dim = r = O(10)
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 6 / 17
Solution
Standard Solution Discretization Levels
(I) PDE dim =∞
(II) FEM dim = N = O(106)
(III) ROM dim = r = O(10)
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 6 / 17
Solution
Standard Solution Discretization Levels
(I) PDE dim =∞
(II) FEM dim = N = O(106)
(III) ROM dim = r = O(10)
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 6 / 17
Solution
Standard Solution Model
ur =r∑
j=1
aj ϕj
(•
ur ,ϕi
)=
(f(ur) ,ϕi
)i = 1, . . . , r
Galerkin ROM 1
1NSE•a = A a + a> B a
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 7 / 17
Solution
Standard Solution Model
ur =r∑
j=1
aj ϕj
(•
ur ,ϕi
)=
(f(ur) ,ϕi
)i = 1, . . . , r
Galerkin ROM 1
1NSE•a = A a + a> B a
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 7 / 17
Solution
Standard Solution Model
ur =r∑
j=1
aj ϕj
(•
ur ,ϕi
)=
(f(ur) ,ϕi
)i = 1, . . . , r
Galerkin ROM 1
1NSE•a = A a + a> B a
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 7 / 17
Solution
New Solution Discretization Levels
(I) PDE dim =∞
(II) FEM dim = N = O(106)
(III) FEM-ROM dim = d = O(103)
(IV) ROM dim = r = O(10)
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 8 / 17
Solution
New Solution Discretization Levels
(I) PDE dim =∞
(II) FEM dim = N = O(106)
(III) FEM-ROM dim = d = O(103)
(IV) ROM dim = r = O(10)
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 8 / 17
Solution
New Solution Discretization Levels
(I) PDE dim =∞
(II) FEM dim = N = O(106)
(III) FEM-ROM dim = d = O(103)
(IV) ROM dim = r = O(10)
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 8 / 17
Solution
New Solution Discretization Levels
(I) PDE dim =∞
(II) FEM dim = N = O(106)
(III) FEM-ROM dim = d = O(103)
(IV) ROM dim = r = O(10)
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 8 / 17
Solution
New Solution Model
ud =d∑
j=1
aj ϕj
(•
ud ,ϕi
)=
(f(ud) ,ϕi
)i = 1, . . . , r
(•
ur ,ϕi
)=
(f(ur) ,ϕi
)+
[(f(ud) ,ϕi
)−(
f(ur) ,ϕi
)]i = 1, . . . , r
Galerkin ROM + correction
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 9 / 17
Solution
New Solution Model
ud =d∑
j=1
aj ϕj
(•
ud ,ϕi
)=
(f(ud) ,ϕi
)i = 1, . . . , r
(•
ur ,ϕi
)=
(f(ur) ,ϕi
)+
[(f(ud) ,ϕi
)−(
f(ur) ,ϕi
)]i = 1, . . . , r
Galerkin ROM + correction
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 9 / 17
Solution
New Solution Model
ud =d∑
j=1
aj ϕj
(•
ud ,ϕi
)=
(f(ud) ,ϕi
)i = 1, . . . , r
(•
ur ,ϕi
)=
(f(ur) ,ϕi
)+
[(f(ud) ,ϕi
)−(
f(ur) ,ϕi
)]i = 1, . . . , r
Galerkin ROM + correction
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 9 / 17
Solution
New Solution Model
ud =d∑
j=1
aj ϕj
(•
ud ,ϕi
)=
(f(ud) ,ϕi
)i = 1, . . . , r
(•
ur ,ϕi
)=
(f(ur) ,ϕi
)+
[(f(ud) ,ϕi
)−(
f(ur) ,ϕi
)]i = 1, . . . , r
Galerkin ROM + correction
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 9 / 17
Solution
New Solution Data-Driven Model
[(f(ud) ,ϕi
)−(
f(ur) ,ϕi
)]data-driven≈
(g(ur) ,ϕi
)
(•
ur ,ϕi
)=
(f(ur) ,ϕi
)+
(g(ur) ,ϕi
)i = 1, . . . , r
Data-Driven Filtered ROM (DDF-ROM) 2
2NSE•a =
(A + Ã
)a + a>
(B + B̃
)a
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 10 / 17
Solution
New Solution Data-Driven Model
[(f(ud) ,ϕi
)−(
f(ur) ,ϕi
)]data-driven≈
(g(ur) ,ϕi
)
(•
ur ,ϕi
)=
(f(ur) ,ϕi
)+
(g(ur) ,ϕi
)i = 1, . . . , r
Data-Driven Filtered ROM (DDF-ROM) 2
2NSE•a =
(A + Ã
)a + a>
(B + B̃
)a
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 10 / 17
Solution
New Solution Data-Driven Model
[(f(ud) ,ϕi
)−(
f(ur) ,ϕi
)]data-driven≈
(g(ur) ,ϕi
)
(•
ur ,ϕi
)=
(f(ur) ,ϕi
)+
(g(ur) ,ϕi
)i = 1, . . . , r
Data-Driven Filtered ROM (DDF-ROM) 2
2NSE•a =
(A + Ã
)a + a>
(B + B̃
)a
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 10 / 17
Solution
DDF-ROM Reproductive
Re = 1000
0.5 1 1.5 2 2.5 3
t
0.605
0.61
0.615
0.62
En
erg
y
r=4
0.5 1 1.5 2 2.5 3
t
2.5
3
3.5
4
4.5
Dra
g
N=4
0.5 1 1.5 2 2.5 3
t
-3
-2
-1
0
1
2
3
Lift
N=4
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 11 / 17
Solution
DDF-ROM + Constraints Predictive
Re = 1000
50% data
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t
0.599
0.6
0.601
0.602
0.603
0.604
0.605
0.606
En
erg
y
r=3
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 12 / 17
Connections
Correction = Numerical Dissipation
eddy viscosity ROMs
mixing length: Lumley et al
Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al
variational multiscale: Iollo et al, Wang et al , . . .
dynamic subgrid-scale: Wang et al
numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .
Schneier Friday talk
Stabile Thursday talk
Ali, Hijazi Wednesday poster
Taddei Wednesday poster
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17
Connections
Correction = Numerical Dissipation
eddy viscosity ROMs
mixing length: Lumley et al
Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al
variational multiscale: Iollo et al, Wang et al , . . .
dynamic subgrid-scale: Wang et al
numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .
Schneier Friday talk
Stabile Thursday talk
Ali, Hijazi Wednesday poster
Taddei Wednesday poster
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17
Connections
Correction = Numerical Dissipation
eddy viscosity ROMs
mixing length: Lumley et al
Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al
variational multiscale: Iollo et al, Wang et al , . . .
dynamic subgrid-scale: Wang et al
numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .
Schneier Friday talk
Stabile Thursday talk
Ali, Hijazi Wednesday poster
Taddei Wednesday poster
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17
Connections
Correction = Numerical Dissipation
eddy viscosity ROMs
mixing length: Lumley et al
Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al
variational multiscale: Iollo et al, Wang et al , . . .
dynamic subgrid-scale: Wang et al
numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .
Schneier Friday talk
Stabile Thursday talk
Ali, Hijazi Wednesday poster
Taddei Wednesday poster
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17
Connections
Correction = Numerical Dissipation
eddy viscosity ROMs
mixing length: Lumley et al
Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al
variational multiscale: Iollo et al, Wang et al , . . .
dynamic subgrid-scale: Wang et al
numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .
Schneier Friday talk
Stabile Thursday talk
Ali, Hijazi Wednesday poster
Taddei Wednesday poster
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17
Connections
Correction = Numerical Dissipation
eddy viscosity ROMs
mixing length: Lumley et al
Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al
variational multiscale: Iollo et al, Wang et al , . . .
dynamic subgrid-scale: Wang et al
numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .
Schneier Friday talk
Stabile Thursday talk
Ali, Hijazi Wednesday poster
Taddei Wednesday poster
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17
Connections
Correction = Numerical Dissipation
eddy viscosity ROMs
mixing length: Lumley et al
Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al
variational multiscale: Iollo et al, Wang et al , . . .
dynamic subgrid-scale: Wang et al
numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .
Schneier Friday talk
Stabile Thursday talk
Ali, Hijazi Wednesday poster
Taddei Wednesday poster
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17
Connections
Correction = Numerical Dissipation
eddy viscosity ROMs
mixing length: Lumley et al
Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al
variational multiscale: Iollo et al, Wang et al , . . .
dynamic subgrid-scale: Wang et al
numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .
Schneier Friday talk
Stabile Thursday talk
Ali, Hijazi Wednesday poster
Taddei Wednesday poster
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17
Connections
Correction = Numerical Dissipation
eddy viscosity ROMs
mixing length: Lumley et al
Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al
variational multiscale: Iollo et al, Wang et al , . . .
dynamic subgrid-scale: Wang et al
numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .
Schneier Friday talk
Stabile Thursday talk
Ali, Hijazi Wednesday poster
Taddei Wednesday poster
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17
Connections
Correction = Numerical Dissipation
eddy viscosity ROMs
mixing length: Lumley et al
Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al
variational multiscale: Iollo et al, Wang et al , . . .
dynamic subgrid-scale: Wang et al
numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .
Schneier Friday talk
Stabile Thursday talk
Ali, Hijazi Wednesday poster
Taddei Wednesday poster
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17
Connections
Correction = Numerical Dissipation
eddy viscosity ROMs
mixing length: Lumley et al
Smagorinsky: Noack et al, Ullman & Lang, Wang et al, Rozza et al
variational multiscale: Iollo et al, Wang et al , . . .
dynamic subgrid-scale: Wang et al
numerical stabilizationNoack et al, Iollo et al, Karniadakis et al, Farhat et al, Amsallem et al, Carlberg et al,Kalashnikova et al, Balajewicz et al, Wells et al,. . .
Schneier Friday talk
Stabile Thursday talk
Ali, Hijazi Wednesday poster
Taddei Wednesday poster
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 13 / 17
Connections
Galerkin ROM Stable, Not Accurate
stable
‖unr ‖2 + ν ∆tM∑
n=1
‖∇unr ‖2 ≤ data
not accurate
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 14 / 17
Connections
Galerkin ROM Stable, Not Accurate
stable
‖unr ‖2 + ν ∆tM∑
n=1
‖∇unr ‖2 ≤ data
not accurate
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 14 / 17
Connections
Galerkin ROM Stable, Not Accurate
stable
‖unr ‖2 + ν ∆tM∑
n=1
‖∇unr ‖2 ≤ data
not accurate
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 14 / 17
Connections
Connections
Large Eddy Simulationu u
h
Variational Multiscale
Nonlinear Galerkin
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 15 / 17
Connections
Connections
Large Eddy Simulationu u
h
Variational Multiscale
Nonlinear Galerkin
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 15 / 17
Connections
Connections
Large Eddy Simulationu u
h
Variational Multiscale
Nonlinear Galerkin
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 15 / 17
Connections
Related Work
no ROM
20th century Adrian, Moser
21st century Iaccarino, Duraisamy, Ling, Xiao
ROM
21st century Chorin, Duraisamy, Codina, Majda, San, Liu
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 16 / 17
Connections
Related Work
no ROM
20th century Adrian, Moser
21st century Iaccarino, Duraisamy, Ling, Xiao
ROM
21st century Chorin, Duraisamy, Codina, Majda, San, Liu
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 16 / 17
Connections
Related Work
no ROM
20th century Adrian, Moser
21st century Iaccarino, Duraisamy, Ling, Xiao
ROM
21st century Chorin, Duraisamy, Codina, Majda, San, Liu
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 16 / 17
Connections
Related Work
no ROM
20th century Adrian, Moser
21st century Iaccarino, Duraisamy, Ling, Xiao
ROM
21st century Chorin, Duraisamy, Codina, Majda, San, Liu
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 16 / 17
Connections
Related Work
no ROM
20th century Adrian, Moser
21st century Iaccarino, Duraisamy, Ling, Xiao
ROM
21st century Chorin, Duraisamy, Codina, Majda, San, Liu
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 16 / 17
Conclusions
Conclusions
Galerkin ROM + correction
correction data-driven
Re = 100, Re = 500, Re = 1000
reproductive X
predictive (cross-validation) X
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 17 / 17
Conclusions
Conclusions
Galerkin ROM + correction
correction data-driven
Re = 100, Re = 500, Re = 1000
reproductive X
predictive (cross-validation) X
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 17 / 17
Conclusions
Conclusions
Galerkin ROM + correction
correction data-driven
Re = 100, Re = 500, Re = 1000
reproductive X
predictive (cross-validation) X
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 17 / 17
Conclusions
Conclusions
Galerkin ROM + correction
correction data-driven
Re = 100, Re = 500, Re = 1000
reproductive X
predictive (cross-validation) X
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 17 / 17
Conclusions
Conclusions
Galerkin ROM + correction
correction data-driven
Re = 100, Re = 500, Re = 1000
reproductive X
predictive (cross-validation) X
T. Iliescu (Mathematics) DDF-ROM MoRePaS 2018 17 / 17
ProblemSolutionConnectionsConclusions