Randomized Model Order Reduction · Randomized Model Order Reduction Alessandro Alla work in...
Transcript of Randomized Model Order Reduction · Randomized Model Order Reduction Alessandro Alla work in...
Randomized Model Order Reduction
Alessandro Allawork in collaboration with J. Nathan Kutz
Data-Driven Methods for Reduced-Order Modelingand Stochastic Partial Differential Equations
Banff, January 30, 2016
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Problem Settings
Dynamical SystemM y(t) = Ay(t) + f(t ,y(t)), t ∈ (0,T ],
y(0) = y0.
Assumptions
y0 ∈ Rn is a given initial data,M,A ∈ Rn×n given matrices,f : [0,T ]× Rn → Rn a continuous function in both arguments andlocally Lipschitz-type with respect to the second variable.
WARNING: High dimensional problems are computationally expensive.
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Introduction
Low-rank approximation requires:Solutions of the original high-dimensional system (snapshots),Dimensionality-reduction produced by SVD,Galerkin projection of the dynamics on the low-rank subspace.
WARNING:Offline stages are exceptionally expensive, but enable the (cheap)online stage to potentially run in real time.
GOAL: To improve the efficiency of the offline stageRandomized techniques attempt to construct low-rank matrixdecompositions, fast and accurate approximations of QR and SVD.
A. Alla (Florida State University) Randomized MOR 3 / 42
Outline
Outline
1 Model Order ReductionProper Orthogonal DecompositionDiscrete Empirical Interpolation MethodDynamic Mode DecompositionCoupling POD and DMD methods
2 Randomized Linear Algebra in Model Order ReductionCompressed Model Order Reduction TechniquesCompressed PODCompressive Sampling DMD
3 Numerical Tests
A. Alla (Florida State University) Randomized MOR 4 / 42
Model Order Reduction
Outline
1 Model Order ReductionProper Orthogonal DecompositionDiscrete Empirical Interpolation MethodDynamic Mode DecompositionCoupling POD and DMD methods
2 Randomized Linear Algebra in Model Order ReductionCompressed Model Order Reduction TechniquesCompressed PODCompressive Sampling DMD
3 Numerical Tests
A. Alla (Florida State University) Randomized MOR 5 / 42
Model Order Reduction Proper Orthogonal Decomposition
Proper Orthogonal Decomposition and SVD
Proper Orthogonal Decomposition (POD), [L. Sirovich ’87]
Simulate at different time instances,Take snapshots of the state,Perform POD (Proper Orthogonal Decomposition) using SVD(Singular Value Decomposition),Use the POD basis functions as (non local) FEM ansatz functions.
A. Alla (Florida State University) Randomized MOR 6 / 42
Model Order Reduction Proper Orthogonal Decomposition
Proper Orthogonal Decomposition and SVD
Given snapshots (y(t0), . . . , y(tn)) ∈ Rm
We look for an orthonormal basis ψi`i=1 in Rm with ` minn,m s.t.
J(ψ1, . . . , ψ`) =n∑
j=1
αj
∥∥∥∥∥yj −∑i=1
〈yj , ψi〉ψi
∥∥∥∥∥2
=d∑
i=`+1
σ2i
reaches a minimum where αjnj=1 ∈ R+.
min J(ψ1, . . . , ψ`) s.t .〈ψi , ψj〉 = δij
Singular Value Decomposition: Y = ΨΣV T .
For ` ∈ 1, . . . ,d = rank(Y ), ψi`i=1 are called POD basis of rank `.
ERROR INDICATOR: E(`) =
∑i=1
σi
d∑i=1
σi
with σi singular values of the SVD.
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Model Order Reduction Proper Orthogonal Decomposition
Reduced Order System
POD-Galerkin ansatz
y(t) ≈ ΨPODy`(t), ΨPOD ∈ Rn×`.
POD dynamical systemM`y`(t) = A`y`(t) + (ΨPOD)T f (t ,ΨPODy`(t)),
y`(0) = y`0.
Dimension of the entries
(M`)ij = 〈Mψi ,ψj〉 ∈ R`×`,(A`)ij = 〈Aψi ,ψj〉 ∈ R`×` ,
y`0 = (ΨPOD)T y0 ∈ R`.
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Model Order Reduction Discrete Empirical Interpolation Method
Discrete Empirical Interpolation Method
Problem:Reduction of the nonlinearity is NOT independent from n:
F(t ,y`(t)) := (ΨPOD)T f(t ,ΨPODy`(t)) = 〈f(t ,y(t)),ΨPOD〉.
IDEA:Do not evaluate the nonlinearity everywhere, but select the mostimportant points via the greedy procedure.
ReferencesM. Barrault, Y. Maday, N.C. Nguyen, A.T. Patera, An empirical interpolationmethod: application to efficient reduced-basis discretization of partial differentialequations, 2004.
S. Chatarantabut, D. Sorensen, Nonlinear Model Reduction via DiscreteEmpirical Interpolation, 2010.
A. Alla (Florida State University) Randomized MOR 9 / 42
Model Order Reduction Discrete Empirical Interpolation Method
Discrete Empirical Interpolation Method
Compute y(tj) from the dynamical system,Evaluate f(tj ,y(tj)),U ∈ Rn×k the POD basis function of rank k of the nonlinear part.The DEIM approximation of f(t ,y(t)) is as follows
fDEIM(t ,yDEIM(t)) := U(ST U)−1f(t ,yDEIM(t))
where S ∈ Rn×k and yDEIM(t) = STΨPODy`(t) ∈ Rk .
Interpolation Points: Matrix S tells where evaluate the nonlinearity.LU decomposition with pivoting (Chatarantabut, Sorensen, 2010),QR decomposition with pivoting (Drmac, Gugercin, 2015).
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Model Order Reduction Discrete Empirical Interpolation Method
Discrete Empirical Interpolation Method
Let us define ΨDEIM := U(ST U)−1 ∈ Rn×k .The reduced nonlinearity may be expressed as:
(ΨPOD)T fDEIM(t ,yDEIM(t)) = (ΨPOD)TΨDEIMf(t ,yDEIM)
where we only select a small (sparse) number of rows of ΨPODy`(t).
Computational expenses
STΨPOD ∈ Rk×`, (ST U)−1 ∈ Rk×k and (ΨPOD)TΨDEIM ∈ R`×k
RemarkPrecomputed quantities are independent of the full dimension n.
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Model Order Reduction Discrete Empirical Interpolation Method
Discrete Empirical Interpolation Method
POD dynamical systemM`y`(t) = A`y`(t) + (ΨPOD)T f (t ,ΨPODy`(t)),
y`(0) = y`0
POD-DEIM dynamical systemM`y`(t) = A`y`(t) + (ΨPOD)TΨDEIMf(t ,yDEIM)y`(0) = y`0.
low-rank approximation of the nonlinear term.
Error Estimation
‖f− fDEIM‖2 ≤ c‖(I− UUT )f‖2, c = ‖(ST U)−1‖2with different performances depending on the matrix S.
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Model Order Reduction Dynamic Mode Decomposition
Introduction to DMD
DMD is an equation-free, data-driven method capable of providingaccurate model for complex system, and short-time future estimates ofsuch a systems.
It traces its origins to pioneering work of Bernard Koopman in 1931.Koopman theory is a dynamical systems tool that provides informationabout a nonlinear dynamical system via an associatedinfinite-dimensional linear system.
DMD method was proposed as a data-driven algorithm for modelingcomplex flows as a special case of Koopman theory.(Brunton, Kutz, Mezic, Noack, Rowley, Schmid, Tu, . . . )
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Model Order Reduction Dynamic Mode Decomposition
Dynamic Mode Decomposition
Dynamic Mode DecompositionSuppose we have a dynamical system and compute snapshots(y(t0), . . . , y(tm) and two sets of data
Y=
y(t0) y(t1) · · · y(tm−1)
, Y′=
y(t1) y(t2) · · · y(tm)
with y(tj) an initial condition of the dynamical system and y(tj+1) itscorresponding output⇒ Y′ = AYY with AY ∈ Rn×n unknown.
The DMD modes are eigenvectors of
Ay = Y′Y†
where † denotes the Moore-Penrose pseudoinverse.A. Alla (Florida State University) Randomized MOR 14 / 42
Model Order Reduction Dynamic Mode Decomposition
Dynamic Mode Decomposition
Ay is a finite dimensional approximation of the Koopman operatorfor a linear observable.The definition of DMD produces a regression procedure wherebythe data snapshots in time are used to produce the best-fit lineardynamical system for the data Y.
The DMD procedure constructs the proxy, approximate linear evolution
d ydt
= Ayy
with y(0) = y0 and whose solution is: y(t) =∑n
i=1 biψi exp(ωi t),ψi and ωi are the eigenfunctions and eigenvalues of the matrix Ay.
SOLUTIONDMD circumvents the eigendecomposition of Ay by considering arank-reduced representation in terms of a POD-projected matrix Ay.
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Model Order Reduction Dynamic Mode Decomposition
Dynamic Mode Decomposition
DMD algorithm
Require: Snapshots y(t0), . . . ,y(tm),1: Set Y = [y(t0), . . . ,y(tm−1)] and Y′ = [y(t1), . . . ,y(tm)],2: Compute the (reduced) SVD of Y, Y = UΣVT
3: Define Ay := U∗Y′VΣ−1
4: Compute eigenvalues and eigenvectors of AyW = WΛ.5: Set ΨDMD = Y′VΣ−1W
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Model Order Reduction Dynamic Mode Decomposition
Example: DMD-Galerkin approximation
yt (x , t) + yx (x , t) = 0 (x , t) ∈ [0,4]× [0,3],
y(x ,0) = y0(x) x ∈ [0,4],
y(0, t) = 0 = y(4, t) t ∈ [0,T ],
where y0(x) = sin(πx) if 0 ≤ x ≤ 1 and 0 elsewhere.
DMD Ansatz: y(t) ≈ ΨDMDyDMD(t).
50 100 150 200 250 300
10−15
10−10
10−5
100
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Model Order Reduction Dynamic Mode Decomposition
Figure: Reduced approximation with rank=5,10,15. POD approximation(top), DMD-Galerkin (middle), DMD (data-driven) (bottom)
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Model Order Reduction Dynamic Mode Decomposition
Example: DMD-Galerkin approximation
0 10 20 30 40 50 6010
−14
10−12
10−10
10−8
10−6
10−4
10−2
100
Number of Basis Functions
POD
DMD−Galerkin
DMD
0 0.5 1 1.5 2 2.5 3 3.5 4−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
POD
DMD−Galerkin
DMD
0 0.5 1 1.5 2 2.5 3 3.5 4−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
POD
DMD−Galerkin
DMD
Figure: Error analysis with respect to the Frobenius norm (left), first mode(middle), second mode (right).
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Model Order Reduction Coupling POD and DMD methods
POD-DMD method
MAIN IDEAThe evaluation of the nonlinearity is the most expensive part in modelorder reduction. We aim faster approximation of the nonlinear term.
We need snapshots!
y(t0), . . . ,y(tm), to compute the POD basis functions,f(t0,y(t0)), . . . , f(tm,y(tm)) to compute the DMD basis functions.
POD-DMD method (NO EVALUATION OF THE NONLINEARITY)(A., Kutz, 2016) M`y`(t) = A`y`(t) + (ΨPOD)TΨDMDdiag(eω
DMDt )b,
y`(0) = y0`.
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Model Order Reduction Coupling POD and DMD methods
Example: Semi-Linear Parabolic Equation
yt − θ∆y + µ(y − y3) = 0 (x , t) ∈ Ω× [0,T ],
y(x ,0) = y0(x) x ∈ Ω,
y(·, t) = 0 x ∈ ∂Ω, t ∈ [0,T ],
Parameters:Ω = [0,1]× [0,1],T = 3,y0(x) = 0.1 if 0.1 ≤ x1x2 ≤ 0.6 and 0 elsewhere.
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Model Order Reduction Coupling POD and DMD methods
Example: Semi-Linear Parabolic Equation
0
0.5
1
0
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1
0
0.02
0.04
0.06
0.08
0.1
0
0.5
1
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0.25
0
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1
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1
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1
Figure: Solution at time t = 0,0.1 (top) and t = 1.5,3 (bottom)
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Model Order Reduction Coupling POD and DMD methods
Example: Semi-Linear Parabolic Equation
0 5 10 15 20 25 30 35 4010
−4
10−3
10−2
10−1
100
101
FULL
POD
POD−DEIM
POD−DMD
DMD
0 5 10 15 20 25 30 35 4010
−10
10−8
10−6
10−4
10−2
100
POD
POD−DEIM
POD−DMD
DMD
Figure: CPU-time online stage (left) and Relative Error wrt Frobenius norm.Number of POD modes and DEIM/DMD points are the same
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Model Order Reduction Coupling POD and DMD methods
Example: Semi-Linear Parabolic Equation
0 5 10 15 20 25 30 35 4010
−3
10−2
10−1
100
POD−DMD
POD−DEIM
0 5 10 15 20 25 30 35 4010
−4
10−3
10−2
10−1
100
POD−DMD
POD−DEIM
0 5 10 15 20 25 30 35 4010
−5
10−4
10−3
10−2
10−1
100
POD−DMD
POD−DEIM
Figure: Relative Error for 5 POD basis functions (left), 10 POD basis (middle),15 POD basis (right)
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Model Order Reduction Coupling POD and DMD methods
Example: Semi-Linear Parabolic Equation
A fair comparison
10−6
10−5
10−4
10−3
10−2
10−1
100
10−3
10−2
10−1
100
RELATIVE ERROR
CP
U T
IME
POD
POD−DEIM
POD−DMD
DMD
A. Alla (Florida State University) Randomized MOR 25 / 42
Randomized Linear Algebra in Model Order Reduction
Outline
1 Model Order ReductionProper Orthogonal DecompositionDiscrete Empirical Interpolation MethodDynamic Mode DecompositionCoupling POD and DMD methods
2 Randomized Linear Algebra in Model Order ReductionCompressed Model Order Reduction TechniquesCompressed PODCompressive Sampling DMD
3 Numerical Tests
A. Alla (Florida State University) Randomized MOR 26 / 42
Randomized Linear Algebra in Model Order Reduction
Randomized SVD (Haiko, Martinsson, Tropp, 2011)
Computational Costs for Y ∈ Rm×n:
SVD: O(mn2)
Randomized SVD: O(mn`),
First Steps:
Choose desired target rank ` m,n,Create a random (gaussian) sampling matrix Ω ∈ Rn×`,Sampled matrix X ∈ Rn×` is computed as: X = YΩ.
Remarks:If the matrix Y has exact rank `, then the sampled matrix X spans,with high probability, a basis for the column space.In practice, it is favorable to slightly oversample ` = `+ p, were pdenotes the number of additional samples.
A. Alla (Florida State University) Randomized MOR 27 / 42
Randomized Linear Algebra in Model Order Reduction
Randomized SVD
Second Steps: Obtain low-rank SVD from a compressed matrix
Compute X = QR, Q ∈ Rn×` orthonormal,Y is projected into this low-dimensional space B ∈ R`×m:
B = QᵀY,
Compute the (cheap) SVD B = UΣVT ,
Set U = QU
Remark: if ` is large enough:
Y ≈QQᵀY≈QB
≈QUΣVT
≈UΣVT
A. Alla (Florida State University) Randomized MOR 28 / 42
Randomized Linear Algebra in Model Order Reduction
Randomized SVD
0 20 40 60 80 100 12010
−14
10−12
10−10
10−8
10−6
10−4
10−2
100
102
104
Singular Values
FULL
randomized l=r/2
randomized l=r
randomized l=2r
A. Alla (Florida State University) Randomized MOR 29 / 42
Randomized Linear Algebra in Model Order Reduction Compressed Model Order Reduction Techniques
How to use RSVD in model reduction?
Model order reduction techniques:are based on snapshots of the dynamical system.SVD decomposition of the snapshots matrix provides alow-dimension projector operator that allows one to obtainsurrogate models.WARNING: SVD may be computationally expensive to reducedthe offline cost of the method.
IDEA:Compute basis functions from a few spatially incoherentmeasurements (not from the full set of measurements)
A. Alla (Florida State University) Randomized MOR 30 / 42
Randomized Linear Algebra in Model Order Reduction Compressed POD
Compressed POD
we collect the snapshot set,we solve the POD optimization problem,optimality conditions provide eigenvalue problems,RSVD computes compressed POD basis functions in asignificantly faster way.
Algorithm
Require: Snapshot Matrix Y ∈ Rn×m, ` number of basis functions., pnumber of measurements.
1: Compute the Randomized SVD [U,Σ,V] = rsvd(Y,p)2: Set Ψi = Ui for i = 1, . . . , `.
A. Alla (Florida State University) Randomized MOR 31 / 42
Randomized Linear Algebra in Model Order Reduction Compressed POD
Compressed POD
Error (d rank of Y, p number of samples)
E
m∑j=1
αj
∥∥∥∥∥y(tj)−∑i=1
〈y(tj),ψi〉ψi
∥∥∥∥∥2 =
(1 +
√`
p − 1
)σ2`+1 +
√`+ pp
d∑j=`+1
σ2j .
Remarks:we consider the expectation value of the errorerror depends on the computation of the set of snapshots and p,if the singular values of Y decay rapidly a few samples drives theerror close to the theoretically minimum value.if the singular values do not decay rapidly we can lose accuracy.we suppose Y ∈ Rm×n, such that m ≈ n(eigenvalue problem is also expensive)
A. Alla (Florida State University) Randomized MOR 32 / 42
Randomized Linear Algebra in Model Order Reduction Compressive Sampling DMD
Compressive DMD (Brunton, Proctor, Kutz, 2015)
GOAL:Compute DMD and apply as Galerkin projection method.
Algorithm
Require: Snapshots y(t0), . . . ,y(tm), C ∈ Rp×m
1: Set Y = [y(t0), . . . ,y(tm−1)] and Y ′ = [y(t1), . . . ,y(tm)],2: X = CY, X′ = CY′
3: Compute the SVD of X, X = UΣVT
4: Define Ax := U∗Y′VΣ−1
5: Compute eigenvalues and eigenvectors of AxW = WΛ.6: Set ΨDMD = X′VΣ−1W
A. Alla (Florida State University) Randomized MOR 33 / 42
Numerical Tests
Outline
1 Model Order ReductionProper Orthogonal DecompositionDiscrete Empirical Interpolation MethodDynamic Mode DecompositionCoupling POD and DMD methods
2 Randomized Linear Algebra in Model Order ReductionCompressed Model Order Reduction TechniquesCompressed PODCompressive Sampling DMD
3 Numerical Tests
A. Alla (Florida State University) Randomized MOR 34 / 42
Numerical Tests
Test 1: Semi-Linear Parabolic Equation
yt − θ∆y + µ(y − y3) = 0 (x , t) ∈ Ω× [0,T ],
y(x ,0) = y0(x) x ∈ Ω,
y(·, t) = 0 x ∈ ∂Ω, t ∈ [0,T ],
Parameters:Ω = [0,1]× [0,1],T = 3,y0(x) = 0.1 if 0.1 ≤ x1x2 ≤ 0.6 and 0 elsewhere.
A. Alla (Florida State University) Randomized MOR 35 / 42
Numerical Tests
Test 1: Semi-Linear Parabolic Equation
0 5 10 15 20 2510
−1
100
101
102
CPU TIME
Number Of Basis Functions
POD
rPOD
POD−DEIM
rPOD−rDEIM
DMD
cDMD
0 5 10 15 20 2510
−8
10−6
10−4
10−2
100
RELATIVE ERROR
Number Of Basis Functions
POD
cPOD
POD−DEIM
rPOD−rDEIM
DMD
cDMD
Figure: CPU-time online stage (left) and Relative Error wrt Frobenius norm.Number of POD modes and DEIM/DMD points are the same
A. Alla (Florida State University) Randomized MOR 36 / 42
Numerical Tests
Test 1: Semi-Linear Parabolic Equation
1000 2000 3000 4000 5000 6000
10−2
100
102
104
CPU TIME
Dimension of the full problem
POD
cPOD
POD−DEIM
rPOD−rDEIM
DMD
cDMD
1000 2000 3000 4000 5000 600010
−4
10−3
10−2
10−1
100
RELATIVE ERROR
Dimension of the full problem
PODcPODPOD−DEIMrPOD−rDEIMDMDcDMD
Figure: CPU-time online stage (left) and Relative Error wrt Frobenius norm.Number of POD modes and DEIM/DMD points are the same
A. Alla (Florida State University) Randomized MOR 37 / 42
Numerical Tests
Test 2: Parametric Elliptic Equation
−∆u(x , y) + s(u(x , y);µ) = f (x , y) (x , y) ∈ Ω
u(x , y) = 0 (x , y) ∈ ∂Ω
Parameters:
Ω = [0,1]× [0,1], µ = (µ1, µ2) ∈ D = [0.01,10]2
s(u, µ) =µ1
µ2(eµ2u − 1), f (x , y) = 100 sin(2πx) sin(2πy).
A. Alla (Florida State University) Randomized MOR 38 / 42
Numerical Tests
Test 2: Parametric Elliptic Equations
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0.1
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−1
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0
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
Figure: Test 2: Solution of problem for different parametric configurations(top) and countour lines (bottom)
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Numerical Tests
Test 2: Parametric Elliptic Equations
0 5 10 15 2010
−1
100
101
102
CPU TIME
Number Of Basis Functions
POD
cPOD
POD−DEIM
cPOD−cDEIM
DMD
cDMD
0 5 10 15 2010
−8
10−6
10−4
10−2
100
RELATIVE ERROR
Number Of Basis Functions
POD
cPOD
POD−DEIM
cPOD−cDEIM
DMD
cDMD
Figure: Test 2: CPU-time of the offline-online stages (left) and Relative Errorin Frobenius norm (right). We compare the following methods: POD (red),cPOD (magenta), POD-DEIM (yellow), cPOD-cDEIM (blue, DMD (black),cDMD (green). Number of model are always the same for all the methods.
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Numerical Tests
Conclusions
Model order reduction is a successful technique that projectsnonlinear high dimensional dynamical systems and PDEs into lowdimensional surrogate modelsCompressed (randomized) techniques are a promising approachto circumventing expensive offline stages in model orderreduction.DMD works successfully in a Galerkin projection framework
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Numerical Tests
Thank you for your attention
A. Alla, J.N. Kutz, Randomized Model Order Reduction, 2016.A. Alla, J.N. Kutz, Nonlinear Model Reduction via DMD, 2016.M. Barrault, Y. Maday, N.C. Nguyen, A.T. Patera, An empirical interpolationmethod: application to efficient reduced-basis discretization of partial differentialequations, 2004.P. Benner, S. Gugercin and K. Willcox, A Survey of Projection-Based ModelReduction Methods for Parametric Dynamical Systems, 2015.S. L. Brunton, J. L. Proctor, and J. N. Kutz, Compressive sampling and dynamicmode decomposition. 2015.S. Chatarantabut, D. Sorensen, Nonlinear Model Reduction via DiscreteEmpirical Interpolation. 2010.Z. Drmac, S. Gugercin, A new selection operator for the discrete empiricalinterpolation method - improved a priori error bound and extensions, 2016.N. Halko, P.-G. Martinsson and J. Tropp, Finding Structure with Randomness:Probabilistic Algorithms for Constructing Approximate Matrix Decompositions,2011.P. Schmid, Dynamic mode decomposition of numerical and experimental data,2010.J. Tu, C. Rowley, D. Luchtenberg, S. Brunton, and J. N. Kutz, On Dynamic ModeDecomposition: Theory and Applications, 2014.S. Volkwein, Model Reduction using Proper Orthogonal Decomposition, 2013.
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