Design, optimization and reduced-order models

Design, optimization and reduced-order models as part of the SMC4 M2 course Piotr Breitkopf, Florian De Vuyst Université de Technologie de Compiègne Master M2 Mention Ingénierie des Systèmes Complexes (ISC) Parcours SMC (resp. Alain Rassineux)

December 21, 2018

F. De Vuyst

Design, optimization and reduced-order models

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Part II Model order reduction (MOR) & Proper Orthogonal Decomposition (POD)

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Data dimensionality reduction

Consider a set of vector (dataset) in Rd : n o S N = x1 , x2 , ..., xN ∈ Rd Context: high-dimensional data (d 1), N = O(100) samples. In all cases, we will assume that N ≤ d.

Goal ˜ i of elements xi of S N , We look for high-fidelity approximations x expressed in a vector space of low dimension (dimensionality reduction).

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Singular Value Decomposition (SVD)

Singular Value Decomposition theorem Let X ∈ MdN (R). There exists a matrix U of size d × d, U T U = Id , a matrix V of size N × N , V T V = IN , and a matrix Σ of size d × N , Σ = diag(σj ) with σj ≥ 0 ∀j such that X = |{z} U |{z} Σ |{z} VT . |{z} d×N

d×d d×N N ×N

The quantities σj are called the singular values of matrix X.

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SVD (2) Application: consider X = x1 , x2 , ..., xN . Notations: h i U = u1 , u2 , ..., ud ,

V = v 1 , v 2 , ..., v N .

SVD : X = U ΣV T , then xi =

N X

σj hv j , ei i uj

∀i ∈ {1, ..., N }

j=1

with ei = (0, ..., 0, |{z} 1 , 0, ..., 0)T . (to check as exercise). pos. i

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Introducing reduction Let us assume that the σj are arranged in decreasing order: σ1 ≥ σ2 ≥ ... ≥ σN ≥ 0.

Projection Consider E K le vector space of dimension K spanned by the K first eigenvectors uj : E K = span u1 , u2 , ..., uK . Consider the orthogonal projection over E K defined by ΠK x =

K X

hx, uj i uj .

j=1

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Reduction (2) In particular, for x = xi we have K X Π x = hxi , uj i uj K

i

j=1

=

K X

σj hv j , ei i uj .

j=1

Projection error : N X

xi − ΠK xi =

σj hv j , ei i uj .

j=K+1

Then i

K

i 2

kx − Π x k =

N X

σj2 hv j , ei i2 .

j=K+1 F. De Vuyst


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Reduction (3)

As a consequence, N X

i

K

i 2

kx − Π x k =

i=1

N N X X

σj2 hv j , ei i2

i=1 j=K+1

=

N X

σj2

N X

hei , v j i2

|i=1 {z

j=K+1

=kei k2 =1

=

N X

}

σj2 .

j=K+1

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Reduction (3) Because the singular values σj are arranged in decreasing order, PN PN i K i 2 2 i=1 kx − Π x k cannot be lower than j=K+1 σj . Actually we have the optimality principle E K = arg

min

V K vector space, dim(V K )=K

N X

K

kxi − projV (xi )k2

i=1

with E K = span(u1 , ..., uK ).

Definition The vectors u1 , ..., uK are called the K first principal components of X. Related vocabulary: Principal Component Analysis (or PCA). F. De Vuyst


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Relative error estimation Recall that xi =

N X

σj hv j , ei i uj

j=1

so that

N X

kxi k2 =

i=1

N X

σj2 .

j=1

Then we have N X

i

K

N X

i 2

kx − Π x k

i=1 N X

= i 2

kx k

i=1

F. De Vuyst

σj2

j=K+1 N X

σj2

j=1


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Choice of the rank K Optimal choice of K according to a tolerance ε > 0: Choose K = K(ε) that realizes K.

min K∈N /

PN

2 j=K+1 σj ≤ε 2 j=1 σj

PN

Then from the previous estimate, N X

kxi − ΠK xi k2

i=1 N X

≤ ε. i 2

kx k

i=1

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Rate of decay the singular value spectrum

The more the decay of the singular value spectrum is, the smaller the truncation rank K is. Surpinsingly, for many dataset coming from real applications, Principal Component Analysis shows an actual low-order dimension of the data. Even for high-dimensional data (dimension of order O(106 )), we may have K = O(10) for reasonable accuracy (ε = O(10−2 )).

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Correlation matrix

From X, consider the “small”, symmetric matrix S defined by S = X T X ∈ MN (R) called the correlation matrix (Sij = hxi , xi i, S is a Gram matrix). From the SVD of X, we get S = V ΣT Σ V T . Thus, the vector v 1 , ..., v N from V are the eigenvectors of S with eigenvalues λj = σj2 , j = 1, ..., N.

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Computation of the uj from X and V From X = U ΣV T , we have XV = U Σ. At column j, j = 1, ..., N we have Xv j = σ uj , or in other words 1 uj = p Xv j , λj

F. De Vuyst

j = 1, ..., N .


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Summary: SVD algorithm 1. Consider the data set X = [x1 , ..., xN ]; 2. Compute the correlation matrix S = X T X ∈ MN (R); 3. Compute the eigenvalues λj and eigenvectors v j of S; 4. Fix tolerance ε, compute truncation rank K = K(ε); 5. Compute the K principal components uj , j = 1, ..., K as 1 uj = p Xv j . λj 6. Dimensionality reduction (by projection over E K ): ˜= x

K X

hx, uj i uj .

j=1

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Example in Matlab Consider the snaphost matrix computed from the following script: 1

3

5

7

9

11

clear all ; d = 200; N = 1000; x = l i n s p a c e ( 0 , 1 , d ) ; x=x ’ ; f o r j =1:N X ( : , j ) = s i n ( 2 ∗ p i ∗ x )+ 0 . 3 ∗ r a n d n ( d , 1 ) ; end ; [ U , S , V ] = s v d (X , ’ e c o n ’ ) ; % C a l l t o SVD d e c o m p o s i t i o n sigma = d i a g (S) ; % Diagonal o f s i n g u l a r v a l u e s f i g u r e ( 1 ) ; p l o t ( sigma , ’ o− ’ ) ; f i g u r e ( 2 ) ; p l o t ( x , U ( : , 1 ) , ’− ’ ) ;

The matrix X should have a main principal component, close to the vector sin(2*pi*x). F. De Vuyst


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Example in Matlab (2)

Figure: Left: Spectrum of the correlation matrix. Right: First principal component

⇒ PCA can be used as a denoising algorithm. F. De Vuyst


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Application to Finite Element discretized functions Notations Consider a polyhedral domain Ωh of R2 . Consider a triangulation T h of Ωh . Consider P 1 finite elements of triangles K ∈ T h , degrees of freedom at nodes. Let V h be associated (discrete) finite element space. dim(V h ) = d. Functional Hilbert space: V defined on Ωh , with scalar product (., .)V . Conformal FE space: V h ⊂ V . Finite element Lagrange basis ’hat’ functions : ωi (x). ( ) d X h h h h V = v ∈ V , v (x) = vi ωi (x) . i=1

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Bridges between functional and vector spaces h

h

v ∈V ,

h

v (x) =

d X

vi ωi (x).

i=1

Let us denote v = (v1 , v2 , ..., vd )T ∈ Rd . We have correspondence between v h and v. Let v h , wh ∈ V h , with respective vector representations v, w ∈ Rd . We have d d X X (v , w )V = ( vi ωi , wj ωj )V h

h

i=1

=

d X d X i=1 j=1

j=1

vi wj (ωi , ωj )V | {z } :=Mij

T

= w M v = hv, M wiRd = hv, wiM . F. De Vuyst


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Functional and vector correspondence As a summary, (v h , wh )V = hv, wiM . Ex: V = L2 (Ωh ), Mij = (ωi , ωj )L2 then M is the FE mass matrix. In particular, kv h k2V = kvk2M .

Choleski factorization of M The matrix M is a Gram, symmetric positive definite matrix. Then there exists a lower triangular matrix B with diagonal elements Bii > 0 such that M = BB T . Then we also have (v h , wh )V = hB T v, B T wiRd . F. De Vuyst


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PCA on Finite Element functions Let S N be a set of FE functions in V h : n o S N = f 1,h , f 2,h , ..., f N,h ∈ V h . Vector representation of the set: n o S N = f 1 , f 2 , ..., f N ∈ Rd . Consider also

n o T N = x1 , x2 , ..., xN ∈ Rd

with xi = B T f i .

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Scalar products

From function-vector correspondence properties, we have (f i,h , f j,h )V = hf i , f j iM = hxi , xj i = (xj )T xi . We have also kf i,h − f j,h k2V = kf i − f j k2M = kxi − xj k2 .

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PCA in vectors (xi )i Let us apply the SVD decomposition on the set T N : considering the matrix X = [x1 , x2 , ..., xN ], we have the SVD decomposition of X: X = U ΣV T. Consider the orthogonal projection over the vector space E K = span(u1 , u2 , ..., uK ): ΠK x =

K X

hx, ui i ui .

i=1

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Projection for the (f i )i, i = 1, ..., N Consider the vectors Φi defined by Φi = B −T ui . We have the orthogonality property hΦi , Φj iM = hB −T ui , B −T uj iM = hui , uj i = δij One can define the projection formula ΠK f =

K X hf , Φi iM Φi . i=1

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Projection for the f i,h From the definition of ΠK f , we can finally define a projection operator acting on FE functions: K h

Π f (x) =

K X

(f h , ϕi,h )V ϕi,h (x)

i=1

where i,h

ϕ (x) =

d X

(Φi )j ωj (x).

j=1

Proper Orthogonal Decomposition modes The functions ϕi,h (x) are called the Proper Orthogonal Decomposition modes (or POD modes).

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Proper Orthogonal Decomposition

The POD modes have the orthogonality property in V : (ϕi,h , ϕj,h )V = hΦi , Φj iM = δij .

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POD projection error Since h

kf i,h − ΠK f i kV = kf i − ΠK f j kM = kxi − ΠK xi k and similarly kf i,h kV = kxi k from the PCA error control K = K(ε) such that N X

kxi − ΠK xi k2

i=1 N X

≤ ε, kxi k2

i=1

we have the same projection error estimate for the FE functions. F. De Vuyst


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POD projection error (2)

As a summary, N X

kf i,h − ΠK f i,h k2V

i=1 N X

≤ ε, kf i,h k2V

i=1

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Practical algorithm Practically, one can skip the Choleski factorization of the matrix M . Indeed, remember that xi = B T f i , so that X = [x1 , x2 , ..., xN ] = [B T f 1 , B T f 2 , ..., B T f N ] := B T F. Consider the correlation matrix S = X T X: Sij = hxi , xj i = hB T f i , B T f j i = hf i , f j iM . From S, we compute the eigenvalues λj and the eigenvectors v j .

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Practical algorithm (2)

Moreover, it is possible to reconstruct the eigenvectors Φj without the need of B. Indeed, recall that 1 uj = p Xv j , λj then

Φi = B −T ui ,

X = B T F,

1 Φj = p F v j . λj

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Final algorithm 1. Compute the snapshot matrix F = f 1 , f 2 , ..., f N ; 2. Compute the correlation matrix S = FT MF and the associated eigenvalues λj and eigenvectors v j ; 3. Compute the dual eigenvectors Φj such that 1 Φj = p F v j . λj 4. Compute the truncation rank K = K(ε) according to the projection error formula. F. De Vuyst


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Final algorithm (2)

5. The K first POD modes ϕi,h ∈ V h , i = 1, ..., K are given by the formula K X i,h (Φi )j ωj (x). ϕ (x) = j=1

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Computation of POD coefficients For f h ∈ V h , we have ΠK f h (x) =

K X (f h , ϕk,h )V ϕk,h (x) k=1

or equivalently (vector representation) K

Π f=

K X

hf , Φk iM Φk .

k=1

So the POD projection coefficients ak (f ) for f are ak (f ) = hf , Φk iM = (Φk )T M f . ⇒ Does it really need the knowledge of M for computation ? F. De Vuyst


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Computation of POD coefficients : TRICK Actually, it is sufficient to know the correlation matrix S = F T M F . Remember that 1 Φj = p F v j . λj Let us denote Φr = [Φ1 , ..., ΦK ], V r = [v 1 , ..., v K ], Λr = diag(λ1 , ..., λK ). Then Φr = F V r (Λr )−1/2 and [a1 , a2 , ..., aN ] = (Φr )T M F = (Λr )−1/2 (V r )T F T M F = (Λr )−1/2 (V r )T S. F. De Vuyst


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Summary The matrix of reduced snapshot POD coefficients is given by [a1 , a2 , ..., aN ] = (Λr )−1/2 (V r )T S. (important formula). NB : computation of the POD coefficients of complexity O(KN 2 ), which is far reasonable for N = O(100) and K = O(10). Reconstruction in the physical (high-dimensional) space: f˜i = Φr ai ,

i = 1, ..., N

(requires high-dimensional operations).

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Application to partial differential parametrized problems

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Problems of interest

ξ : design parameter vector, ξ ∈ X ⊂ Rp u(.; ξ) ∈ V : function, solution of a (nonlinear) partial differential problem: N (u(.; ξ)) = f (.; ξ). Quantities of interest: fi (ξ) = ì (u(.; ξ)).

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Design of Computer Expreriment (DoCE)

1. Consider a sampling {ξi }i=1,...,N of the design set X ; 2. Compute each solution ui,h (x) = uh (x; ξi ) from a Finite Element solver (offline phase, computationally expensive). Vector representation: ui ∈ Rd ; 3. Create a database X = u1 , u2 , ..., uN ∈ MdN (R). 4. Compute the POD modes ϕi,h (x) associated to X.

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Model order reduction From the POD modes, one can define a low-order approximation u ˜h (x; ξ) of uh (x; ξ) by h

u ˜ (x; ξ) =

K X

ak (ξ) ϕk,h (x).

k=1

Vector-based representation of the same formula: u ˜h (x; ξ) = [ϕ1,h(x) , ϕ2,h (x), ..., ϕN,h (x)] a(ξ) with a(ξ) = [a1 (ξ), a2 (ξ), ..., aK (ξ)]T ∈ RK vector of coefficients.

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Determination of the coefficients

Because the POD modes ϕk,h are orthonormal, we have ak (ξ) = (uh (.; ξ), ϕk,h )V .

(1)

Online phase: from a new evaluation at parameter vector ξ, we want to get an approximation u ˜h (x; ξ). But ak (ξ) cannot be computed from (1) because we do not know uh (.; ξ). ⇒ Compute a metamodel a ˜k (ξ) of ak (ξ). Q: how to do ?

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Back to be offline stage During the offline stage, the solutions ui,h = uh (.; ξi ) have been computed. So one can compute ak (ξi ) = (ui,h , ϕk,h )V = hui , Φk iM and form the vectors ai = (ak (ξi ))k ∈ RK ,

∀ i = 1, ..., N

So summarize, we obtain a cloud of points (ξi , ai ) ∈ Rp × RK , i ∈ {1, 2, ..., N } . ⇒ Find the manifold mapping that interpolates (or approximates) the points (ξi , ai ).

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Kernel-based interpolation One can once again use kernel-based interpolation to compute the nonlinear mapping: ˜ (ξ) = a

N X

wj k(ξ, ξj ).

j=1

where wj ∈ RK for all j = 1, ..., N . Interpolation property: we want ˜ (ξi ) = a(ξ) ∀i = 1, ..., N. a This leads to the matrix identity A=KW

⇒

W = K −1 A

where A = [a(ξ1 ), ...a(ξN )], K = (k(ξi , ξj ))ij is the spd kernel matrix and W is the matrix of unknown vector coefficients. F. De Vuyst


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Final reduced-order model

We obtain the reduced-order model u ˜h (x; ξ) =

K X

˜ (ξ) ϕk,h (x) a

k=1

where ˜ (ξ) = a

N X

wj k(ξ, ξj )

j=1

with the matrix of vector coefficients W = K −1 A,

A = [a(ξ1 ), ...a(ξN )].

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Metamodels for the objective functions

Recall that each objective function fi (ξ) is a linear or nonlinear function of uh (.; ξ): fi (ξ) = ì (uh (.; ξ)). Few examples: fi (ξ) = min uh (x; ξ) x∈Ωh

or

1 fi (ξ) = h |Ω |

Z

(non linear)

uh (x; ξ) dx

(linear).

Ωh

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Case of a linear functional

In the case of linear functional, fi (ξ) can be approximated by f˜i (ξ) = ì (˜ uh (.; ξ)) = ì

K X

! a ˜k (ξ)ϕk,h (.)

k=1

=

K X k=1

a ˜k (ξ) ì (ϕk,h (.x)) {z } |

F. De Vuyst

to compute offline


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Case of a nonlinear functional

In the nonlinear case, unfortunately, the evaluation of ! K X k,h f˜i (ξ) = ì a ˜k (ξ)ϕ (.) k=1

requires high-dimensional operations and offline pre-computations are irrevelant in this case. ⇒ Requires the design of a new (nonlinear) metamodel ˜ (ξ) ∈ RK −→ f˜i (ξ) ∈ Rm . ξ ∈ Rp −→ a

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PRACTICAL WORK (Freefem++ and Matlab)

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Practical Work

Software environment: 1. Finite Element solver : Freefem++ 2. SVD, PCA, visualization : Matlab + PDE Toolbox 3. Freefem++-to-Matlab interface (by Julien Dambrine)

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Freefem++ to Matlab interface

See : http://www.tobias-rienmueller.de/post/freefem2matlab.html

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Step 1 – First experiment Run the Freefem++ script freefem-example.edp. 1

3

5

7

9

// s o l v e P o i s s o n ( u , v , s o l v e r=LU) = i n t 2 d ( Th ) ( dx ( u ) ∗ dx ( v ) + dy ( u ) ∗ dy ( v ) ) − i n t 2 d ( Th ) ( s o ∗v ) + on ( a , b , c , d , e , f , u=0) ; p l o t ( Th , u ) ; s a v e m e s h ( Th , ” G i l g a m e s h . msh” ) ; // f r e e f e m { o f s t r e a m f i l e ( ” Heat . bb ” ) ; f i l e 0. Vector form: x(t) = (x(t), y(t)), x˙ = f (x),

t>0

NB: for µ λ, we have x(t) = exp(−λt)x0 and y(t) ≈ (x(t))2 .

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Nonlinear case: example NB: for µ λ, we have x(t) = exp(−λt)x0 and y(t) ≈ (x(t))2 . Consider τ > 0 and xk = x(kτ ). Then xk = exp(−λkτ )x0 ,

y k ≈ (xk )2

and xk+1 = exp(−λτ )xk ,

y k+1 ≈ (xk+1 )2 = exp(−2λτ )(xk )2 .

A linear model in the form k+1 k x x =A y k+1 yk is not suitable for such a linear system. BUT ... F. De Vuyst


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Nonlinear case: example (2) BUT ... including the new observable z k = (xk )2 , we observe that xk+1 = exp(−λτ )xk , z k+1 = exp(−2λτ )z k , z k+1 = exp(−2λτ )z k , that can be written in matrix-vector form   k  k+1   x x exp(−λτ ) 0 0   y k+1  =  0 0 exp(−2λτ ) yk  k+1 0 0 exp(−2λτ ) zk z Conclusion: by variable augmentation, it has been able to transform the linear system into a linear one.

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Another example (continuous in time) Consider the next example (λ > 0) x˙ = −λx2 x0 = x0 > 0. Introducing x2 = x2 , we have x˙ = −λx2 ,

(x˙2 ) = −2λx3 .

Introducing x3 = x3 , we have (x˙2 ) = −2λx3 ,

(x˙3 ) = −3λx4 ,

etc...

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Infinite dimensional system

Actually we get an infinite-dimensional system    0 −λ 0 x    0 0 −2λ x d  2   = 0 dt x3  0 0 .. .. .. .. . . . .

but linear differential

0 0 −3λ .. .

  x ... x2  . . .     . . .  x3  .. .. . .

Rem: x2 = x2 , x3 = x3 ,... are called the features in Data Science or in Artificial Intelligence (AI).

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Another example (continuous in time) (3)

Remark: consider the n-th augmented variable xn = xn with n big enough. We have the differential equation n+1 (x˙n ) = −nλxn+1 = −nλ (xn ) n

then

1 (x˙n ) = −nλ (xn )1+ n

which is a weakly nonlinear differential equation. Then we can expect that DMD gives reasonable results in this case.

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Open problem

Up to today’s knowledge, for a general nonlinear system, there is no algorithm able to automatically select the optimal nonlinear observables (features) to transform (or approximate) the problem into a linear one of minimal size. Any idea ?

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Mixing POD and DMD ...

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Framework

Consider the following partial differential problem ∂t u − κ∆u =

L X

v` (t)ψ` (x)

in Ω × (0, T ],

`=1

dv = Bv(t), t ∈ (0, T ], dt u(., t) = 0 on ∂Ω, u(., t = 0) = u0 (.)

in Ω,

v(0) = v 0 . with v(t) = (v1 (t), v2 (t), ..., vL (t))T .

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Semi-discretization in space Using Finite Element spatial discretization, suppose that the resulting dynamical system reads du = Au + Ψv, t ∈ (0, T ], dt dv = Bv, dt u(0) = u0 , v(0) = v 0 where u(t) = (u1 (t), u2 (t), ..., ud (t))T . Rem: functional correspondence h

u (x, t) =

d X

ui (t) ωi (x).

i=1

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Vector form

d dt Initial condition:

u A Ψ u = v 0 B v

0 u u (t = 0) = . v v0

We have a linear differential problem of high dimension (d + L).

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Step 1: POD on the heat equation Using different snapshots of the FE solution (uh (., ti ))i=1,...,N , one can find a POD basis made of K POD modes ϕ1,h , ϕ2,h , ... ϕK,h . Vector correspondence : Φ1 , Φ2 , ..., ΦK ∈ Rd . POD matrix: Φ = Φ1 , Φ2 , ..., ΦK ∈ Md×K

(ΦT Φ = I).

Reduced-order approximate solution: we look for a solution in the form L X ˜ = u(t) ak (t)Φk = Φa(t). k=1

˜ Rem: we also have a(t) = ΦT u(t). F. De Vuyst


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Reduced-order system

Considering x(t) = (a(t), v(t))T ∈ RK+L (small size), we look for a matrix A such that x˙ = Ax. Initial condition: x(0) = ΦT u0 , v 0

T

.

⇒ The matrix A can be identified from snaphot data matrices X and Y following the DMD methodology.

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Whole algorithm 1. Choose a sampling time τ > 0. 2. Compute K spatial POD modes Φk from snaphosts computed by the fine solver 3. Compute snapshots in the reduced-order space: ak = a(τ k), k = 1, ..., N . 4. Compute also v k = v(τ k), k = 1, ..., N . 5. Consider the variables xk = (ak , v k )T , k = 1, ..., N . 6. Assemble two snapshots matrices X = [x0 , x1 , ..., xN −1 ],

Y = [x1 , x2 , ..., xN ].

7. Compute A according to the DMD formula: A = Y X †. 8. We get a dynamical ROM: ˜ k+1 = A˜ ˜ k+1 = Φ˜ x ak , xk+1 = (ak+1 , v k+1 )T , u ak+1 . F. De Vuyst


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Example Example: consider the ODE-PDE harmonic oscillator + 1d heat equation system with boundary condition coupling: 2 ∂t u − κ∂xx u=0

in (0, 1) × (0, T ],

v˙ = w, w˙ = −v, u(0, t) = 0,

u(1, t) = v(t) t ∈ [0, T ],

u(x, t = 0) = 0

in (0, 1)

v(0) = 0, w(0) = 1. Rem: we have v(t) = sin(t), w(t) = cos(t), but we don’t know analytical solutions for u.

F. De Vuyst


94 / 110

Semi-discretization in space Consider centered finite differences for spatial discretization: ui (t) ≈ u(xi , t), xi = ih, h =

1 . N +1

We have u0 (t) = 0 (left boundary condition), dui (t) κ + 2 (−ui−1 (t) + 2ui (t) − ui+1 (t)) = 0, (t) h and

i = 1, ..., N − 1,

duN (t) κ κ + 2 (−uN −1 (t) + 2uN (t)) = 2 v(t). dt h h

Vector form : let us introduce u(t) = (u1 (t), ..., uN (t))T ∈ RN .

F. De Vuyst


95 / 110

Semi-discretization in space (2) Then by considering the state vector u(t) u(t) x(t) = , a(t) = , a(t) v(t) we have the following linear differential system of size (N + 2): d u(t) −A B u(t) = [0] C a(t) dt a(t) with κ A = 2 tridiag(−1, 2, −1), h and

C=

0 1 −1 0

T 0 0 ... 0 κ/h2 B= . 0 0 ... 0 0 F. De Vuyst


96 / 110

Matlab code

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4

6

8

% edoedp .m % ( Heat e q u a t i o n + h a r m o n i c o s c i l l a t o r ) N = 200; M = 15; dt = 0 . 1 ; D = 0 . 0 5 ; % Thermal c o n d u c t i v i t y h = 1 / (N+1) ; % ...

F. De Vuyst


97 / 110

Matlab code (2)

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% ... w h i l e ( t < T) % t i m e l o o p k = k + 1; t = k∗ dt ; F (N+2 ,1) = s i n ( t ) ; % r i g h t hand s i d e u = A \ ( ( C−B) ∗u + F ) ; % l i n e a r s y s t e m t o s o l v e ( h e a t ) s n a p p d e = [ snappde , u ] ; % g e t s n a p s h o t s o f h e a t s o l s n a p o d e = [ snapode , [ s i n ( t ) ; c o s ( t ) ] ] ; % g e t s n a p s h o t s % of the o s c i l l a t o r p l o t ( x , u , ’ .− ’ ) ; a x i s ( [ 0 1 −1 1 ] ) ; g r i d ; drawnow ; end ; % ...

F. De Vuyst


98 / 110

Matlab code (3)

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% POD + b u i l d d a t a m a t r i c e s f o r DMD [ U , S , V ] = s v d ( snappde , 0 ) ; P s i = U ( : , 1 :M) ; % T a i l l e N∗M a = P s i ’ ∗ s n a p p d e ; % Reduced v a r i a b l e s , r e d u c e d m a t r i x % snaps = [ a ; snapode ] ; % Concatenation X = s n a p s ( : , 1 : end −1) ; % e n t r i e s Y = s n a p s ( : , 2 : end ) ; % o u t p u t s A = Y∗ p i n v (X) ; % DMD f o r m u l a

F. De Vuyst


99 / 110

Matlab code (4)

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% Then p l a y t h e d y n a m i c a l s o l u t i o n d i s p ( ” Reduced o r d e r model . . . ” ) % I n i t i a l state a0 = [ a ( : , 1 ) ; 0 ; 1 ] ; ak = a0 ; clf ; f o r k =1:1200 % t i m e l o o p ( s i m u l a t i o n ) ak = A∗ ak ; % Time a d v a n c e apde = ak ( 1 :M, 1 ) ; % g e t POD c o e f f i c i e n t s u = P s i ∗ apde ; % P l o t h e a t s o l u t i o n (ROM) p l o t ( x , u , ’ .− ’ ) ; a x i s ( [ 0 1 −1 1 ] ) ; g r i d ; drawnow ; end ; % f o r k %

F. De Vuyst

Design, optimization and reduced-order models 100 / 110

Modulus of the eigenvalues of A

>> abs(eig(A))’ ans = Columns 1 through 8 0.9518

0.8204

0.6366

0.4339

0.2339

0.0322

0.1864

0.4074

0.8573

0.9229

0.9921

0.9612

0.9820

1.0000

Columns 9 through 16 0.6025

0.7532

Column 17 1.0000

F. De Vuyst


Case of inhomogeneous systems

Suppose that the dynamics is given by the (linear) inhomogeneous system of equations du + Au = f (t), dt u(0) = u0 At time t = τ , the solution is given by Z τ 0 u(τ ) = exp(−Aτ ) u + exp(−A(τ − s))f (s) ds . | {z } 0 | {z } =A unpredictable !

F. De Vuyst


Case of a constant source term In case of a constant f (independent of time), i.e., du + Au = f , dt the solution at time t = τ is 0

u(τ ) = exp(−Aτ ) u + | {z } =A

Z |0

τ

exp(−A(τ − s))) ds f . {z } b

We get the discrete dynamical system uk+1 = Auk + b. We can try to identify both matrix A and vector b from data.

F. De Vuyst


Extended DMD approach for inhomogeneous equations Consider the extended vector k u k x = ∈ Rd+1 , 1 then the snapshot matrices X = [x0 , ..., xN −1 ],

Y = [u1 , ..., uN ].

Then find AE ∈ Md,d+1 (R) that minimizes kAE X − Y k2F . The solution is again AE = Y X † . NB: the (d + 1)-th column of AE gives b and AE = (A|b). F. De Vuyst


Applications of the inhomogeneous model

Possible applications: linear heat equation with heat source f being constant in time or piecewise constant in time, e.g. f (x, t) = f1 (x) 1tt0 (t). ⇒ Requires the derivation of a DMD model for each constant component of f . NB: This framework can be used for the project on switched control heating system.

F. De Vuyst


HOMEWORK on DMD

F. De Vuyst


Homework Exercise 1. - Consider the two mass-spring coupled system x˙1 = u1 , m1 u˙1 = −k1 x1 + k2 (x2 − x1 ), x˙2 = u2 , m2 u˙2 = −k2 (x2 − x1 ) − k3 x2 , x1 (0) = 0,

x2 (0) = 0,

F. De Vuyst

u1 (0) = 1,

u2 (0) = 0.


Mass-spring system (2) Setting x = (x1 , x2 , u1 , u2 )T , we have    0 0 1 0 0 0  0 1 0  0 0 0    0 0 m1 0  x˙ = −(k1 + k2 ) k2 k2 −(k2 + k3 ) 0 0 0 m2 | {z =A

1 0 0 0

 0 1  x, 0 0 }

and x(0) = (0, 0, 1, 0T ). Rem: the total energy E of the system 1 1 1 1 1 E = k1 x21 + k3 x22 + k2 (x2 − x1 )2 + m1 u21 + m2 u22 2 2 2 2 2 is conserved. F. De Vuyst


Exercise 1. Compute the eigenvalues of A 2. Numerical parameters: m1 = 1, m2 = 1/2, k1 = k2 = 1, k3 = 1/2, x0 = (0, 0, 1, 0)T ; 3. Use ode45() from Matlab to solve the system. Use a sample time τ = 0.05 and generate 1000 sample states xk 4. Apply DMD methodology to compute a DMD matrix A such that xk+1 ≈ Axk . 5. Use the DMD model to simulate the system and compare the results to the solution given by ode45(). 6. Check the value of the total energy E during time iterations using the DMD model. 7. Do the real time visualization of the mass-spring dynamics F. De Vuyst


Solutions given by the DMD model

NB: if the fine solver preserves the total energy, then the DMD model keeps the property (for noiseless data). F. De Vuyst


Design, optimization and reduced-order models

Design, optimization and reduced-order models

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