Proper Orthogonal Decomposition (POD)-driven procedure

Decom-position is new. We refer to[8]for a different presentation, based on singular value decomposition (SVD).

First mode

Suppose, for a soft start, that we want to find a reduced basis containing only one item: {ζ₁}. Without loss of generality we can always suppose that||ζ₁||= 1. It seems reasonable to make the choice which makes the least summed squares error when projecting orthogonally the solutionsu_µ (for all µ ∈ P) onto the reduced space Span{ζ₁}. Said with formulas, we seekζ₁satisfying:

ζ₁= arginf

The functionζ₁ will be called ourfirst POD mode.

Unfortunately, the integral appearing in the objective function below is, in gen-eral, analytically untractable. We can approximate it using a finite sample of parametersΞ⊂ P; the integral then gets replaced by a finite sum overΞ:

ζ₁= arginf

The set of solutions{u_µ,µ∈Ξ}is called thesnapshotsensemble.

The objective function can then be rewritten:

X The objective function in (13) is a quadratic form in p, that has to be maxi-mized under a "unity norm" constraint. We know this can be reduced to an eigenproblem.

More specifically, let us introduce a basis {φ₁, . . . ,φ_N} ofX. As we stressed

in the introduction,N is large. We enumerate the snapshots parameters: Ξ = {µ₁, . . . ,µ_N

snap}.

Each snapshotu(i) =u_µi (i=1, . . . ,N_snap) has an expansion in the basis of X:

u(i) = XN

j=1

u(i)_jφ_j

and we can write downM, the matrix of theu(i)_j coefficients:

M=









u(1)₁ u(2)₁ . . . u(N_snap)₁ u(1)₂ u(2)₂ . . . u(N_snap)₂

... ... . . . ... u(1)_N u(2)_N . . . u(N_snap)_N









Denote by Ω, the (symmetric, positive definite) matrix of the scalar product

<,>inX.

We can write down our objective function as:

X

µ∈Ξ

<u_µ,p>²=p^TΩ^TM M^TΩp

The condition of optimality for (13) is then given by the Lagrange multipliers theorem: there has to existλ∈R, so that:

Ω^TM M^TΩζ₁=λΩζ₁ (14)

andλis the optimal value of the problem (as can be seen by left-multiplying (14) byζ₁^T, and using||ζ₁||²=1) – which should be made as large as possible.

Now left-multiply (14) byΩ^−T = Ω⁻¹:

M M^TΩζ₁=λζ₁ (15)

so that ζ₁ is the unit eigenvector associated with the greatest eigenvalue of M M^TΩ.

So we have reduced our problem as an eigenproblem of dimensionN, because M M^TΩis of sizeN ×N. SinceN can be large, this could turn out to be highly unpractical. Fortunately, we are to see that we can solve the same problem using eigenproblem of sizeN_snap×N_snap.

The key is to look forζ₁as a linear combinationof the snapshots, instead of the

"generic" basis items{φ_i}.

We write:

ζ₁=

NXsnap

i=1

z_1iu(i) =M z₁

wherez₁= (z₁1, . . . ,z_1,Nsnap)^T. Settingζ₁=M z₁into (15) yields:

M M^TΩM z₁=λM z₁ and a sufficient condition for this to hold is:

M^TΩM z₁=λz₁

so that our new unknownz₁∈R^Nsnap is an eigenvector ofM^TΩM (which of size N_snap×N_snap) associated with its greatest eigenvalue.

The absence of optimality loss when looking for ζ₁ in Range(M) (in other words,ζ₁ = M z₁ with z₁ as defined below satisfies (13)) follows clearly from these facts:

1. if z is a nonzero eigenvector of M^TΩM (associated with λ 6= 0), then ζ=M zis a nonzero eigenvector ofM M^TΩassociated withλ;

2. if ζ is a nonzero eigenvector of M M^TΩ (associated with λ 6= 0), then z=M^TΩζis a nonzero eigenvector ofM^TΩM associated withλ;

3. M^TΩM and M M^TΩ have the same nonzero eigenvalues, each with the same multiplicity in one and the other.

Proof: To see 1., first note that if we hadM z=0, thenM^TΩM z=0 and soz could not be an eigenvector associated with a nonzero eigenvalue of M^TΩM; thus M z6=0. Now M M^TΩ(M z) = M(M^TΩM z) =Mλz =λ(M z) so that M z is an eigenvector of M M^TΩ associated withλ. Point 2. is 1. mutatis mutan-dis. Point 3.: it is clear from 1. and 2. that the two matrices have the same nonzero eigenvalues. To get the statement about the multiplicities, suppose that Span{z₁, . . . ,z_k} is the eigenspace of M^TΩM associated with eigenvalue λ. Since M^TΩM is self-adjoint with respect to the Euclidean inner product (i.e., is symmetric), we can suppose that{z₁, . . . ,z_k}is orthogonal (for the Eu-clidean inner product). Now, from 2., Span{M z₁, . . . ,M z_k}is contained in the eigenspace ofM M^TΩassociated withλ. Moreover, for everyi6= j:

<M z_i,M z_j>=z^T_j M^TΩM z_i=λz^T_j z_i=0

so that{M z₁, . . . ,M z_k}is<,>-orthogonal, and therefore linearly independent, and soλhas multiplicitykinM M^TΩ. ƒ

Second mode

Now let’s say we are looking for a set of two functions as our reduced basis:

{ζ₁,ζ₂}. As earlier, we want(ζ₁,ζ₂)to minimize:

In other words, we aim to minimize the overall error (over our sample of pa-rameters Ξ) made when orthogonally projecting u_µ onto Span{ζ₁,ζ₂}, with {ζ₁,ζ₂}an orthonormal family.

Mimicking what we have done earlier, we rewrite our objective function as:

X

The objective function, apart from the terms that are independent of ζ₁ and ζ₂, is a sum of a function dependent on ζ₁ only and a function dependent onζ₂ only. Thus the joint optimization on(ζ₁,ζ₂) is equivalent to successive optimization onζ₁, then onζ₂. Optimization onζ₁ has been done previously.

Optimization inζ₂is the following:

ζ₂= argmax

p∈X,||p||=1,<p,ζ₁>=0

X

µ∈Ξ

<u_µ,p>²

The objective function has exactly the same matrix expression as the one of the problem defining the first POD mode:

X

µ∈Ξ

<u_µ,p>²=p^TΩ^TM M^TΩp

The Lagrange multiplier theorem now gives that there existλ₂∈Rso that:

Ω^TM M^TΩζ₂=λ₂Ωζ₂ (16) As before, left multiplying byζ₂^T gives thatλ₂ is the optimal value of the prob-lem, which should be made as large as possible.

Left-multiplying (16) byΩ^−T we get that (16) is equivalent to:

M M^TΩζ₂=λζ₂ soλ₂ is an eigenvalue ofM M^TΩ.

Now we have, for every x,y∈X:

<M M^TΩx,y> = y^TΩM M^TΩx

= y^T€

M M^TΩŠT

Ωx

= <x,M M^TΩy >

so that M M^TΩ is self-adjoint with respect to <,>. So we can find a unit eigenvector ζ₂, orthogonal to ζ₁ associated with the second greatest eigen-value (note that this eigeneigen-value is equal toλ₁ ifλ₁has multiplicity>1).

As earlier, we can use the fact thatM M^TΩandM^TΩM have the same nonzero eigenvalues, to getζ₂= M z₂ wherez₂ is an eigenvector of the smaller matrix M^TΩM. The fact that multiplicities of nonzero eigenvalues in M M^TΩ and M^TΩM are the same ensures that this procedure is correct even ifλ₁=λ₂. Following modes, and summary for POD

We can readily repeat what we have done forζ₂ to prove:

Let Ξ = {µ₁, . . . ,µ_Nsnap} be a finite part of P. Let M be the matrix whose columns are the coefficients ofu_µ1, . . . ,u_µ

Nsnap with respect to some basis ofX. LetΩbe the matrix of the inner product inX.

Then, the family(ζ₁, . . . ,ζ_N)of vectors inX that minimizes:

X

µ∈Ξ

u(µ)− <u_µ,ζ₁> ζ₁+<u_µ,ζ₂> ζ₂+. . .+<u_µ,ζ_N> ζ_N

2

under the constraints< ζ_i,ζ_j>=

( 1 ifi= j 0 else.

is defined by: 









ζ₁ = 1

||M z₁||M z₁ ...

ζ_N = 1

||M z_n||M z_n

wherez_k (k=1, . . . ,N) is a nonzero eigenvector, associated withk-th largest eigenvalue (counting repeated nonsimple eigenvalues) ofM^TΩM.

The statement above holds for everyN∈N^∗so that theN-th largest eigenvalue ofM^TΩM is nonzero.

The functions(ζ₁, . . . ,ζ_N)defined above are called the first N POD modes as-sociated with the snapshot ensembleΞ.

A note about the (offline) complexity of the POD: for aN-sized reduced basis, it requires computation ofN_snap "expensive" solutionsu_µ and extraction of the N leading eigenelements of anN_snap×N_snapsymmetric matrix.

5 Case of a scalar output

One is often not interested in the full solution u_µ, but rather in an "output"

calculated fromu_µ:

s_µ=η(u_µ) whereη:X→Ris a (µ-independent) functional.

We present here some approaches for computation of a surrogate outputes_µ fromeu_µ.

5.1 Primal approach

A trivial approach to the question is to computeeu_µ, and then, online:

es_µ=η(ue_µ) (17) Done naively, this procedure might be dimX-dependent, since evaluation ofη may require access to every component ofeu_µ in the originalX space.

A way of overcoming this problem, ifηis a linear functional, is to write that:

e s_µ=η

XN n=1

c_n(µ)ζ_n

!

= XN n=1

c_n(µ)η(ζ_n)

and the offline/online procedure becomes:

• offline: compute and storeη(ζ_n),n=1, . . . ,N

• online: findc_n(µ)(n=1, . . . ,N) satisfying (2) using procedure described in2.2and compute

e s_µ=

XN n=1

c_n(µ)η(ζ_n) (18)

And now the online phase is still dimX-independent.

For a quadratic output functional, one proceeds the same way using:

e s_µ=

XN n=1

XN n^′=1

c_n(µ)c_n^′(µ)η_b(ζ_n,ζ_n^′)

whereη_bis the symmetric bilinear form associated withη.

This time, one has to compute and store (offline)η_b(ζ_n,ζ_n^′), forn,n^′=1, . . . ,N.

The same procedure can easily be generalized for polynomial outputs.

Error bound for the output

A straightforward majoration for |s_µ−es_µ|can be found under a Lipschitz hy-pothesis forη: let L>0 be so that, whateverv,w∈X:

|η(v)−η(w)| ≤L||v−w||

(for example, ifηis a continuous linear functional, Lcan be chosen as the dual norm ofη).

Then we have:

|s_µ−es_µ| ≤ Lǫ_µ (19)

Dans le document Reduced-basis solutions of affinely-parametrized linear partial differential equations (Page 17-23)