Chapter 4 Reduced order modeling of parameter dependent, lin-
4.2 Basis emulation and DEIM extension
For each input/parameter ⇠⇠⇠ the snapshot matrix U(⇠⇠⇠) is obtained from the FOM and the basisVr(⇠⇠⇠) is constructed according to section subsection 4.1.2.
To perform an analysis w.r.t. the inputs, this procedure is computation- ally prohibitive. A global basis across the parameter space of interest [133] can be constructed by computing a set of snapshot matrices U(⇠⇠⇠j) for ⇠⇠⇠j 2
X, j = 1, . . . , n. The vi(⇠⇠⇠) are extracted from a global snapshot matrix
The global basis method uses information only from the “truth approx- imation”, i.e., the FOM. The optimality of the POD method, on the other hand, is violated since the snapshots used to derive the basis do not pertain to the parameter value of interest (the particular dynamical system under consideration) during the online phase. Furthermore, the range of validity of the global basis could be limited for complex mappings between the parame- ters and the outputs [134]. Interpolation methods (and the method proposed here) violate the truth approximation in the sense that the snapshots or quan- tities derived therein are not obtained from the original model. In contrast to the global basis, however, these methods attempt to construct more accurate ROMs during the online phase. The main limitation is the accuracy of the interpolation or emulation, which depends on the data available and on the method itself. Moreover, it may not be possible to obtain sharp error bounds using such methods (in cases where the underlying PDE problem is amenable to a rigorous analysis).
Another problem associated with the standard POD-Galerkin approach is that the computational efficiency is compromised when f(·;⇠⇠⇠) 2 Rd is a
strong nonlinearity, since the evaluation offr in Eq. (4.4) has a computational
complexity that depends on d [135]. The DEIM [136] seeks a set of vectors wi(⇠⇠⇠)2Rd, i= 1, . . . , d, such that the subspace span(w1(⇠⇠⇠), . . . ,ws(⇠⇠⇠))⇢Rd
for some s ⌧ d well approximates f(u(t;⇠⇠⇠);⇠⇠⇠) for an arbitrary t. That is, an approximationf(u(t;⇠⇠⇠);⇠⇠⇠)⇡W(⇠⇠⇠)h(t;⇠⇠⇠), where W(⇠⇠⇠) = [w1(⇠⇠⇠). . .ws(⇠⇠⇠)]
and h(t;⇠⇠⇠) 2 Rs. The basis {w
i(⇠⇠⇠)}di=1 is constructed from snapshots of the nonlinearity{fi(⇠⇠⇠)}mi=1, wherefi(⇠⇠⇠) =f(ui(⇠⇠⇠);⇠⇠⇠), from which the matrixF(⇠⇠⇠) =
[f1(⇠⇠⇠). . .fm(⇠⇠⇠)] is formed. A PCA on F(⇠⇠⇠)F(⇠⇠⇠)T or SVD of F(⇠⇠⇠) yields the
{wi(⇠⇠⇠)}di=1, arranged such that the corresponding eigenvalues decay with i. Since the systemf(u(t;⇠⇠⇠);⇠⇠⇠) =W(⇠⇠⇠)h(t;⇠⇠⇠) is overdetermined inh(t;⇠⇠⇠), the DEIM selects s of the d equations to obtain an ‘optimal’ solution. The matrix P = [ep1. . .eps] 2 R
d⇥s is introduced, where e
pi is the standard Eu-
clidean basis vector inRd with nonzero entry located at the p
AssumingPTW(⇠⇠⇠) is nonsingular, one obtain:
fr(a(t;⇠⇠⇠);⇠⇠⇠)⇡Vr(⇠⇠⇠)TW(⇠⇠⇠)h(t;⇠⇠⇠) = Vr(⇠⇠⇠)TW(⇠⇠⇠)(PTW(⇠⇠⇠)) 1PTf(u(t;⇠⇠⇠);⇠⇠⇠)
=Vr(⇠⇠⇠)TW(⇠⇠⇠)(PTW(⇠⇠⇠)) 1f(PTu(t;⇠⇠⇠);⇠⇠⇠)
(4.5) assuming that the functionf(·;⇠⇠⇠) acts pointwise. The indicespi 2{1,2, . . . , d},
i= 1, . . . , sare specified by a greedy algorithm [136] that satisfies the following error bound (for a given s):
||f bf||||(PTW(⇠⇠⇠)) 1|| ||(I W(⇠⇠⇠)W(⇠⇠⇠)T)f|| (4.6) where|| · || is the standard Euclidean norm andbf :=W(⇠⇠⇠)(PTW(⇠⇠⇠)) 1PTf is
the DEIM approximation off. This estimate is valid for a givent (considering f as a function of t) by virtue of the second factor on the r.h.s., which is the error in the best 2-norm approximation off in Range(W(⇠⇠⇠)).
In this chapter, a systematic and rigorous method to approximate the local basis and the nonlinearity is introduced by first approximating the snap- shots{ui(⇠⇠⇠)}mi=1 and{fi(⇠⇠⇠)}mi=1 for an arbitrary input⇠⇠⇠ using Bayesian nonlin- ear regression. These snapshots lie in very high-dimensional spaces and thus a recently developed method is used that exploits manifold learning to yield a computationally feasible Gaussian process (GP) model. Below the compo- nents of this emulation method are described and subsequently explain how it can be used for a POD analysis of parameterized, dynamic problems.
4.2.1
Formulation and solution of the learning problem
For an arbitrary input ⇠⇠⇠, consider the mapping ⌘⌘⌘ : X ! O ⇢ Rmd defined
below:
y=⌘⌘⌘(⇠⇠⇠) = u1(⇠⇠⇠)T, . . . ,um(⇠⇠⇠)T T2Rmd (4.7)
i.e., a vectorial rearrangement of snapshots {ui(⇠⇠⇠)}mi=1 for the given value of
⇠⇠⇠. A similar map yf =⌘⌘⌘f(⇠⇠⇠) for snapshots of the nonlinearity {f
i(⇠⇠⇠)}mi=1 can be defined. The emulation procedure mirrors that described below for the snapshots {ui(⇠⇠⇠)}mi=1.
⌘⌘⌘(⇠⇠⇠j)2O(in a high dimensional space) fordesign points ⇠⇠⇠j 2X,j = 1, . . . , n.
One of the main methods for dealing with such high dimensional outputs is to define approximate outputs in anq dimensional subset Oq ⇢O (q⌧md)
using PCA and independently emulate theqcoefficients of the points inOq for
new values of⇠⇠⇠[43]. Shah and co-workers [137, 138] extended the latter method by replacing PCA with manifold learning methods, making it applicable to a broader class of output spaces O. In this chapter the method of [137, 138] is employed with kernel PCA (kPCA), which is outlined in subsection 2.1.1, together with an approximation of the inverse map. kPCA [102] defines a map
q :O!Fq, whereFqis aq-dimensional feature space. The coordinateszi(y)
of points q(y) inFq define composite maps from the input spaceX toR, i.e.,
zi(⇠⇠⇠) := zi(⌘⌘⌘(⇠⇠⇠)), i = 1, . . . , q. Independent GP priors are placed over these
maps, justified by the properties of kPCA.
The approximation of ⌘⌘⌘ : X ! O given the training points {yj}nj=1 is then substituted for independent approximations of the coefficients zi(⇠⇠⇠),
i= 1, . . . , q, given training data {zi(⇠⇠⇠j) =zi(⌘⌘⌘(⇠⇠⇠j)}nj=1. The value of zi(⇠⇠⇠) for
a new input⇠⇠⇠ is inferred from scalar GP emulation (outlined in section 3.2) as the mean of a posterior distribution. Given{zi(⇠⇠⇠)}qi=1, an approximation of the inverse q1 :Fq ! O yields an approximation of y =⌘⌘⌘(⇠⇠⇠) 2 O, from which
{ui(⇠⇠⇠)}mi=1 can be obtained using definition (4.7). GP emulation is exact at the training points if there are no (spurious) errors in the training data. In the present case, an error is introduced in the pre-image map so that the training snapshots will not be recovered exactly. This error, however, is negligible (section 4.3). I should be noted that the size ofmd is not a limitation for the manifold learning methods employed in this chapter, in which the eigenvalue problems are primarily dependent on the number of training pointsn.
4.2.2
Main Algorithm
Once the snapshots {ui(⇠⇠⇠)}mi=1 (and {fi(⇠⇠⇠)}mi=1 for nonlinear problems) are obtained using the procedure outlined in sections 4.1 and 4.2.2 for a new input
⇠⇠⇠, POD can be performed in the usual manner (with the extended DEIM for nonlinear problems). The entire procedure is outlined in Algorithm 1. It has to be mentioned that kPCA can be replaced with other manifold learning
Algorithm 1kGPE-POD (steps 1a-7a) and kGPE-POD-DEIM (steps 1a-7a and 1b-7b).
1a: Snapshots from FOM:
uj(⇠⇠⇠i)T, i= 1, . . . , n, j= 1, . . . , m 2a: Set: yi ⌘⌘⌘(⇠⇠⇠i)
(u1(⇠⇠⇠i)T, . . . ,um(⇠⇠⇠i)T)T,i= 1, . . . , n
3a: Do kPCA for {yi}ni=1
!{(z1(yi), . . . , zq(yi))T}ni=1 4a: for j 1to q do
{⌘(⇠⇠⇠i) zj(⇠⇠⇠i) zj(yi)}ni=1 Perform scalar GPE: zj(⇠⇠⇠) E[⌘(⇠⇠⇠)]
end for 5a: Inverse map:
⌘⌘⌘(⇠⇠⇠) Pj2Jyj (dj,⇤)/Pi2J (di,⇤) 6a: Snapshots for input ⇠⇠⇠:
(u1(⇠⇠⇠)T, . . . ,um(⇠⇠⇠)T)T ⌘⌘⌘(⇠⇠⇠) 7a: Perform POD with {ui(⇠⇠⇠)}mi=1
1b: Collect nonlinearity snapshots: fj(⇠⇠⇠i), i= 1, . . . , n,j= 1, . . . , m 2b: Set: yfi ⌘⌘⌘f(⇠⇠⇠ i) (f1(⇠⇠⇠i)T, . . . ,fm(⇠⇠⇠i)T)T, i= 1, . . . , n 3b: Do kPCA for {yfi}n i=1 !{(z1f(yfi), . . . , zqf(yif))T}ni=1 4b: for j 1 toq do {⌘f(⇠⇠⇠ i) zfj(⇠⇠⇠i) zfj(yfi)}ni=1
Perform scalar GPE: zfj(⇠⇠⇠)
E[⌘f(⇠⇠⇠)] end for 5b: Inverse map: ⌘⌘⌘f(⇠⇠⇠) P j2Jy f j (dj,⇤)/Pi2J (di,⇤)
6b: Snapshots for nonlinear term:
(f1(⇠⇠⇠)T, . . . ,fm(⇠⇠⇠)T)T ⌘⌘⌘f(⇠⇠⇠)
7b: Perform DEIM on {fi(⇠⇠⇠)}mi=1
methods, e.g., di↵usion maps or Isomap [137, 138]. The terminology ‘kGPE- POD’ is introduced to denote the method of Algorithm 1 without the extended DEIM (i.e, steps 1a-7a alone). Similarly, the terminology ‘kGPE-POD-DEIM’ to denote the method of Algorithm 1 with the extended DEIM (steps 1a-7a and steps 1b-7b together) is used.