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A New Algorithm of Proper Generalized Decomposition
for Parametric Symmetric Elliptic Problems
M. Azaïez, F. Ben Belgacem, J. Casado-Díaz, T. Chacón Rebollo, F. Murat
To cite this version:
M. Azaïez, F. Ben Belgacem, J. Casado-Díaz, T. Chacón Rebollo, F. Murat. A New Algorithm of
Proper Generalized Decomposition for Parametric Symmetric Elliptic Problems. SIAM Journal on
Mathematical Analysis, Society for Industrial and Applied Mathematics, 2018, 50 (5), pp.5426-5445.
�hal-01939854�
A new Algorithm of Proper Generalized Decomposition
for parametric symmetric elliptic problems
M. Aza¨ıez
∗
F. Ben Belgacem
†
J. Casado-D´ıaz
‡
T. Chac´on Rebollo
§
F. Murat
¶
July 18, 2018
Abstract
We introduce in this paper a new algorithm of Proper Generalized Decomposition for para-
metric symmetric elliptic partial differential equations. For any given dimension, we prove the
existence of an optimal subspace of at most that dimension which realizes the best approx-
imation –in mean parametric norm associated to the elliptic operator– of the error between
the exact solution and the Galerkin solution calculated on the subspace. This is analogous to
the best approximation property of the Proper Orthogonal Decomposition (POD) subspaces,
excepting that in our case the norm is parameter-depending.
We apply a deflation technique to build a series of approximating solutions on finite-dimensional
optimal subspaces, directly in the on-line step, and we prove that the partial sums converge to
the continuous solution in mean parametric elliptic norm.
We show that the standard PGD for the considered parametric problem is strongly related
to the deflation algorithm introduced in this paper. This opens the possibility of computing the
PGD expansion by directly solving the optimization problems that yield the optimal sub-spaces.
∗
I2M, IPB (UMR CNRS 5295), Universit´e de Bordeaux, 33607 Pessac (France).
†
Sorbonne Universit´es, UTC, EA 2222, Laboratoire de Math´ematiques Appliqu´ees de Compi`egne, 60205 Compi`egne
(France).
‡
Departamento EDAN & IMUS, Universidad de Sevilla, C/Tarfia, s/n, 41012 Sevilla (Spain).
§
I2M, IPB (UMR CNRS 5295), Universit´e de Bordeaux, 33607 Pessac (France), and Departamento EDAN & IMUS,
Universidad de Sevilla, C/Tarfia, s/n, 41012 Sevilla (Spain).
¶
Laboratoire Jacques-Louis Lions, Boˆıte courrier 187, Sorbonne Universit´es, 75252 Paris cedex 05 (France).
1
2
1 Introduction
The Karhunen-Lo`eve’s expansion (KLE) is a widely used tool, that provides a reliable procedure
for a low dimensional representation of spatiotemporal signals (see [13, 23]). It is referred to
as the principal components analysis (PCA) in statistics (see [15, 17, 30]), or called singular
value decomposition (SVD) in linear algebra (see [14]). It is named the proper orthogonal
decomposition (POD) in mechanical computation, where it is also widely used (see [5]). Its
use allows large savings of computational costs, and make affordable the solution of problems
that need a large amount of solutions of parameter-depending Partial Differential Equations
(see [4, 10, 16, 21, 30, 31, 32, 34]).
However the computation of the POD expansion requires to know the function to be ex-
panded, or at least its values at the nodes of a fine enough net. This makes it rather expensive
to solve parametric elliptic Partial Differential Equations (PDEs), as it requires the previous
solution of the PDE for a large enough number of values of the parameter (“snapshots”) (see
[18]), even if these can be located at optimal positions (see [20]). Galerkin-POD strategies are
well suited to solve parabolic problems, where the POD basis is obtained from the previous
solution of the underlying elliptic operator (see [19, 26]).
An alternative approach is the Proper Generalized Decomposition that iteratively computes
a tensorized representation of the parameterized PDE, that separates the parameter and the
independent variables, introduced in [3]. It has been interpreted as a Power type Generalized
Spectral Decomposition (see [27, 28]). It has experienced a fast development, being applied to
the low-dimensional tensorized solution of many applied problems. The mathematical analysis
of the PGD has experienced a relevant development in the last years. The convergence of a
version of the PGD for symmetric elliptic PDEs via minimization of the associated energy has
been proved in [22]. Also, in [11] the convergence of a recursive approximation of the solution
of a linear elliptic PDE is proved, based on the existence of optimal subspaces of rank 1 that
minimize the elliptic norm of the current residual.
The present paper is aimed at the direct determination of a variety of reduced dimension
for the solution of parameterized symmetric elliptic PDEs. We intend to on-line determine an
optimal subspace of given dimension that yields the best approximation in mean (with respect
to the parameter) of the error (in the parametric norm associated to the elliptic operator)
between the exact solution and the Galerkin solution calculated on the subspace. The optimal
POD sub-spaced can no longer be characterized by means of a spectral problem for a compact
self-adjoint operator (the standard POD operator) and thus the spectral theory for compact
self-adjoint operators does no apply. We build recursive approximations on finite-dimensional
optimal subspaces by minimizing the mean parametric error of the current residual, similar
to the one introduced in [11], that we prove to be strongly convergent in the “intrinsic”mean
parametric elliptic norm. For this reason we call “intrinsic”PGD the method introduced.
In addition, we prove that the method introduced is a genuine extension of both POD and
PGD methods, when applied to the solution of parametric elliptic equations. In particular it
is strongly related to the PGD method in the sense that the standard formulation of the PGD
method actually provides the optimality conditions of the minimization problem satisfied by
the optimal 1D sub-spaces. As a consequence of the analysis developed in the paper, the PGD
expansion is strongly convergent to the targeted solution in parametric elliptic norm, whenever
it is implemented in such a way that all modes are optimal. Furthermore, the characterization
of the modes by means of optimization problems opens the door to their computation by using
optimization techniques, in addition to the usual Power Iteration algorithm.
The abstract framework considered includes several kind of problems of practical interest,
3
to which the PGD has been and continues being applied. This is the case of the design analysis
in computational mechanics. For instance in the design of energy efficient devices (HVACs) or
buildings, it is mandatory to address the heat equation with several structural parameters, for
instance the thermal diffusivity or transmittance, and the geometric shape of the device, among
others. Also, the optimal design of heterogeneous materials with linear behavior law fits into
the framework considered, as the parameters model the structural configuration of the various
materials (cf. [29, 33]). Moreover, in practice the structural configuration that optimizes a
certain predefined criterion (e.g. construction costs, benefits, etc.) needs to take into account
the unavoidable uncertainties in the structural performance. This leads to elliptic problems
including modeling of the targeted uncertainty that, when the PDE model is linear, also fits
into the abstract framework considered. In addition classical homogenization problems governed
by linear symmetric elliptic PDEs formally also fit into this general framework, although the
kind of approximation of the solution that is proposed in this work is different than the usual
one, that looks for a limit averaged solution. Here we rather approximate the whole family of
parameter-depending solutions by a function series.
The method, however, does not apply, for instance, to non-symmetric elliptic forms, neither
to non-linear problems.
The present paper focuses on theoretical aspects: We study the existence of the intrinsic
POD, and give a convergence result for the deflation algorithm. We keep the quantitative
analysis of the convergence as well as numerical investigations for future works.
The paper is structured as follows: In Section 2 we state the general problem of finding
optimal subspaces of a given dimension. We prove in Section 3 that there exists a solution for
1D optimal subspaces, characterized as a maximization problem with a non-linear normalization
restriction. We extend this existence result in Section 4 to general dimensions. In Section 5 we
use the results in Sections 3 and 4 to build a deflation algorithm to approximate the solution of
a parametric family of elliptic problems and we show the convergence. Section 6 explains why
the method introduced is a genuine extension of both POD and PGD algorithms, and provides
a theoretical analysis for the latter. Finally in Section 7 we present the main conclusions of the
paper.
4
2 Statement of the problem
Let H be a separable Hilbert space endowed with the scalar product (·, ·). The related norm is
denoted by k · k. We denote by B
s
(H) the space of bilinear, symmetric and continuous forms
in H.
Assume given a measure space (Γ, B, µ), with standard notation, so that µ is σ-finite.
Let a ∈ L
∞
(Γ, B
s
(H); dµ) be such that there exists α > 0 satisfying
α kuk
2
≤ a(u, u; γ), ∀u ∈ H, dµ-a.e. γ ∈ Γ. (1)
For µ−a.e γ ∈ Γ, the bilinear form a(·, ·; γ) determines a norm uniformly equivalent to the norm
k · k. Moreover, a ∈ B
s
(L
2
(Γ, H; dµ)) defined by
a(v, w) =
Z
Γ
a(v(γ), w(γ); γ) dµ(γ), ∀v, w ∈ L
2
(Γ, H; dµ) (2)
defines an inner product in L
2
(Γ, H; dµ) which generates a norm equivalent to the standard one
in L
2
(Γ, H; dµ).
Let be given a data function f ∈ L
2
(Γ, H
0
; dµ). We are interested in the variational problem:
Find u(γ) ∈ H such that a(u(γ), v; γ) = hf(γ), vi, ∀v ∈ H, dµ-a.e. γ ∈ Γ, (3)
where h·, ·i denotes the duality pairing between H
0
and H.
By Riesz representation theorem, problem (3) admits a unique solution for dµ-a.e. γ ∈ Γ.
On the other hand, we claim that ˜u solution of
˜u ∈ L
2
(Γ, H; dµ), ¯a(˜u, ¯v) =
Z
Γ
hf(γ), ¯v(γ)i dµ(γ), ∀ ¯v ∈ L
2
(Γ, H; dµ), (4)
also satisfies (3): Indeed taking ¯v = vχ
B
, with v ∈ H fixed and B ∈ B arbitrary, implies that
there exists a subset N
v
∈ B with µ(N
v
) = 0 such that
a(˜u(γ), v; γ) = hf (γ), vi, ∀ γ ∈ Γ \ N
v
.
The separability of H implies that N
v
can be chosen independent of v, which proves the claim.
By the uniqueness of the solution of (3) this shows that
˜u = u dµ-a.e. γ ∈ Γ. (5)
This proves that u defined by (3) belongs to L
2
(Γ, H; dµ) and provides an equivalent definition
of u,namely, that u is the solution of (4).
Given a closed subspace Z of H, let us denote by u
Z
(γ) the solution of the Galerkin approx-
imation of problem (3) on Z, which is defined as
u
Z
(γ) ∈ Z, a(u
Z
(γ), z; γ) = hf (γ), zi, ∀z ∈ Z, dµ-a.e. γ ∈ Γ, (6)
or equivalently as
u
Z
∈ L
2
(Γ, Z; dµ), ¯a(u
Z
, z) =
Z
Γ
hf(γ), z(γ)i dµ(γ), ∀ z ∈ L
2
(Γ, Z; dµ). (7)