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Journal ArticleDOI

A new algorithm of proper generalized decomposition for parametric symmetric elliptic problems

11 Oct 2018-Siam Journal on Mathematical Analysis (Society for Industrial and Applied Mathematics)-Vol. 50, Iss: 5, pp 5426-5445

TL;DR: It is shown that the standard PGD for the considered parametric problem is strongly related to the deflation algorithm introduced in this paper, and the partial sums converge to the continuous solution in the mean parametric elliptic norm.
Abstract: We introduce a new algorithm of proper generalized decomposition (PGD) for parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optima...

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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 , B, µ), with standard notation, so that µ is σ-finite.
Let a L
, B
s
(H); ) be such that there exists α > 0 satisfying
α kuk
2
a(u, u; γ), u H, -a.e. γ Γ. (1)
For µa.e γ Γ, the bilinear form a(·, ·; γ) determines a norm uniformly equivalent to the norm
k · k. Moreover, a B
s
(L
2
, H; )) defined by
a(v, w) =
Z
Γ
a(v(γ), w(γ); γ) (γ), v, w L
2
, H; ) (2)
defines an inner product in L
2
, H; ) which generates a norm equivalent to the standard one
in L
2
, H; ).
Let be given a data function f L
2
, H
0
; ). We are interested in the variational problem:
Find u(γ) H such that a(u(γ), v; γ) = hf(γ), vi, v H, -a.e. γ Γ, (3)
where , ·i denotes the duality pairing between H
0
and H.
By Riesz representation theorem, problem (3) admits a unique solution for -a.e. γ Γ.
On the other hand, we claim that ˜u solution of
˜u L
2
, H; ), ¯a(˜u, ¯v) =
Z
Γ
hf(γ), ¯v(γ)i (γ), ¯v L
2
, H; ), (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 -a.e. γ Γ. (5)
This proves that u defined by (3) belongs to L
2
, H; ) 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, -a.e. γ Γ, (6)
or equivalently as
u
Z
L
2
, Z; ), ¯a(u
Z
, z) =
Z
Γ
hf(γ), z(γ)i (γ), z L
2
, Z; ). (7)

Citations
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Chady Ghnatios1, Emmanuelle Abisset2, Amine Ammar2, Elías Cueto3  +2 moreInstitutions (4)
TL;DR: The proposed technique achieves a fully separated representation for layered domains with interfaces exhibiting waviness or deviating from planar surfaces, parallel to the coordinate plane, to make possible a simple separated representation, equivalent to others.
Abstract: This work aims at proposing a new procedure for parametric problems whose separated representation has been considered difficult, or whose SVD compression impacted the results in terms of performance and accuracy. The proposed technique achieves a fully separated representation for layered domains with interfaces exhibiting waviness or – more generally – deviating from planar surfaces, parallel to the coordinate plane. This will make possible a simple separated representation, equivalent to others, already analyzed in some of our former works. To prove the potentialities of the proposed approach, two benchmarks will be addressed, one of them involving an efficient space–time separated representation achieved by considering the same rationale.

6 citations


Cites background from "A new algorithm of proper generaliz..."

  • ...For s ∈ [0, 1]  X = x0 + νt, x = sh1(X), For s ∈ [1, 2]  X = x0 + νt, x = (s− 1)hg + h1(X), For s ∈ [2, 3]  X = x0 + νt, x = (s− 2)h2(X) + hg + h1(X), (44)...

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  • ...s ∈ [0, 1]  X = r, x = sh1(X) = sh1(r), s ∈ [1, 2]  X = r, x = (s− 1)l + h1(X) = (s− 1)l + h1(r), s ∈ [2, 3]  X = r, x = (s− 2)h2(X) + l + h1(X) = (s− 2)h2(r) + l + h1(r), (36)...

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  • ...The interested reader can refer to [2, 7, 11, 20, 23, 25, 26] and the references therein for practical details on the computer implementation of separated representations....

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Abstract: In the last decades, numerical simulation has experienced tremendous improvements driven by massive growth of computing power. Exascale computing has been achieved this year and will allow solving ever more complex problems. But such large systems produce colossal amounts of data which leads to its own difficulties. Moreover, many engineering problems such as multiphysics or optimisation and control, require far more power that any computer architecture could achieve within the current scientific computing paradigm. In this chapter, we propose to shift the paradigm in order to break the curse of dimensionality by introducing decomposition to reduced data. We present an extended review of data reduction techniques and intends to bridge between applied mathematics community and the computational mechanics one. The chapter is organized into two parts. In the first one bivariate separation is studied, including discussions on the equivalence of proper orthogonal decomposition (POD, continuous framework) and singular value decomposition (SVD, discrete matrices). Then, in the second part, a wide review of tensor formats and their approximation is proposed. Such work has already been provided in the literature but either on separate papers or into a pure applied mathematics framework. Here, we offer to the data enthusiast scientist a description of Canonical, Tucker, Hierarchical and Tensor train formats including their approximation algorithms. When it is possible, a careful analysis of the link between continuous and discrete methods will be performed.

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Journal ArticleDOI
Abstract: Space separation within the Proper Generalized Decomposition—PGD—rationale allows solving high dimensional problems as a sequence of lower dimensional ones. In our former works, different geometrical transformations were proposed for addressing complex shapes and spatially non-separable domains. Efficient implementation of separated representations needs expressing the domain as a product of characteristic functions involving the different space coordinates. In the case of complex shapes, more sophisticated geometrical transformations are needed to map the complex physical domain into a regular one where computations are performed. This paper aims at proposing a very efficient route for accomplishing such space separation. A NURBS-based geometry representation, usual in computer aided design—CAD—, is retained and combined with a fully separated representation for allying efficiency (ensured by the fully separated representations) and generality (by addressing complex geometries). Some numerical examples are considered to prove the potential of the proposed methodology.

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Journal ArticleDOI
01 Mar 2020
TL;DR: This paper proves the existence of an optimal subspace of at most that dimension which realizes the best approximation—in mean parametric norm associated to the elliptic operator—of the error between the exact solution and the Galerkin solution calculated on the subspace.
Abstract: In a recent paper (Azaiez et al. in SIAM J Math Anal 50(5):5426–5445, https://doi.org/10.1137/17m1137164, 2018) a new algorithm of Proper Generalized Decomposition for parametric symmetric elliptic partial differential equations has been introduced. For any given dimension, this paper proves the existence of an optimal subspace of at most that dimension which realizes the best approximation—in mean parametric norm associated to the elliptic operator—of the error between the exact solution and the Galerkin solution calculated on the subspace. When the dimension is equal one and making use of a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, the method turns to be a classical progressive proper generalized decomposition. In this contribution we prove the linear convergence of the Power Iterate method applied to compute the modes of the PGD expansion, for both symmetric and non-symmetric problems, when the data are small. We also find a spectral convergence ratio of the PGD expansion in the mean parametric norm, for meaningful parametric elliptic problems.

References
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Book
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"A new algorithm of proper generaliz..." refers background in this paper

  • ...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])....

    [...]


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"A new algorithm of proper generaliz..." refers background in this paper

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    [...]


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Abstract: It has often been remarked that turbulence is a subject of great scientific and technological importance, and yet one of the least understood (e.g. McComb 1990). To an outsider this may seem strange, since the basic physical laws of fluid mechanics are well established, an excellent mathematical model is available in the Navier-Stokes equations, and the results of well over a century of increasingly sophisticated experiments are at our disposal. One major difficulty, of course, is that the governing equations are nonlinear and little is known about their solutions at high Reynolds number, even in simple geometries. Even mathematical questions as basic as existence and uniqueness are unsettled in three spatial dimensions (cf Temam 1988). A second problem, more important from the physical viewpoint, is that experiments and the available mathematical evidence all indicate that turbulence involves the interaction of many degrees of freedom over broad ranges of spatial and temporal scales. One of the problems of turbulence is to derive this complex picture from the simple laws of mass and momentum balance enshrined in the NavierStokes equations. It was to this that Ruelle & Takens (1971) contributed with their suggestion that turbulence might be a manifestation in physical

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  • ...It is named proper orthogonal decomposition (POD) in mechanical computation, where it is also widely used (see [5])....

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Journal ArticleDOI
Abstract: In this work error estimates for Galerkin proper orthogonal decomposition (POD) methods for linear and certain non-linear parabolic systems are proved. The resulting error bounds depend on the number of POD basis functions and on the time discretization. Numerical examples are included.

536 citations


"A new algorithm of proper generaliz..." refers background in this paper

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    [...]


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