Computer Methods in Applied Mechanics and Engineering
Fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains☆
Highlights
► We propose a tensor-based method for the solution of PDEs defined on uncertain parameterized domains. ► We use a fictitious domain approach to obtain a formulation in a tensor product space. ► We use a PGD algorithm for the construction of a tensor approximation of the solution. ► We introduce a constrained SVD of the parameterized indicator function which preserves positivity. ► We analyze errors due to fictitious domain formulations and approximations of indicator functions.
Introduction
Uncertainty quantification has become a critical issue in computational and prediction science. Uncertainty may reflect inherent variabilities in physical systems which have to be incorporated in mathematical models, or some level of ignorance that yields an imprecise or incomplete characterization of a model. When adopting a probabilistic modeling of uncertainties, their impact on a model output may be classically evaluated by means of classical sampling techniques if one is interested in probabilistic or statistical quantities of interest. In the last two decades, a growing attention has been given to an alternative functional view of uncertainty quantification, where uncertain (random) quantities are seen as functionals of parameters characterizing the input uncertainties. This functional view, combined with approximation theory and numerical analysis, has led to the development of a family of numerical methods, called spectral stochastic methods, for the propagation of uncertainties through a model, yielding a complete characterization of uncertain model outputs (see recent reviews [19], [30], [14], [13]). The introduction of model uncertainty may also be required for different purposes such as model design, identification or optimization, where one is interested in the quantification of model outputs for a certain range of input parameters. In this context of parametric analyses, spectral methods provide an explicit representation of the output with respect to input parameters, thus allowing a posteriori parametric analyses. Therefore, they constitute efficient alternatives to traditional methods that require many evaluations of the initial model for particular values of input parameters (e.g. corresponding to an experimental design, iterates of an optimization procedure…).
Spectral methods for uncertainty quantification have been successfully applied to many problems in science and engineering. In particular, many works have considered the uncertainty propagation through models involving partial differential equations (PDEs) with uncertain operators and source terms (see e.g. [9], [4], [15], [28]). A few works have been recently devoted to numerical methods for PDEs defined on uncertain domains [31], [29], [5], [23], [2], [16], [10], [22]. The explicit characterization of output quantities with respect to input shape parameters is of great interest in various analyses: impact of random perturbations of a shape, shape optimization in model design, inverse analysis in non-destructive testing (location of a defect)… The above mentioned works start with a reformulation of the problem on a fixed deterministic domain. In [31], [29], [16], a random mapping maps the random domain into a deterministic domain, thus transforming a PDE defined on an uncertain domain into a PDE defined on a fixed domain with uncertain operator and right-hand-side depending on the mapping and its derivatives. In [5], [23], fictitious domain methods are introduced and consist in embedding the uncertain domain into a fixed domain, the geometry being characterized with a level-set technique or a mapping technique.
In this paper, we present an efficient method for the numerical solution of PDEs defined on uncertain parameterized domains Ω(ξ), with ξ ∈ Ξ being parameters (eventually random), which combines a fictitious domain approach and a tensor-based method, namely the Proper Generalized Decomposition, for the construction of optimal separated representations of the solution. For the proposed tensor-based method to be computationally tractable, additional technical ingredients are introduced in order to recast the problem in a suitable tensor format. These ingredients consist of specific treatments of the random geometry, more precisely of the indicator function representing the random domain. The impact of these approximations of the geometry are carefully analyzed.
As a model example, we consider a simple diffusion equation −Δu = f defined on Ω(ξ). The paper is limited to the case of Neumann conditions on uncertain parts of the boundary.
A fictitious domain approach is first adopted for the reformulation of the problem on a fixed domain Ω□, which introduces a prolongation of the solution u. It yields a weak formulation of the parametric (stochastic) problem in a tensor product space (product of space functions and parametric functions),1 with and .
Model reduction techniques based on the construction of separated representations are receiving a growing interest in scientific computing. A family of methods, recently called Proper Generalized Decomposition (PGD) methods have been introduced for the a priori construction of separated representations of the solution of problems defined in tensor product spaces [1], [12], [20], [6], [8]. PGD methods can be interpreted as generalizations of Proper Orthogonal Decomposition (or Singular Value Decomposition, or Karhunen–Loeve Decomposition) for the a priori construction of a separated representation of the solution. In the context of uncertainty propagation, this method has been initially introduced as a generalization of spectral decompositions [17] for the a priori construction of a separated representation of the solution of stochastic PDEs. Here, the solution is searched under the formwith and . Several definitions of separated representations have been proposed. In this paper, we introduce a particular progressive definition of the PGD, based on successive best approximation problems for the progressive definition of the couples of functions , which are constructed with an alternated minimization procedure.2 This construction only requires the solution of independent subproblems defined on (parametric algebraic equations) and subproblems defined on (non parametric PDEs). Let us note that PGD methods have also been extended to uncertainty quantification problems for high-dimensional parametric models [7], [21] (i.e. involving a large number of parameters ) by using separated variables representations of the solution . These methods allow the a priori construction of a separated representation of a solution defined on a very high-dimensional parametric space. The method can handle problems with such a dimension that their solution is unfeasible with classical spectral stochastic approximation techniques. For the sake of simplicity, in this article, we restrict the presentation to the construction of a separation of type (1), without any additional separation of parameters ξ = (ξ1, … , ξr). However, following [21], the methodology could be easily extended to the case of multidimensional separations.
The outline of the paper is as follows. In Section 2, we introduce a fictitious domain formulation of partial differential equations defined on uncertain parameterized domains, resulting in a weak formulation defined in a tensor product space, where variational forms involve the indicator function of the parameterized domain. In Section 3, we introduce and illustrate the Proper Generalized Decomposition (PGD) method for the construction of a separated representation (1) of the solution. In order to obtain an efficient method outperforming traditional solution techniques, variational forms must also admit accurate low rank separated representations. For this purpose, in Section 4, we introduce separated representations of the indicator function in order to obtain accurate separated representations of variational forms. Smoothing of indicator functions is introduced in order to improve the convergence rate of their separated representations. Moreover, a method is proposed for the construction of a constrained tensor product approximation which preserves positivity and therefore ensures well-posedness of problems associated with approximate indicator functions. Finally, in Section 5, a second example illustrates the overall methodology.
Section snippets
Partial differential equation defined on uncertain parameterized domain
Let be a finite dimensional probability space representing the uncertainties on a geometry,3 where is the set of elementary events.
Proper Generalized Decomposition
We consider the solution of problem (Pη) with the Proper Generalized Decomposition (PGD) method. The idea is to find an approximation of the solution of (Pη) under the form8where um is called a rank-m separated representation of uη. This can be interpreted as a simultaneous construction of reduced bases of spatial functions and stochastic functions , which are optimal (in some sense to be
Separated representations of the indicator functions
When directly applied to problem (Pη), the PGD method is not computationally tractable since bilinear form Aη and linear form L do not have a “separated form”. In practice, when the approximation spaces and are introduced, the solutions of problems and require a fine integration of the bilinear and linear forms, which limits the use of the method to relatively coarse approximation spaces at both spatial and stochastic levels. This point was also the limiting point of the
Description of the problem
The overall methodology is now conducted on the Poisson problem (2) with f = 1 and a random domain Ω(ξ) delineated with two vertical lines and two random sinusondal curves (see Fig. 21). The domain is characterized bywithand where ξ1 and ξ2 are two independent uniform random variables on Ξ1 = (0, 1) and Ξ2 = (0, 1), respectively. The homogeneous Dirichlet boundary ΓD is composed
Conclusion
A numerical methodology has been proposed for the solution of partial differential equations defined on uncertain parameterized domains. This methodology first relies on a fictitious domain approach, yielding a formulation of the problem in a tensor product space. This tensor product structure is then exploited by applying the Proper Generalized Decomposition method, allowing the a priori construction of a tensor product approximation of the solution. This PGD method can be seen as a model
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This work is supported by the French National Research Agency (Grant ANR-2010-COSI-006-01).