Sparse Finite Elements for Stochastic Elliptic Problems ? Higher Order Moments

Schwab, Christoph; Todor, Radu Alexandru

doi:10.1007/s00607-003-0024-4

Cited by 70 publications

(58 citation statements)

References 10 publications

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“…Higher-order moments then involve larger tensor products (Schwab & Todor, 2003b). This approach extends to stochastic diffusion functions and more general PDEs with stochastic coefficient functions as well as to stochastic domains (Harbrecht et al, 2008a;Harbrecht, 2010).…”

Section: Introductionmentioning

confidence: 99%

Approximation of bi-variate functions: singular value decomposition versus sparse grids

Griebel

Harbrecht

2013

IMA Journal of Numerical Analysis

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We compare the cost complexities of two approximation schemes for functions f ∈ H p (Ω 1 × Ω 2 ) which live on the product domain Ω 1 × Ω 2 of sufficiently smooth domains Ω 1 ⊂ R n 1 and Ω 2 ⊂ R n 2 , namely the singular value/Karhunen-Lòeve decomposition and the sparse grid representation. Here, we assume that suitable finite element methods with associated fixed order r of accuracy are given on the domains Ω 1 and Ω 2 . Then, the sparse grid approximation essentially needs only O(ε −q ), with q = max{n 1 , n 2 }/r, unknowns to reach a prescribed accuracy ε, provided that the smoothness of f satisfies p r((n 1 + n 2 )/max{n 1 , n 2 }), which is an almost optimal rate. The singular value decomposition produces this rate only if f is analytical, since otherwise the decay of the singular values is not fast enough. If p < r((n 1 + n 2 )/max{n 1 , n 2 }), then the sparse grid approach gives essentially the rate O(ε −q ) with q = (n 1 + n 2 )/p, while, for the singular value decomposition, we can only prove the rate O(ε −q ) with q = (2 min{r, p} min{n 1 , n 2 } + 2p max{n 1 , n 2 })/(2p − min{n 1 , n 2 }) min{r, p}. We derive the resulting complexities, compare the two approaches and present numerical results which demonstrate that these rates are also achieved in numerical practice.

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Section: Introductionmentioning

confidence: 99%

Approximation of bi-variate functions: singular value decomposition versus sparse grids

Griebel

Harbrecht

2013

IMA Journal of Numerical Analysis

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“…The following lemma, proven in [35,39], tells us that the approximation power in the sparse tensor product space is nearly as good as in the full tensor product space, provided that the given function has some extra regularity in terms of bounded mixed derivatives. Lemma 1.…”

Section: Sparse Second Moment Analysismentioning

confidence: 99%

“…This linearization technique has already been applied to random diffusion coefficients or even to elliptic equations on random domains in [18,22,23]. In difference to these previous works, we do not explicitly use wavelets [23,34,35] or multilevel frames [18,22] for the discretization in a sparse tensor product space. Instead, we define the complement spaces which enter the sparse tensor product construction by Galerkin projections.…”

Section: Introductionmentioning

confidence: 99%

Combination Technique Based Second Moment Analysis for Elliptic PDEs on Random Domains

Harbrecht

Peters

2016

Lecture Notes in Computational Science and Engineering

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In this article, we propose the sparse grid combination technique for the second moment analysis of elliptic partial differential equations on random domains. By employing shape sensitivity analysis, we linearize the influence of the random domain perturbation on the solution. We derive deterministic partial differential equations to approximate the random solution's mean and its covariance with leading order in the amplitude of the random domain perturbation. The partial differential equation for the covariance is a tensor product Dirichlet problem which can efficiently be determined by Galerkin's method in the sparse tensor product space. We show that this Galerkin approximation coincides with the solution derived from the combination technique provided that the detail spaces in the related multiscale hierarchy are constructed with respect to Galerkin projections. This means that the combination technique does not impose an additional error in our construction. Numerical experiments quantify and qualify the proposed method.

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“…Here one is facing the difficulty that for Λ corresponding to a sparse grid, A Λ ,Λ itself does not have Kronecker product structure. The basis of the scheme introduced by Schwab and Todor in [43] for applying A Λ ,Λ to a vector are partitions…”

Section: Matrix-vector Products For Hyperbolic Wavelet Discretizationsmentioning

confidence: 99%

Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation

Bachmayr

2012

ESAIM: M2AN

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Abstract. In the framework of an explicitly correlated formulation of the electronic Schrödinger equation known as the transcorrelated method, this work addresses some fundamental issues concerning the feasibility of eigenfunction approximation by hyperbolic wavelet bases. Focusing on the two-electron case, the integrability of mixed weak derivatives of eigenfunctions of the modified problem and the improvement compared to the standard formulation are discussed. Elements of a discretization of the eigenvalue problem based on orthogonal wavelets are described, and possible choices of tensor product bases are compared especially from an algorithmic point of view. The use of separable approximations of potential terms for applying operators efficiently is studied in detail, and estimates for the error due to this further approximation are given.Mathematics Subject Classification. 35B65, 35J10, 65T60, 81Q05.

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Sparse Finite Elements for Stochastic Elliptic Problems ? Higher Order Moments

Cited by 70 publications

References 10 publications

Approximation of bi-variate functions: singular value decomposition versus sparse grids

Approximation of bi-variate functions: singular value decomposition versus sparse grids

Combination Technique Based Second Moment Analysis for Elliptic PDEs on Random Domains

Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation

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