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

Griebel, Michael; Harbrecht, Helmut

doi:10.1093/imanum/drs047

Cited by 38 publications

(62 citation statements)

References 38 publications

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“…The convergence rates for the singular values have a similar flavor to univariate approximation theory: the smoother the function, the faster the singular values decay. Some results to this effect can be found in [17,23,41], and we hope to discuss the convergence question further in a separate publication. Numerically, a rank k approximant to f can be computed by sampling it on a n × n Chebyshev tensor grid, taking the matrix of sampled values, and computing its matrix singular value decomposition.…”

Section: Low Rank Function Approximation Given a Continuous Bivariatmentioning

confidence: 81%

An Extension of Chebfun to Two Dimensions

Townsend¹,

Trefethen²

2013

SIAM J. Sci. Comput.

108

135

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Abstract. An object-oriented Matlab system is described that extends the capabilities of Chebfun to smooth functions of two variables defined on rectangles. Functions are approximated to essentially machine precision by using iterative Gaussian elimination with complete pivoting to form "chebfun2" objects representing low rank approximations. Operations such as integration, differentiation, function evaluation, and transforms are particularly efficient. Global optimization, the singular value decomposition, and rootfinding are also extended to chebfun2 objects. Numerical applications are presented.

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Section: Low Rank Function Approximation Given a Continuous Bivariatmentioning

confidence: 81%

An Extension of Chebfun to Two Dimensions

Townsend¹,

Trefethen²

2013

SIAM J. Sci. Comput.

108

135

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“…Especially, we introduce here the related function spaces which are used in the rest of this article. For further details on the KarhunenLoève expansion in general and also on computational aspects, we refer to [10,11,17,29]. …”

Section: Reformulation On the Reference Domainmentioning

confidence: 99%

“…Then, there holds the following Lemma 2 The operators S and S given by (11) and (12), respectively, are bounded with Hilbert-Schmidt norms…”

Section: Reformulation On the Reference Domainmentioning

confidence: 99%

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Analysis of the domain mapping method for elliptic diffusion problems on random domains

2016

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In this article, we provide a rigorous analysis of the solution to elliptic diffusion problems on random domains. In particular, based on the decay of the Karhunen-Loève expansion of the domain perturbation field, we establish decay rates for the derivatives of the random solution that are independent of the stochastic dimension. For the implementation of a related approximation scheme, like quasi-Monte Carlo quadrature, stochastic collocation, etc., we propose parametric finite elements to compute the solution of the diffusion problem on each individual realization of the domain generated by the perturbation field. This simplifies the implementation and yields a non-intrusive approach. Having this machinery at hand, we can easily transfer it to stochastic interface problems. The theoretical findings are complemented by numerical examples for both, stochastic interface problems and boundary value problems on random domains.

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“…Results on the decay of the eigenvalues have been established for periodic functions already in . Nevertheless, because we do not want to restrict ourselves to this situation, we refer here to the more general results in , Theorem 3.3,Theorem 3.5]. Theorem Let

a \in L_{double-struckP}^{2} ((), normalΩ; H^{p} (D))

with p > d /2.…”

Section: Approximation Of Random Fieldsmentioning

confidence: 99%

Efficient approximation of random fields for numerical applications

Harbrecht

Peters

Siebenmorgen

2015

Numerical Linear Algebra App

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We consider the rapid computation of separable expansions for the approximation of random fields. We compare approaches based on techniques from the approximation of non-local operators on the one hand and based on the pivoted Cholesky decomposition on the other hand. We provide an a-posteriori error estimate for the pivoted Cholesky decomposition in terms of the trace. Numerical examples validate and quantify the considered methods. The Karhunen-Loève expansionLet (Ω, F, P) be a separable, complete probability space and let D ⊂ R d be a Lipschitz domain. In the sequel, we consider random fields a ∈ L 2 P Ω; L 2 (D) . For a given random field a, we denote the related centered field according to a 0 (ω, x) := a(ω, x) − E[a](x).Moreover, we define the Hilbert-Schmidt operator associated to a 0 asOne can show that the covariance operator C := SS is of trace-class, i.e. Tr C :One readily verifiesAdditionally, C is symmetric and positive semi-definite. Therefore, by the spectral theorem, C exhibits a representation of the formwhere {(λ i , φ i )} i∈I denote the corresponding eigen-pairs.is called Karhunen-Loève expansion with respect to a. Herein, the random variables {X m } m∈I are given according to Finite element approximationFor the approximation of spatial functions in L 2 (D), we employ (parametric) finite elements of order s. To that end, we introduce a family of quasi-uniform triangulations T h for D with mesh width h and define the spacesThen, given a function v ∈ H t (D) with 0 ≤ t ≤ s, it holds due to the Bramble-Hilbert lemma the approximation estimateuniformly in h.In the sequel, we assume that the random field a exhibits additional spatial regularity, i.e. a ∈ L 2 P (Ω; H p (D) for some p > 0. Then, we may consider the spatial approximationIn terms of the trace, we obtain the following approximation result in V s h . Theorem 2.1. Let a ∈ L 2 P Ω; H p (D) . Then, the spatially approximated, centered random field Q h a 0 satisfies the error estimateBy the application of the theorem and the approximation estimate (1) it is straightforward to show the following Corollary 2.2. The trace error satisfiesand its covariance according to C h,M := P h C h P h . We arrive at the subsequent approximation result., then the random field a h,M given by (2) satisfies the error estimateThe theorem indicates that, after fixing the ansatz space V s h , the approximation error of the stochastic field is controllable in terms of the discretized operators C h and C h,M . The optimal choice of P h in terms of minimizing the trace error is the orthogonal projection onto the dominant invariant subspace of C h , i.e. U M,h := span{φ 1,h , . . . , φ M,h } corresponding to the M dominant eigenvalues of C h . The related Karhunen-Loève expansion then readswhere the random variables are given according toIn this setting, the discretization of the stochastic field implies a change of the stochastic model induced by (5). The pivoted Cholesky decompositionBased on the observation in Theorem 2.3 and the subsequent discussion, we ...

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

Cited by 38 publications

References 38 publications

An Extension of Chebfun to Two Dimensions

An Extension of Chebfun to Two Dimensions

Analysis of the domain mapping method for elliptic diffusion problems on random domains

Efficient approximation of random fields for numerical applications

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