HT-AWGM: a hierarchical Tucker–adaptive wavelet Galerkin method for high-dimensional elliptic problems

Ali, Mazen; Urban, Karsten

doi:10.1007/s10444-020-09797-9

Cited by 7 publications

(16 citation statements)

References 31 publications

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“…A similar approach was performed in [3] and [2]. In [3] the authors controlled the error only in L 2 but generally observed convergence in H 1 as well.…”

Section: Exponential Sumsmentioning

confidence: 94%

Singular Value Decomposition in Sobolev Spaces: Part II

Ali,

Nouy

2019

Preprint

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Under certain conditions, an element of a tensor product space can be identified with a compact operator and the singular value decomposition (SVD) applies to the latter. These conditions are not fulfilled in Sobolev spaces. In the previous part of this work (part I), we introduced some preliminary notions in the theory of tensor product spaces. We analyzed low-rank approximations in H 1 and the error of the SVD performed in the ambient L 2 space.In this work (part II), we continue by considering variants of the SVD in norms stronger than the L 2 -norm. Overall and, perhaps surprisingly, this leads to a more difficult control of the H 1 -error. We briefly consider an isometric embedding of H 1 that allows direct application of the SVD to H 1 -functions. Finally, we provide a few numerical examples that support our theoretical findings.

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“…A similar approach was performed in [3] and [2]. In [3] the authors controlled the error only in L 2 but generally observed convergence in H 1 as well.…”

Section: Exponential Sumsmentioning

confidence: 94%

Singular Value Decomposition in Sobolev Spaces: Part II

Ali,

Nouy

2019

Preprint

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“…is the minimal subspace of u as a function in H (0,1) (resp. H (1,0) ) and U j IV (u) is the minimal subspace of u as a function in H 1 mix (Ω). Since we want to consider precisely u ∈ H 1 (Ω), we consider u ∈ H (1,0) and choose the variant U 1 II (u) for the left minimal subspace, and u ∈ H (0,1) with the variant U 2 II (u) for the right minimal subspace.…”

Section: Minimal Subspaces and Tensormentioning

confidence: 99%

“…does not make sense in H 1 (Ω) in general, only in H (1,0) . Similarly, we can interpret u ∈ H 1 (Ω) as a Hilbert Schmidt operator u :…”

Section: Minimal Subspaces and Tensormentioning

confidence: 99%

“…k be the SVD of u interpreted as an element of H (0,0) , H (1,0) and H (0,1) respectively. Then, we have for all r ≥ 0…”

Section: Minimal Subspaces and Tensormentioning

confidence: 99%

“…There we require the additional assumption (3.15). (1,0) /H (0,1) projections. Given the singular functions {ψ 1 k } k∈N and {φ 1 k } k∈N associated with H (1,0) and H (0,1) SVDs of u respectively, we consider the finite dimensional subspaces…”

Section: Minimal Subspaces and Tensormentioning

confidence: 99%

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Singular Value Decomposition in Sobolev Spaces: Part I

Ali,

Nouy

2018

Preprint

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A well known result from functional analysis states that any compact operator between Hilbert spaces admits a singular value decomposition (SVD). This decomposition is a powerful tool that is the workhorse of many methods both in mathematics and applied fields. A prominent application in recent years is the approximation of high-dimensional functions in a low-rank format. This is based on the fact that, under certain conditions, a tensor can be identified with a compact operator and SVD applies to the latter. One key assumption for this application is that the tensor product norm is not weaker than the injective norm. This assumption is not fulfilled in Sobolev spaces, which are widely used in the theory and numerics of partial differential equations. The aim of this work is the analysis of the SVD in Sobolev spaces. We show that many properties are preserved in Sobolev spaces. Moreover, to an extent, SVD can still provide "good" low-rank approximations for Sobolev functions. We present 3 variants of SVD that can be applied to a function in a Sobolev space. First, we can apply the SVD in the ambient L 2 space. Second, the Sobolev space is an intersection of spaces which are the product of a Sobolev space in one coordinate and L 2 spaces in the complementary variables, and we can apply SVD on each of the spaces. Third, with additional regularity, we can apply SVD on the space of functions with mixed smoothness. We conclude with a few numerical examples that support our theoretical findings.

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A posteriori error analysis and adaptivity for high-dimensional elliptic and parabolic boundary value problems

Merle

Prohl

2023

Numer. Math.

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We derive a posteriori error estimates for the (stopped) weak Euler method to discretize SDE systems which emerge from the probabilistic reformulation of elliptic and parabolic (initial) boundary value problems. The a posteriori estimate exploits the use of a scaled random walk to represent noise, and distinguishes between realizations in the interior of the domain and those close to the boundary. We verify an optimal rate of (weak) convergence for the a posteriori error estimate on deterministic meshes. Based on this estimate, we then set up an adaptive method which automatically selects local deterministic mesh sizes, and prove its optimal convergence in terms of given tolerances. Provided with this theoretical backup, and since corresponding Monte-Carlo based realizations are simple to implement, these methods may serve to efficiently approximate solutions of high-dimensional (initial-)boundary value problems.

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HT-AWGM: a hierarchical Tucker–adaptive wavelet Galerkin method for high-dimensional elliptic problems

Cited by 7 publications

References 31 publications

Singular Value Decomposition in Sobolev Spaces: Part II

Singular Value Decomposition in Sobolev Spaces: Part II

Singular Value Decomposition in Sobolev Spaces: Part I

A posteriori error analysis and adaptivity for high-dimensional elliptic and parabolic boundary value problems

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