Quantized tensor-structured finite elements for second-order elliptic PDEs in two dimensions

Kazeev, Vladimir; Schwab, Christoph

doi:10.1007/s00211-017-0899-1

Cited by 34 publications

(38 citation statements)

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“…In [33], the QTT-structured h-FE approximation is shown to converge exponentially with respect to the number of QTT parameters for functions defined on curvilinear polygons and having B 2 β -type singularities at some of the vertices. The basic approach remains the same, as in the one-dimensional setting of the present paper.…”

Section: Approximation In H-and Hp-spacesmentioning

confidence: 99%

Approximation of Singularities by Quantized‐Tensor FEM

Kazeev

Schwab

2015

Proc Appl Math and Mech

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In d dimensions, first-order tensor-product finite-element (FE) approximations of the solutions of second-order elliptic problems are well known to converge algebraically, with rate at most 1/d in the energy norm and with respect to the number of degrees of freedom. On the other hand, FE methods of higher regularity may achieve exponential convergence, e.g. global spectral methods for analytic solutions and hp methods for solutions from certain countably normed spaces, which may exhibit singularities.In this note, we revisit, in one dimension, the tensor-structured approach to the h-FE approximation of singular functions. We outline a proof of the exponential convergence of such approximations represented in the quantized-tensor-train (QTT) format. Compared to special approximation techniques, such as hp, that approach is fully adaptive in the sense that it finds suitable approximation spaces algorithmically. The convergence is measured with respect to the number of parameters used to represent the solution, which is not the dimension of the first-order FE space, but depends only polylogarithmically on that. We demonstrate the convergence numerically for a simple model problem and find the rate to be approximately the same as for hp approximations.

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Section: Approximation In H-and Hp-spacesmentioning

confidence: 99%

Approximation of Singularities by Quantized‐Tensor FEM

Kazeev

Schwab

2015

Proc Appl Math and Mech

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“…Low-rank approximability assumptions. For the case of one or two dimensions, a low-rank approximation analysis for the solution of the problem (2.2) under certain analyticity assumptions on the coefficients and right-hand side, following from the regularity analysis developed in [2,3], is available in [28,33,34]. The following result can be obtained as an immediate consequence of [34, Theorem 5.16].…”

Section: Complexity Of Solversmentioning

confidence: 99%

Stability of Low-Rank Tensor Representations and Structured Multilevel Preconditioning for Elliptic PDEs

Bachmayr

Kazeev

2020

Found Comput Math

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Folding grid value vectors of size 2 L into Lth order tensors of mode size 2 × · · · × 2, combined with low-rank representation in the tensor train format, has been shown to result in highly efficient approximations for various classes of functions. These include solutions of elliptic PDEs on nonsmooth domains or with oscillatory data. This tensor-structured approach is attractive because it leads to highly compressed, adaptive approximations based on simple discretizations. Standard choices of the underlying bases, such as piecewise multilinear finite elements on uniform tensor product grids, entail the well-known matrix ill-conditioning of discrete operators. We demonstrate that, for low-rank representations, the use of tensor structure itself additionally introduces representation ill-conditioning, a new effect specific to computations in tensor networks. We analyze the tensor structure of a BPX preconditioner for a second-order linear elliptic operator and construct an explicit tensor-structured representation of the preconditioner, with ranks independent of the number L of discretization levels. The straightforward application of the preconditioner yields discrete operators whose matrix conditioning is uniform with respect to the discretization parameter, but in decompositions that suffer from representation ill-conditioning. By additionally eliminating certain redundancies in the representations of the preconditioned discrete operators, we obtain reduced-rank decompositions that are free of both matrix and representation ill-conditioning. For an iterative solver based on soft thresholding of low-rank tensors, we obtain convergence and complexity estimates and demonstrate its reliability and efficiency for discretizations with up to 2 50 nodes in each dimension.

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“…The main benefit of the QTT-format is that it leads to logarithmic complexity to represent the vector of unknowns, if the ranks r k are bounded: we only need to store O(dr 2 ) parameters. For elliptic problems, the upper bounds of QTT-ranks were provided in [5] and extended to the highly oscillating case in [4]. The last case is the most practically interesting, since it is exactly the case when astronomically large grids are needed.…”

Section: Qtt Representation For the One Dimensional Casementioning

confidence: 99%