Low-rank tensor approximation of singularly perturbed boundary value problems in one dimension

Marcati, Carlo; Rakhuba, Maxim; Ulander, Johan E. M.

doi:10.1007/s10092-021-00439-0

Cited by 4 publications

(4 citation statements)

References 31 publications

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“…Similarly to convergence results for ℎ -FEM, they establish exponential-type convergence of approximate solutions with respect to the total number of representation parameters. Comparably efficient approximations are obtained for multiscale problems in Schwab (2017, 2022) and for convection-diffusion problems on intervals in Marcati, Rakhuba and Ulander (2022b). The technique is extended to isogeometric analysis in Markeeva, Tsybulin and Oseledets (2021).…”

Section: Parameter-dependent Elliptic Problemsmentioning

confidence: 99%

Low-rank tensor methods for partial differential equations

Bachmayr¹

2023

Acta Numerica

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Low-rank tensor representations can provide highly compressed approximations of functions. These concepts, which essentially amount to generalizations of classical techniques of separation of variables, have proved to be particularly fruitful for functions of many variables. We focus here on problems where the target function is given only implicitly as the solution of a partial differential equation. A first natural question is under which conditions we should expect such solutions to be efficiently approximated in low-rank form. Due to the highly nonlinear nature of the resulting low-rank approximations, a crucial second question is at what expense such approximations can be computed in practice. This article surveys basic construction principles of numerical methods based on low-rank representations as well as the analysis of their convergence and computational complexity.

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Section: Parameter-dependent Elliptic Problemsmentioning

confidence: 99%

Low-rank tensor methods for partial differential equations

Bachmayr¹

2023

Acta Numerica

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“…In the last equation we obtained a factorization into a product of 2 × 3 block matrix, which we denote as Q 2 , times a 3 × 4 matrix, which we propagate further to Q 3 (see (17) for Q 3 ):…”

Section: Three-dimensional Screened Poisson Equationmentioning

confidence: 99%

“…This problem was formalized in [1] and originates from both ill conditioning of discretized differential operators and the ill conditioning of the tensor representations themselves. To overcome these issues, an explicit QTT representation of BPX-preconditioned systems was proposed in the same work, which was later used for multiscale and singularly-perturbed problems in [13,17]. In [25], a robust and efficient solver based on the alternating direction implicit method (ADI) and explicit inversion formulas for tridiagonal Toeplitz matrices was developed.…”

Section: Introductionmentioning

confidence: 99%

Tensor rank bounds and explicit QTT representations for the inverses of circulant matrices

Vysotsky¹,

Rakhuba²

2022

Preprint

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In this paper, we are concerned with the inversion of circulant matrices and their quantized tensor-train (QTT) structure. In particular, we show that the inverse of a complex circulant matrix A, generated by the first column of the form (a0, . . . , am−1, 0, . . . , 0, a−n, . . . , a−1) admits a QTT representation with the QTT ranks bounded by (m + n). Under certain assumptions on the entries of A, we also derive an explicit QTT representation of A −1 . The latter can be used, for instance, to overcome stability issues arising when numerically solving differential equations with periodic boundary conditions in the QTT format.

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“…This problem was formalized in Reference 18 and originates from both ill conditioning of discretized differential operators and the ill conditioning of the tensor representations themselves. To overcome these issues, an explicit QTT representation of BPX‐preconditioned systems was proposed in the same work, which was later used for multiscale and singularly‐perturbed problems in References 19 and 20. In Reference 21, a robust and efficient solver based on the alternating direction implicit method (ADI) and explicit inversion formulas for tridiagonal Toeplitz matrices was developed.…”

Section: Introductionmentioning

confidence: 99%

Tensor rank bounds and explicit QTT representations for the inverses of circulant matrices

Vysotsky

Rakhuba

2022

Numerical Linear Algebra App

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In this paper, we are concerned with the inversion of circulant matrices and their quantized tensor‐train (QTT) structure. In particular, we show that the inverse of a complex circulant matrix A$$ A $$, generated by the first column of the form false(a0,…,amprefix−1,0,…,0,aprefix−n,…,aprefix−1false)⊤$$ {\left({a}_0,\dots, {a}_{m-1},0,\dots, 0,{a}_{-n},\dots, {a}_{-1}\right)}^{\top } $$ admits a QTT representation with the QTT ranks bounded by false(m+nfalse)$$ \left(m+n\right) $$. Under certain assumptions on the entries of A$$ A $$, we also derive an explicit QTT representation of Aprefix−1$$ {A}^{-1} $$. The latter can be used, for instance, to overcome stability issues arising when numerically solving differential equations with periodic boundary conditions in the QTT format.

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Low-rank tensor approximation of singularly perturbed boundary value problems in one dimension

Cited by 4 publications

References 31 publications

Low-rank tensor methods for partial differential equations

Low-rank tensor methods for partial differential equations

Tensor rank bounds and explicit QTT representations for the inverses of circulant matrices

Tensor rank bounds and explicit QTT representations for the inverses of circulant matrices

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