QTT-isogeometric solver in two dimensions

Markeeva, Larisa; Tsybulin, I. V.; Oseledets, Ivan

doi:10.1016/j.jcp.2020.109835

Cited by 2 publications

(1 citation statement)

References 17 publications

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“…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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TetraFEM: Numerical Solution of Partial Differential Equations Using Tensor Train Finite Element Method

Kornev,

Dolgov,

Perelshtein

et al. 2024

Mathematics

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In this paper, we present a methodology for the numerical solving of partial differential equations in 2D geometries with piecewise smooth boundaries via finite element method (FEM) using a Quantized Tensor Train (QTT) format. During the calculations, all the operators and data are assembled and represented in a compressed tensor format. We introduce an efficient assembly procedure of FEM matrices in the QTT format for curvilinear domains. The features of our approach include efficiency in terms of memory consumption and potential expansion to quantum computers. We demonstrate the correctness and advantages of the method by solving a number of problems, including nonlinear incompressible Navier–Stokes flow, in differently shaped domains.

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QTT-isogeometric solver in two dimensions

Cited by 2 publications

References 17 publications

Low-rank tensor methods for partial differential equations

Low-rank tensor methods for partial differential equations

TetraFEM: Numerical Solution of Partial Differential Equations Using Tensor Train Finite Element Method

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