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Montardini, M.; Sangalli, G.; Tani, M.

A low-rank isogeometric solver based on Tucker tensors. (English) Zbl 1536.65150

Comput. Methods Appl. Mech. Eng. 417, Part B, Article ID 116472, 25 p. (2023).
Summary: We propose an isogeometric solver for Poisson problems that combines (i) low-rank tensor techniques to approximate the unknown solution and the system matrix, as a sum of a few terms having Kronecker product structure, (ii) a Truncated Preconditioned Conjugate Gradient solver to keep the rank of the iterates low, and (iii) a novel low-rank preconditioner, based on the Fast Diagonalization method where the eigenvector multiplication is approximated by the Fast Fourier Transform. Although the proposed strategy is written in arbitrary dimension, we focus on the three-dimensional case and adopt the Tucker format for low-rank tensor representation, which is well suited in low dimension. We show by numerical tests that this choice guarantees significant memory saving compared to the full tensor representation. We also extend and test the proposed strategy to linear elasticity problems.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65D07 Numerical computation using splines
65F55 Numerical methods for low-rank matrix approximation; matrix compression
15A69 Multilinear algebra, tensor calculus

Keywords:

isogeometric analysis; preconditioning; truncated preconditioned conjugate gradient method; Tucker representation; low-rank decomposition

Software:

Tensorlab; TensorToolbox.jl; Chebfun; GeoPDEs
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Full Text: DOI arXiv
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