Volume 62, pp. 188-207, 2024.

Operator-dependent prolongation and restriction for the parameter-dependent multigrid method using low-rank tensor formats

Lars Grasedyck and Tim A. Werthmann

Abstract

Iterative solution methods, such as the parameter-dependent multigrid method, solve linear systems arising from partial differential equations that depend on parameters. When parameters introduce non-smooth dependencies, such as jumping coefficients, the convergence of the parameter-dependent multigrid method declines or even results in divergence. The goal is to enhance robustness of this multigrid method to enable effective solutions of problems involving jumping coefficients.
An operator-dependent prolongation and restriction, inspired by block Gaussian elimination, is derived that fulfills the approximation property under exact arithmetic, thereby enhancing the method's robustness. Using an approximation of this prolongation and restriction directly in a low-rank tensor format is a trade-off between computational cost and guaranteed convergence. Numerical experiments provide empirical support for the effectiveness of the method, even when using a lower accuracy to compute the approximation within the low-rank format. The proposed operator-dependent prolongation and restriction improves the convergence of the parameter-dependent multigrid method in the presence of jumping coefficients.

Full Text (PDF) [316 KB], BibTeX , DOI: 10.1553/etna_vol62s188

Key words

multigrid, transfer operators, partial differential equations, iterative solvers, low-rank tensor format, jumping coefficients

AMS subject classifications

65N55, 15A69

Links to the cited ETNA articles

[6]	Vol. 48 (2018), pp. 348-361 Matthias Bolten, Karsten Kahl, Daniel Kressner, Francisco Macedo, and Sonja Sokolović: Multigrid methods combined with low-rank approximation for tensor-structured Markov chains