A Self-learning Algebraic Multigrid Method for Extremal Singular Triplets and Eigenpairs

TL;DR

This paper introduces a self-learning algebraic multigrid method for efficiently computing extremal singular triplets and eigenpairs, combining multilevel phases with a novel SVD generalization, demonstrating high accuracy and flexibility.

Contribution

It presents a new multigrid algorithm that self-learns interpolation operators and extends existing methods to compute extremal singular triplets and eigenpairs.

Findings

Converges to high accuracy in few iterations

Handles a variety of problems due to self-learning

Effective on model problems from different areas

Abstract

A self-learning algebraic multigrid method for dominant and minimal singular triplets and eigenpairs is described. The method consists of two multilevel phases. In the first, multiplicative phase (setup phase), tentative singular triplets are calculated along with a multigrid hierarchy of interpolation operators that approximately fit the tentative singular vectors in a collective and self-learning manner, using multiplicative update formulas. In the second, additive phase (solve phase), the tentative singular triplets are improved up to the desired accuracy by using an additive correction scheme with fixed interpolation operators, combined with a Ritz update. A suitable generalization of the singular value decomposition is formulated that applies to the coarse levels of the multilevel cycles. The proposed algorithm combines and extends two existing multigrid approaches for symmetric positive definite eigenvalue problems to the case of dominant and minimal singular triplets. Numerical tests on model problems from different areas show that the algorithm converges to high accuracy in a modest number of iterations, and is flexible enough to deal with a variety of problems due to its self-learning properties.