A Proposal of Multigrid Methods for Hermitian Positive Definite Linear Systems enjoying an order relation

TL;DR

This paper develops a multigrid method for efficiently solving Hermitian positive definite linear systems related by an order relation, demonstrating optimality and applicability to structured matrices from differential equations.

Contribution

It introduces a novel multigrid approach leveraging order relations between matrices, ensuring optimality for structured Hermitian positive definite systems.

Findings

Proves Two-Grid method optimality under order relation assumptions

Applies the method to structured matrices like Toeplitz and circulants

Achieves linear time complexity for certain classes of problems

Abstract

Given a multigrid procedure for linear systems with coefficient matrices $A_n$, we discuss the optimality of a related multigrid procedure with the same smoother and the same projector, when applied to properly related algebraic problems with coefficient matrices $B_n$: we assume that both $A_n$ and $B_n$ are positive definite with $A_n\le \vartheta B_n$, for some positive $\vartheta$ independent of $n$. In this context we prove the Two-Grid method optimality. We apply this elementary strategy for designing a multigrid solution for modifications of multilevel structured (Toeplitz, circulants, Hartley, sine ($\tau$ class) and cosine algebras) linear systems, in which the coefficient matrix is banded in a multilevel sense and Hermitian positive definite. In such a way, several linear systems arising from the approximation of integro-differential equations with various boundary conditions can be efficiently solved in linear time (with respect to the size of the algebraic problem). Some numerical experiments are presented and discussed, both with respect to Two-Grid and multigrid procedures.