Localized Estimation of Condition Numbers for MILU Preconditioners on a Graph

TL;DR

This paper introduces a localized measure called LECN for estimating the condition number of MILU preconditioners on graphs, enabling local analysis and broadening applicability to complex matrix structures.

Contribution

The paper develops the LECN measure, providing a local condition number estimate that bounds the global condition number for MILU preconditioned systems on graphs.

Findings

LECN provides an upper bound for the condition number.

The approach simplifies condition number estimation for complex matrices.

Numerical validation confirms LECN's effectiveness on hierarchical grids.

Abstract

This paper proposes a theoretical framework for analyzing Modified Incomplete LU (MILU) preconditioners. Considering a generalized MILU preconditioner on a weighted undirected graph with self-loops, we extend its applicability beyond matrices derived by Poisson equation solvers on uniform grids with compact stencils. A major contribution is, a novel measure, the \textit{Localized Estimator of Condition Number (LECN)}, which quantifies the condition number locally at each vertex of the graph. We prove that the maximum value of the LECN provides an upper bound for the condition number of the MILU preconditioned system, offering estimation of the condition number using only local measurements. This localized approach significantly simplifies the condition number estimation and provides a powerful tool or analyzing the MILU preconditioner applied to previously unexplored matrix structures. To demonstrate the usability of LECN analysis, we present three cases: (1) revisit to existing results of MILU preconditioners on uniform grids, (2) analysis of high-order implicit finite difference schemes on wide stencils, and (3) analysis of variable coefficient Poisson equations on hierarchical adaptive grids such as quadtree and octree. For the third case, we also validate LECN analysis numerically on a quadtree.