A graph-theoretic framework for bounding the condition number of a Modified Incomplete LU (MILU) preconditioned linear system. The coefficient matrix is reinterpreted as the weighted adjacency matrix of a weighted undirected graph with self-loops; a partial order on the vertices defines precursors and successors that drive the recursive MILU diagonal. The Localized Estimator of Condition Number (LECN) is a per-vertex quantity whose maximum provably upper-bounds the preconditioned condition number, extending classical MILU analysis from uniform grids to high-order wide-stencil finite-difference schemes and to variable-coefficient Poisson problems on adaptive quadtree/octree finite-volume grids.
This paper develops a theoretical framework for estimating the condition number of Modified Incomplete LU (MILU) preconditioned linear systems by reinterpreting the coefficient matrix as the weighted…