Localized Estimation of Condition Numbers for MILU Preconditioners on a Graph

TL;DR

The paper develops a graph-based framework for analyzing MILU preconditioners through a Localized Estimator of Condition Number (LECN). By representing SPD M-matrices as weighted graphs with self-loops and decomposing A into L+D+L^T, the authors define a vertex-local metric τ_K that bounds the MILU-conditioned system’s condition number via κ(M^{-1}A) ≤ max_K τ_K. This local approach is then applied to Poisson problems on uniform grids, high-order wide-stencil schemes, and variable-coefficient Poisson equations on adaptive quadtree/octree grids, proving κ(M^{-1}A) = O(ar{h}^{-1}) where ar{h} is the smallest cell size. Numerical experiments validate the theory, showing substantial reductions in κ and faster PCG convergence with MILU on adaptive grids and wide stencils. The work significantly broadens the applicability of MILU analysis to complex matrix structures, nonuniform grids, and hierarchical discretizations, offering a practical tool for predicting preconditioner performance from local graph information.

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.

Figures (6)

• Figure 1: Stencils used for IFD$(1,1)$, IFD$(2,2)$ and HIFD$(2,2)$ at the red-colored grid.
• Figure 1: (a) Illustration of the quadtree order where cells are assigned lexicographical order starting from the bottom-right. For a cell $C$, the two cells highlighted in green are its precursors (a lower order than $C$) and the three neighbor cells highlighted in pink are its successors (a higher order than $C$). (b) An example of a quadtree used to illustrate how the discrepancy in sizes of neighbor cells $C_{1,i}$ and $C_{2,i'}$ causes a jump in the LECN values described in \ref{['eq:example1']} and \ref{['eq:example2']}.
• Figure 2: A logical flow in the proof of \ref{['theorem:2d_octree_main']} and \ref{['corollary:main']}, illustrating the progression from Lemmas and \ref{['proposition:tau_e']} to the main results on condition number estimation for MILU preconditioners on quadtree.
• Figure 3: Visualization of randomly generated initial quadtrees.
• Figure 4: Comparison of $\kappa\left(M^{-1}A\right)$ between Jacobi, ILU, and MILU preconditioners in the random quadtree. The MILU preconditioner reduces the condition number to $\mathcal{O}\left(\bar{h}^{-1}\right)$.

Theorems & Definitions (36)

• Definition 2.1: Order on Graph
• Definition 2.2: Precursor and Successor
• Definition 2.3: MILU on Graph
• Definition 2.4: Localized Estimator of Condition Number (LECN)
• Proposition 2.5: LECN analysis
• Proof 1
• Lemma 2.6
• Proof 2
• Lemma 2.7
• Proof 3