Domain-Decomposition-Type Methods for Computing the Diagonal of a Matrix Inverse

Abstract

This paper presents two methods based on domain decomposition concepts for determining the diagonal of the inverse of specific matrices. The first uses a divide-and-conquer principle and the Sherman–Morrison–Woodbury formula and assumes that the matrix can be decomposed into a $2 \times 2$ block-diagonal matrix and a low-rank matrix. The second method is a standard domain decomposition approach in which local solves are combined with a global correction. Both methods can be successfully combined with iterative solvers and sparse approximation techniques. The efficiency of the methods usually depends on the specific implementation, which should be fine-tuned for different test problems. Preliminary results for some two-dimensional (2D) problems are reported to illustrate the proposed methods.