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

This paper presents two methods based on domain decomposition concepts for determining the diagonal of the inverse of a sparse matrix. The first uses a divide-and-conquer principle and the ShermanMorrison-Woodbury formula, and assumes that the matrix can be decomposed into a 2 × 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 succesfully combined with iterative solvers and sparse approximation techniques. Results of numerical experiments are reported to illustrate the performance of the proposed methods.