Subspace-preserving sparsification of matrices with minimal perturbation to the near null-space. Part I: Basics

Abstract: This is the first of two papers to describe a matrix sparsification algorithm that takes a general real or complex matrix as input and produces a sparse output matrix of the same size. The non-zero entries in the output are chosen to minimize changes to the singular values and singular vectors corresponding to the near null-space of the input. The output matrix is constrained to preserve left and right null-spaces exactly. The sparsity pattern of the output matrix is automatically determined or can be given as input. If the input matrix belongs to a common matrix subspace, we prove that the computed sparse matrix belongs to the same subspace. This works without imposing explicit constraints pertaining to the subspace. This property holds for the subspaces of Hermitian, complex-symmetric, Hamiltonian, circulant, centrosymmetric, and persymmetric matrices, and for each of the skew counterparts. Applications of our method include computation of reusable sparse preconditioning matrices for reliable and efficient solution of high-order finite element systems. The second paper in this series describes our open-source implementation, and presents further technical details.