Emanuel Rubensson: Localized inverse factorization

Abstract:

We will present a localized divide and conquer algorithm for inverse factorization of Hermitian positive definite matrices with localized structure, e.g. exponential decay with respect to some distance function on the index set of the matrix. The problem is recursively broken down into two smaller inverse factorization problems whose solutions are combined using iterative refinement, giving a solution to the original problem. We will present theoretical results regarding convergence, stability, and the decay properties of the involved matrices. The localization properties will be demonstrated in numerical experiments with overlap matrices coming from electronic structure calculations. In this context, the inverse factor is used to transform the quantum mechanical eigenvalue problem to standard form. We will also present results regarding the scaling of the computational cost with system size as well as the parallel performance of our Chunks and Tasks implementation.