A Distributed-Memory Implementation of Schur-Complement PCA Preconditioner for Ill-Conditioned Problems

This paper describes our first effort toward an implementation of Schur-complement principal component analysis (PCA) preconditioner for distributed-memory computing platforms. A distributed binary tree structure for binary hierarchical matrix partitioning is proposed. This implementation targets performance portability by leveraging the abstractions provided by Kokkos and Kokkos Kernels. We evaluate the implementation via an ill-conditioned problem with an iterative solver in terms of effectiveness and efficiency.

I. Introduction

This paper focuses on the solution of linear equation systems , where is a dense, square matrix. In frequency-domain electromagnetics, this problem typically originates from the discretization of the integral form of Maxwell's equations on the surfaces of interest. For large one might use iterative solvers with computational complexities of , where is the condition number of matrix . However, certain problems, e.g. structures with high-quality factor resonances, might yield extremely ill-conditioned system matrices making iterative solves costly as grows.