Non-Linear Least-Squares Optimization of Rational Filters for the Solution of Interior Eigenvalue Problems

Rational filter functions improve convergence of contour-based eigensolvers, a popular algorithm family for the solution of the interior eigenvalue problem. We present an optimization method of these rational filters in the Least-Squares sense. Our filters out-perform existing filters on a large and representative problem set, which we show on the example of FEAST. We provide a framework for (non-convex) weighted Least-Squares optimization of rational filter functions. To this end we discuss: (1) a formulation of the optimization problem that exploits symmetries of the filter function in the Hermitian case, (2) gradient descent and Levenberg-Marquardt solvers that exploit the symmetries, (3) a method to choose the starting position of the optimization that reliably produces good results, (4) constrained optimization that produces filter functions with desirable properties, e.g., poles of the rational filter function with large imaginary parts which benefit convergence of Krylov based solvers for the arising linear system solves.