AUTHOR=Winkelmann Jan , Di Napoli Edoardo TITLE=Non-linear Least-Squares Optimization of Rational Filters for the Solution of Interior Hermitian Eigenvalue Problems JOURNAL=Frontiers in Applied Mathematics and Statistics VOLUME=5 YEAR=2019 URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2019.00005 DOI=10.3389/fams.2019.00005 ISSN=2297-4687 ABSTRACT=

Rational filter functions can be used to improve convergence of contour-based eigensolvers, a popular family of algorithms for the solution of the interior eigenvalue problem. We present a framework for the optimization of rational filters based on a non-convex weighted Least-Squares scheme. When used in combination with a contour based eigensolvers library, our filters out-perform existing ones on a large and representative set of benchmark problems. This work provides a detailed description of: (1) a set up of the optimization process that exploits symmetries of the filter function for Hermitian eigenproblems, (2) a formulation of the gradient descent and Levenberg-Marquardt algorithms that exploits the symmetries, (3) a method to select the starting position for the optimization algorithms that reliably produces effective filters, (4) a constrained optimization scheme that produces filter functions with specific properties that may be beneficial to the performance of the eigensolver that employs them.