Title: Non-Linear Least-Squares Optimization of Rational Filters for the Solution of Interior Eigenvalue Problems,
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 the FEAST 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.