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Rational Spectral Filters with Optimal Convergence Rate
SIAM Journal on Scientific Computing  (IF2.373),  Pub Date : 2021-07-29, DOI: 10.1137/20m1313933
Konrad Kollnig, Paolo Bientinesi, Edoardo A. Di Napoli

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2660-A2684, January 2021.
In recent years, contour-based eigensolvers have emerged as a standard approach for the solution of large and sparse eigenvalue problems. Building upon recent performance improvements through nonlinear least-squares optimization of so-called rational filters, we introduce a systematic method to design these filters by minimizing the worst-case convergence rate and eliminate the parametric dependence on weight functions. Further, we provide an efficient way to deal with the box-constraints which play a central role for the use of iterative linear solvers in contour-based eigensolvers. Indeed, these parameter-free filters consistently minimize the number of iterations and the number of FLOPs to reach convergence in the eigensolver. As a byproduct, our rational filters allow for a simple solution to load balancing when the solution of an interior eigenproblem is approached by the slicing of the sought after spectral interval.