Enhancing the performance and robustness of the FEAST eigensolver

The FEAST algorithm is a subspace iteration method that uses a spectral projector as a rational filter in order to efficiently solve interior eigenvalue problems in parallel. Although the solutions from the FEAST algorithm converge rapidly in many cases, convergence can be slow in situations where the eigenvalues of a matrix are densely populated near the edges of the search interval of interest, which can be detrimental to parallel load balancing. This work introduces two methods that allow one to improve the convergence robustness of the FEAST algorithm in these situations without having to increase the amount of computation. Selected numerical examples are presented and discussed.

[1]  T. Sakurai,et al.  A projection method for generalized eigenvalue problems using numerical integration , 2003 .

[2]  Ping Tak Peter Tang,et al.  Zolotarev Quadrature Rules and Load Balancing for the FEAST Eigensolver , 2014, SIAM J. Sci. Comput..

[3]  Eric Polizzi,et al.  Density-Matrix-Based Algorithms for Solving Eingenvalue Problems , 2009 .

[4]  E. Polizzi,et al.  Non-linear eigensolver-based alternative to traditional SCF methods. , 2012, The Journal of chemical physics.

[5]  Andrew V. Knyazev,et al.  Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method , 2001, SIAM J. Sci. Comput..

[6]  Stephanie Kajpust.  Variations of the FEAST eigenvalue algorithm , 2014 .

[7]  Eric Polizzi,et al.  FEAST fundamental framework for electronic structure calculations: Reformulation and solution of the muffin-tin problem , 2011, Comput. Phys. Commun..

[8]  Ping Tak Peter Tang,et al.  Feast Eigensolver for Non-Hermitian Problems , 2015, SIAM J. Sci. Comput..

[9]  Lloyd N. Trefethen,et al.  Computing Eigenvalues of Real Symmetric Matrices with Rational Filters in Real Arithmetic , 2015, SIAM J. Sci. Comput..

[10]  Ping Tak Peter Tang,et al.  FEAST as Subspace Iteration Accelerated by Approximate Spectral Projection , 2013 .

[11]  Edoardo Di Napoli,et al.  Efficient estimation of eigenvalue counts in an interval , 2013, Numer. Linear Algebra Appl..

[12]  T. Sakurai,et al.  CIRR: a Rayleigh-Ritz type method with contour integral for generalized eigenvalue problems , 2007 .

[13]  Eric Polizzi,et al.  A Density Matrix-based Algorithm for Solving Eigenvalue Problems , 2009, ArXiv.

[14]  Tetsuya Sakurai,et al.  A block Arnoldi-type contour integral spectral projection method for solving generalized eigenvalue problems , 2014, Appl. Math. Lett..