Dissecting the FEAST algorithm for generalized eigenproblems

We analyze the FEAST method for computing selected eigenvalues and eigenvectors of large sparse matrix pencils. After establishing the close connection between FEAST and the well-known Rayleigh-Ritz method, we identify several critical issues that influence convergence and accuracy of the solver: the choice of the starting vector space, the stopping criterion, how the inner linear systems impact the quality of the solution, and the use of FEAST for computing eigenpairs from multiple intervals. We complement the study with numerical examples, and hint at possible improvements to overcome the existing problems.Comment: 11 Pages, 5 Figures. Submitted to Journal of Computational and Applied Mathematic

Zolotarev Quadrature Rules and Load Balancing for the FEAST Eigensolver

The FEAST method for solving large sparse eigenproblems is equivalent to subspace iteration with an approximate spectral projector and implicit orthogonalization. This relation allows to characterize the convergence of this method in terms of the error of a certain rational approximant to an indicator function. We propose improved rational approximants leading to FEAST variants with faster convergence, in particular, when using rational approximants based on the work of Zolotarev. Numerical experiments demonstrate the possible computational savings especially for pencils whose eigenvalues are not well separated and when the dimension of the search space is only slightly larger than the number of wanted eigenvalues. The new approach improves both convergence robustness and load balancing when FEAST runs on multiple search intervals in parallel.Comment: 22 pages, 8 figure

A spectral projection method for transmission eigenvalues

In this paper, we consider a nonlinear integral eigenvalue problem, which is a reformulation of the transmission eigenvalue problem arising in the inverse scattering theory. The boundary element method is employed for discretization, which leads to a generalized matrix eigenvalue problem. We propose a novel method based on the spectral projection. The method probes a given region on the complex plane using contour integrals and decides if the region contains eigenvalue(s) or not. It is particularly suitable to test if zero is an eigenvalue of the generalized eigenvalue problem, which in turn implies that the associated wavenumber is a transmission eigenvalue. Effectiveness and efficiency of the new method are demonstrated by numerical examples.Comment: The paper has been accepted for publication in SCIENCE CHINA Mathematic

ESSEX: Equipping Sparse Solvers for Exascale

The ESSEX project investigates computational issues arising at exascale for large-scale sparse eigenvalue problems and develops programming concepts and numerical methods for their solution. The project pursues a coherent co-design of all software layers where a holistic performance engineering process guides code development across the classic boundaries of application, numerical method and basic kernel library. Within ESSEX the numerical methods cover both widely applicable solvers such as classic Krylov, Jacobi-Davidson or recent FEAST methods as well as domain specific iterative schemes relevant for the ESSEX quantum physics application. This report introduces the project structure and presents selected results which demonstrate the potential impact of ESSEX for efficient sparse solvers on highly scalable heterogeneous supercomputers