9 research outputs found
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
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
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