Implicit application of polynomial filters in a k-step Arnoldi method
SIAM Journal on Matrix Analysis and Applications
Templates for the solution of algebraic eigenvalue problems: a practical guide
Templates for the solution of algebraic eigenvalue problems: a practical guide
A Krylov--Schur Algorithm for Large Eigenproblems
SIAM Journal on Matrix Analysis and Applications
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the 9th International Congress on computational and applied mathematics
BoomerAMG: a parallel algebraic multigrid solver and preconditioner
Applied Numerical Mathematics - Developments and trends in iterative methods for large systems of equations—in memoriam Rüdiger Weiss
hypre: A Library of High Performance Preconditioners
ICCS '02 Proceedings of the International Conference on Computational Science-Part III
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems
ACM Transactions on Mathematical Software (TOMS) - Special issue on the Advanced CompuTational Software (ACTS) Collection
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We propose two iterative numerical methods for eigenvalue computations of large dimensional problems arising from finite approximations of integral operators, and describe their parallel implementation. A matrix representation of the problem on a space of moderate dimension, defined from an infinite dimensional one, is computed along with its eigenpairs. These are taken as initial approximations and iteratively refined, by means of a correction equation based on the reduced resolvent operator and performed on the moderate size space, to enhance their quality. Each refinement step requires the prolongation of the correction equation solution back to a higher dimensional space, defined from the infinite dimensional one. This approach is particularly adapted for the computation of eigenpair approximations of integral operators, where prolongation and restriction matrices can be easily built making a bridge between coarser and finer discretizations. We propose two methods that apply a Jacobi---Davidson like correction: Multipower Defect-Correction (MPDC), which uses a single-vector scheme, if the eigenvalues to refine are simple, and Rayleigh---Ritz Defect-Correction (RRDC), which is based on a projection onto an expanding subspace. Their main advantage lies in the fact that the correction equation is performed on a smaller space while for general solvers it is done on the higher dimensional one. We discuss implementation and parallelization details, using the PETSc and SLEPc packages. Also, numerical results on an astrophysics application, whose mathematical model involves a weakly singular integral operator, are presented.