22 research outputs found
Preconditioned Locally Harmonic Residual Method for Computing Interior Eigenpairs of Certain Classes of Hermitian Matrices
We propose a Preconditioned Locally Harmonic Residual (PLHR) method for
computing several interior eigenpairs of a generalized Hermitian eigenvalue
problem, without traditional spectral transformations, matrix factorizations,
or inversions. PLHR is based on a short-term recurrence, easily extended to a
block form, computing eigenpairs simultaneously. PLHR can take advantage of
Hermitian positive definite preconditioning, e.g., based on an approximate
inverse of an absolute value of a shifted matrix, introduced in [SISC, 35
(2013), pp. A696-A718]. Our numerical experiments demonstrate that PLHR is
efficient and robust for certain classes of large-scale interior eigenvalue
problems, involving Laplacian and Hamiltonian operators, especially if memory
requirements are tight
Graph partitioning using matrix values for preconditioning symmetric positive definite systems
Prior to the parallel solution of a large linear system, it is required to
perform a partitioning of its equations/unknowns. Standard partitioning
algorithms are designed using the considerations of the efficiency of the
parallel matrix-vector multiplication, and typically disregard the information
on the coefficients of the matrix. This information, however, may have a
significant impact on the quality of the preconditioning procedure used within
the chosen iterative scheme. In the present paper, we suggest a spectral
partitioning algorithm, which takes into account the information on the matrix
coefficients and constructs partitions with respect to the objective of
enhancing the quality of the nonoverlapping additive Schwarz (block Jacobi)
preconditioning for symmetric positive definite linear systems. For a set of
test problems with large variations in magnitudes of matrix coefficients, our
numerical experiments demonstrate a noticeable improvement in the convergence
of the resulting solution scheme when using the new partitioning approach