Numerical Publications

3. Analysis of a novel preconditioner for a class of p-dimensional lower rank extracted systems
by S. Salapaka and A. Peirce.
Numerical Linear Algebra with Applications, 13, issue 6, 437-472, 2006.

Abstract: This paper proposes and studies the performance of a preconditioner suitable for solving a class of symmetric positive definite systems, A^x = b, which we call p-level lower rank extracted systems (p-level LRES), by the preconditioned conjugate gradient method. The study of these systems is motivated by the numerical approximation of integral equations with convolution kernels defned on arbitrary p-dimensional domains. This is in contrast to p-level Toeplitz systems which only apply to rectangular domains. The coefficient matrix, A^, is a principal submatrix of a p-level Toeplitz matrix, A, and the preconditioner for the preconditioned conjugate gradient algorithm is provided in terms of the inverse of a p-level circulant matrix constructed from the elements of A. The preconditioner is shown to yield clustering in the spectrum of the preconditioned matrix which leads to a substantial reduction in the
computational cost of solving LRE systems.
2. Analysis of a circulant based preconditioner for a class of lower rank extracted systems
by S. Salapaka, A.Peirce, and M.Dahleh.
Numerical Linear Algebra with Applications, 12, 9-32, 2005.

Abstract:This paper proposes and studies the performance of a preconditioner suitable for solving a class of symmetric positive definite systems, Apx=b, which we call lower rank extracted systems (LRES), by the preconditioned conjugate gradient method. These systems correspond to integral equations with convolution kernels defined on a union of many line segments in contrast to only one line segmentin the case of Toeplitz systems. The p×p matrix, Ap, is shown to be a principal submatrix of a larger N ×N Toeplitz matrix, A_N . The preconditioner is provided in terms of the inverse of a 2N ×2N circulant matrix constructed from the elements of A_N . The preconditioner is shown to yield clusteringin the spectrum of the preconditioned matrix similar to the clustering results for iterative algorithms used to solve Toeplitz systems. The analysis also demonstrates that the computational expense to solve
LRE systems is reduced to O(N log N).
1. On the lack of convergence of unconditionally stable explicit rational Runge-Kutta Schemes
by A. Peirce and J.H. Prevost.
Computer Methods in Applied Mechanics and Engineering, 57, 171-180, 1986.

Abstract:The poor performance of the rational Runge-Kutta (RRK) schemes of Hairer are investigated. By considering two simple model problems, it is demonstrated that this poor performance is in fact due to a lack of convergence. A conceptual model of an unconditionally stable implicit-explicit time-integration scheme is also considered. With the aid of this model, it is possible to establish necessary bounds on the extent of the explicit region for convergence. This demonstrates the limited applicability of such hybrid time-integration schemes.