Colloquium
3:00 p.m., Wednesday (Feb. 26th)
Math Annex 1100
Srinivasa M. Salapaka
Laboratory for Information and Decision Systems
M.I.T.
Efficient Solutions to Lower Rank Extracted Systems
This presentation proposes efficient numerical solutions to a class
of symmetric positive definite linear systems Ax=b, called
Lower Rank Extracted Systems (LRES). These systems appear in a
wide range of scientific and engineering models such as in image
reconstruction in image processing, in the modeling of interacting cracks,
in the modeling of tabular mining excavations, the modeling of planar
array antennae in the field of telecommunications.
These systems appear in the numerical modeling of convolution type
integral equations defined on arbitrary domains. This is in contrast to
the Toeplitz systems which considers only rectangular domains. We compute
their solution using a recursive method called the Preconditioned Conjugate
Gradient Method (PCGM). Our contribution lies in prescribing a
preconditioning matrix and proving that it achieves a substantial
reduction in the computational expense. In the case of integral equations
defined on one dimensional domains, we have proved that our approach reduces
the computational from O(N^2) to O(N log N) operations where N is the size
of the coefficient matrix A. We have generalized this approach to solve
convolution type integral equations defined on a class of higher dimensional
domains called the pdimensional Lower Rank Extracted Systems (pd LRES).
The method that we propose is shown to yield clustering in the spectrum
of the preconditioned matrix which leads to a substantial reduction in
the computational expense.
