# Applied Math Seminar

**Title:** Theory and Application of a Direct Solution Algorithm for Large Dense Matrices of Boundary Element Methods

Master's Talk

Subject Paper:

Martinsson, and Rokhlin. "A Fast Direct Solver for Boundary Integral Equations in Two Dimensions." Journal of Computational Physics 205.1 (2005): 1-23. Web. ISSN: 0021-9991 ; DOI: 10.1016/j.jcp.2004.10.033

In computational science and engineering, the numerical solution of partial differential equations is effected through the solution of extremely large linear systems. Finite element and finite difference methods give rise to sparse matrices that admit iterative solution techniques. Acoustic and electromagnetic scattering problems, however, are often better approached via boundary element methods. These result in huge dense matrices that would be prohibitively expensive to solve conventionally.

The subject paper details a method to construct the matrix inverse directly. The nature of the boundary integrals causes the system matrix to exhibit rank deficiency of blocks further removed from the diagonal. A modified QR algorithm from the literature both reveals the rank and approximates the nullspace basis of such blocks. An algorithm based on the Schur complement is then applied iteratively, inverting selected pivot blocks. The approach is extended to a hierarchical application reminiscent of Greengard and Rokhlin's Fast Multipole Method.

This Master's Degree examination talk will present the theory of the key elements of the method, as well as the performance metrics of the derived algorithms. A sample implementation with numerical results will also be described.

Refreshments will be served.