Applied Math Seminar
Spring Quarter 2009
4:15 p.m.
Sloan Mathematics Corner
Building 380, Room 380-C

Special: Tuesday, May 12, 2009 at 4:15p

Naoki Saito
Department of Mathematics
University of California at Davis

Laplacian Eigenfunctions that do not feel the boundary: Theory, Computation, and Applications


Abstract:

I will discuss Laplacian eigenfunctions defined on a Euclidean domain of general shape, which ``do not feel the boundary.'' One can also cut up the given domain into local pieces using (sharp) characteristic functions without overlaps, and build the Laplacian eigenfunctions on each subdomain; consequently, one can construct a local orthonormal basis, which can be viewed as a generalization of block cosine transform on a rectangular domain.

I will also discuss how to compute such eigenfunctions. In general, directly solving the associated Helmholtz equations on such subdomains or computing the Green's functions satisfying the specific boundary conditions are rather difficult. I found an integral operator commuting with the Laplacian without imposing strict boundary conditions (such as the Dirichlet or Neumann conditions) a priori. Hence, the computation of the Laplacian eigenfunctions turned out to be equivalent to that of the integral operator. Another advantage of this approach is its amenability to modern fast numerical algorithms, such as the Fast Multipole Method (used here as a fast matrix-vector multiplication), which is indispensable for large scale iterative eigenvalue-eigenvector solvers such as the Lanczos iteration.

In addition, I will present applications of my Laplacian eigenfunctions to statistical image analysis and contrast them with Principal Component Analysis.