Applied Math Seminar
Fall Quarter 2006
3:15 p.m.
Sloan Mathematics Corner
Building 380, Room 380-C

Friday, December 8, 2006

James Lambers
Department of Energy Resources, and ICME, Stanford

The Evolution of Krylov Subspace Spectral Methods (or: How the Work of Gene Golub and Joe Oliger Collided)


Abstract:

As graduate students, we learn about how easy it is to work with constant-coefficient differential operators on ``nice'' domains such as intervals, rectangles and circles. We know their eigenvalues and eigenfunctions, and therefore can write down simple formulas for analytical solutions of PDE, thus facilitating the development and analysis of numerical methods for these problems, such as spectral methods that can compute the Fourier components of solutions independently of one another. When I began to work with Joe Oliger on multiresolution methods for time-dependent variable-coefficient PDE, I took it upon myself to try to extend, as much as possible, these desirable aspects of constant-coefficient operators to their variable-coefficient counterparts, particularly self-adjoint second-order operators.

This talk covers the progress that has been made on this undertaking to date, and the new directions currently being explored. First, I will discuss the ongoing development of Krylov Subspace Spectral Methods, which take advantage of Gene Golub's work on modified moments and Gaussian quadrature in the spectral domain, rather than in physical space as with traditional spectral methods, to produce high-order accurate approximate solutions that have a representation similar to those obtained using the Fourier method for constant-coefficient problems. This representation leads to a straightforward approach to deferred correction, and to a method for solving second-order wave equations that, while high-order accurate and explicit, is also surprisingly stable and straightforward to implement.

Second, I will discuss the use of similarity transformations involving unitary pseudo-differential operators to approximately diagonalize self-adjoint differential operators in one space dimension, and ongoing attempts to generalize to higher dimensions. These transformations, inspired by Joe Oliger's desire to use Fefferman's SAK principle to compute approximate solutions of PDE, serve as ideal preconditioners for Krylov Subspace Spectral Methods, but they can also be used to obtain analytical representations of orthogonal bases of approximate eigenfunctions and corresponding approximate eigenvalues.