Research topics

Spectral collocation (pseudospectral) methods for PDEs

Spectral collocation methods use all the available point values of a function to estimate (typically) derivatives of that function. Compared to finite differences, spectral collocation methods use data far more efficiently and can operate on much smaller discretizations. Compared to Galerkin methods, they are easier to implement especially for time-dependent, variable-coefficient, and nonlinear problems.
For MATLAB software based on spectral methods, see the chebfun project.

Radial basis functions for PDEs

RBFs are already well-loved for interpolation in many dimensions.  By differentiating such interpolants, it is possible to simulate PDEs in many dimensions over scattered points without grids. These approximations can combine spectral accuracy with fine resolution in selected regions of activity.

Numerical conformal mapping

This discussion is limited to Schwarz-Christoffel mapping, which has many interesting applications to elliptic problems on polygons.

Eigenmodes of the Laplacian

I'm interested in spectrally accurate methods for this problem on 2D regions, particularly in the presence of corners or other singularities. The best known example is the computation for the famous negative answerto the question, "Can one hear the shape of a drum?" Lately I've gotten involved in a nonlinear eigenvalue problem arising in micro-electromechanical systems (MEMS).

Sampling of rare events

Stanislaw Ulam mused while playing solitaire that while an exact analysis of the probability of winning a game was impossible, one could get a fair picture by keeping track of successes and failures while playing. He then realized that with the new computing power available to the Manhattan Project, one might do the same sort of thing to study nuclear chain reactions. The descendants of his methods are used to simulate many phenomena that defy analysis. I am particularly interested in methods that gather statistics about rarely occurring events, using techniques called importance sampling and multicanonical Monte Carlo.

Numerical software

I have a soft spot for developing numerical tools in MATLAB.

Please see my cv for a complete list of papers.