My research is concerned with the development of efficient algorithms for scientific computing, primarily methods for solving differential and integral equations, and for analyzing very large matrices and datasets. My interests are broad, but there are several recurring themes:
- Randomized sampling to overcome the computational challenges associated with problems set in high dimensional spaces. Applications to data analysis, computational statistics, geometry of data-sets in high dimensional spaces, etc.
- PDE solvers that draw on the full arsenal of techniques provided by classical mathematical physics and harmonic analysis.
- The construction of direct (as opposed to iterative) solvers for linear PDEs. These solvers directly construct an approximation to the relevant solution operator such as, e.g., a Green's function, an evolution operator, or a Dirichlet-to-Neumann operator.
- Design of computational algorithms that are engineered from the ground up to minimize communication. This is essential for performance in modern multi-core and parallel computing environments.
Randomized methods in linear algebra