Speaker:

Per-Gunnar Martinsson, University of Colorado

Title:

Fast numerical methods for solving linear PDEs

Linear boundary value problems occur ubiquitously in many areas of
science and engineering, and the cost of computing approximate
solutions to such equations is often what determines which problems
can, and which cannot, be modelled computationally. Due to advances in
the last few decades (multigrid, FFT, fast multipole methods, etc), we
today have at our disposal numerical methods for most linear boundary
value problems that are "fast" in the sense that their computational
cost grows almost linearly with problem size. Most existing "fast"
schemes are based on iterative techniques in which a sequence of
incrementally more accurate solutions is constructed. In contrast, we
propose the use of recently developed methods that are capable of
directly inverting large systems of linear equations in almost linear
time. Such "fast direct methods" have several advantages over existing
iterative methods:

(1) Dramatic speed-ups in applications involving the repeated solution
     of similar problems (e.g. optimal design, molecular dynamics).
(2) The ability to solve inherently ill-conditioned problems (such as
     scattering problems) without the use of custom designed
     preconditioners.
(3) The ability to construct spectral decompositions of differential
     and integral operators.
(4) Improved robustness and stability.