TITLE:

Fast numerical methods for solving elliptic PDEs

ABSTRACT:

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 modeled
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 solution
    of a sequence 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.