Fast numerical methods for solving linear PDEs

Applied and Computational Mathematics Seminar
Thursday, April 23, 2009 - 1:00pm
1 hour (actually 50 minutes)
Skiles 255
Dept of Applied Mathematics, University of Colorado

Note special day

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.
In the talk, we will also describe how randomized sampling can be used
to rapidly and accurately construct low rank approximations to matrices.
The cost of constructing a rank k approximation to an m x n matrix A
for which an O(m+n) matrix-vector multiplication scheme is available
is O((m+n)*k). This cost is the same as that of the well-established
Lanczos scheme, but the randomized scheme is significantly more robust.
For a general matrix A, the cost of the randomized scheme is O(m*n*log(k)),
which should be compared to the O(m*n*k) cost of existing deterministic