The numerical solution of linear boundary values problems play an
important role in the modeling of physical phenomena. As practitioners continue
to want to solve more complicated problems, it is important to develop robust
and efficient numerical methods. For some linear boundary value problems, it
is possible to recast the problem as an integral equation which sometimes leads
to a reduction in dimensionality. The trade-off for the reduction in dimensionality
is the need to solve a dense linear system. Inverting the dense N by N matrix via
Gaussian elimination has computational cost of O(N^3). This talk presents
solution techniques that exploit the physics in the boundary integral equation
to invert the dense matrix for a cost that scales linearly with N with
small constants. For example, on a laptop computer, a matrix with N=100,000
can be inverted in 90 seconds and applying the solver takes under a tenth of a second.
The speed in which new boundary conditions can be processed makes these
methods ideal applications involving many solves such as optimal design and
inverse scattering. Extensions of the single body direct solver to select
applications will also be presented.