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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.