Numerical Solution of Integral Equations

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Most scientific problems are formulated as partial differential
equations which must be solved numerically. Sometimes the solution can
be expressed as a boundary integral involving an unknown density
function which can also be computed numerically by solving an integral
equation. Thanks to recently developed algorithms such as
iterative linear system solvers and the fast multipole method it is
sometimes more efficient to solve the integral equation. Recent work
has dealt with the numerical solution of integral equations and
stable and efficient methods for evaluating the solution at various
points once the integral representation in known.

A typical problem is to compute the magnetic field induced by
a wire wrapped around a U-shaped recording head and carrying a current.
The procedure for solving the problem is to discretize the surface of
the region using triangles, solve a boundary integral equation to
determine an integral expression for the solution, use this expression
to evaluate the discrete Laplacian at the mesh points of an embedding
cube, and finally apply a fast Poisson solver to determine the value of
the field at all mesh points, both inside and outside the recording head.

It is not unusual to
require a 1000 by 1000 by 1000 regular grid; i.e., one billion mesh points.
These cannot be stored on a single processor, so a multiprocessor must
be used. The 3-D fast Poisson solver has been parallelized and run on
the IBM SP2. This enables the solution of a problem that is p times
as large on p processors, in almost the same amount of time that the
original problem requires on one processor.

For more details, see the references:

* Fast Parallel Iterative Solution of Poisson's and the Biharmonic
Equations on Irregular Regions *

** SIAM J. Sci. Stat. Comput., 13 (1992), pp. 101-117 **,

by A. Mayo and A. Greenbaum.

* Laplace's Equation and the Dirichlet-Neumann Map in Multiply
Connected Domains*

** J. Comput. Phys., 105 (1993), pp. 267-278 **,

by A. Greenbaum, L. Greengard, and G. McFadden.