Fast Multipole Method
Eric Darve, Stanford
I will present recent developments on the fast multipole method (FMM).
The first part of the talk will describe a new variant for the
Helmholtz kernel where Fourier basis are used instead of spherical
harmonics. This reduces significantly the cost of shifting multipole
expansions up and down the oct-tree. We will discuss the algorithm, an
error analysis and will present some numerical results on various test
cases. The second part will consider a black box multipole method where
the user only provides a routine which evaluates the kernel at given
points. In many applications, the kernel is not analytically known or
is very complicated. Consequently, deriving an FMM from analytical
formulas is not always possible or practical. A new black box approach
will be presented based on Chebyshev polynomial interpolation and
singular value decompositions of kernels. We will compare this
algorithm and the analytical FMM in terms of accuracy and run time.
This will be illustrated with applications in dislocation dynamics.