Fast Multipole Method
Eric Darve, Stanford University

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.