Multipole translation theory in the wavelet domain
Johannes Tausch
Department of Mathematics
Southern Methodist University
When the Fast Multipole Method is applied to scattering
problems, the expansion order must be doubled in each step to the
next-coarser level of the oct-tree. Thus the computational cost of
naive multipole translation operations becomes prohibitive even for
moderate frequencies.
Since multipole translations are convolutional, the commonly applied
fix is to perform these operations in the Fourier domain, where
translation operators are diagonal. However, the diagonal forms
involve n-th order Hankel functions which grow like n^n.
Therefore this approach can suffer from numerical instabilities.
This talk will discuss an alternative way to perform fast multipole
translations, which promises to be both stable and efficient at a
large range of frequencies. We replace sources and potentials in a
cube by an equivalent source- or potential distribution on a sphere
surrounding the cube. Translation operators between equivalent
sources/potentials are integral operators which have sparse
representations in wavelet space. We will analyze complexity and
accuracy of wavelet-based translation operators.