Numerical Synthesis of Translation Operators for the Multi-Level Fast Multipole Method

A key operation in the fast multipole method is the translation of propagating plane waves from source groups to observation groups. For calculating the translation functions, the classical approach typically requires an exact knowledge of the system, such as given for the free-space Green's function. The resulting operators are commonly defined over all propagating plane-wave directions on the Ewald sphere. A relatively less explored alternative, however, is a numerical approach in which the translation operators are synthesized from known point-point interactions. This approach also provides an opportunity to reduce the matrix vector multiplication time. It is demonstrated that numerically synthesized translation operators are not only accurate enough but also greatly improve the performance of multi-level fast multipole method based electromagnetic solvers.