Approach to the Implementation of Laplace and a Broadband Helmholtz Fast Multipole Method as an Application Independent Library

In this paper, we propose an approach to develop an application independent library of Laplace and Helmholtz fast multipole method (FMM) that can be used in different applications. For this purpose, we consider a generalized problem and a corresponding canonical problem (defined below). In the first main contribution, we show that it is possible to capture the essential characteristics of the canonical summation from sampling the values of certain potentials or signature functions. In the second main contribution, we show that partial derivatives of arbitrary orders acting on the far field can be represented as product of sparse matrices within the library, transparent to the user. Combining the two ideas, we show that once the FMM is configured to compute the canonical summation, the same setup can be used to work with a much wider, general class of problems.