A parallel fast multipole method for elliptic difference equations
This work provides an efficient and parallelizable solver for elliptic difference equations on unbounded domains, which is important for applications like crystal structures and PDE discretizations.
The paper presents a parallel fast multipole method for solving elliptic difference equations on unbounded domains, achieving O(N) complexity by using a kernel-independent interpolation-based approach with FFTs. Numerical experiments on the discrete 3D Poisson equation demonstrate accuracy and parallel efficiency.
A new fast multipole formulation for solving elliptic difference equations on unbounded domains and its parallel implementation are presented. These difference equations can arise directly in the description of physical systems, e.g. crystal structures, or indirectly through the discretization of PDEs. In the analog to solving continuous inhomogeneous differential equations using Green's functions, the proposed method uses the fundamental solution of the discrete operator on an infinite grid, or lattice Green's function. Fast solutions $\mathcal{O}(N)$ are achieved by using a kernel-independent interpolation-based fast multipole method. Unlike other fast multipole algorithms, our approach exploits the regularity of the underlying Cartesian grid and the efficiency of FFTs to reduce the computation time. Our parallel implementation allows communications and computations to be overlapped and requires minimal global synchronization. The accuracy, efficiency, and parallel performance of the method are demonstrated through numerical experiments on the discrete 3D Poisson equation.