An O(N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles

TL;DR

This paper introduces a parallel, kernel-independent Fast Multipole Method that reduces the computational complexity of integral problems from O(N^2) to O(N), enabling efficient large-scale simulations in physics and engineering.

Contribution

The authors develop a scalable parallel approach using a kernel-independent FMM to evaluate integral equations in O(N) time, significantly improving computational efficiency.

Findings

Achieved O(N) computational complexity for integral evaluations.

Demonstrated accuracy and scalability through two physics-based examples.

Enabled large-scale simulations of electromagnetic and electrostatic systems.

Abstract

Large classes of materials systems in physics and engineering are governed by magnetic and electrostatic interactions. Continuum or mesoscale descriptions of such systems can be cast in terms of integral equations, whose direct computational evaluation requires O(N^2) operations, where N is the number of unknowns. Such a scaling, which arises from the many-body nature of the relevant Green's function, has precluded wide-spread adoption of integral methods for solution of large-scale scientific and engineering problems. In this work, a parallel computational approach is presented that relies on using scalable open source libraries and utilizes a kernel-independent Fast Multipole Method to evaluate the integrals in O(N) operations, with O(N) memory cost, thereby substantially improving the scalability and efficiency of computational integral methods. We demonstrate the accuracy, efficiency, and scalability of our approach in the contest of two examples. In the first, we solve a boundary value problem for a ferroelectric/ferromagnetic volume in free space. In the second, we solve an electrostatic problem involving polarizable dielectric bodies in an unbounded dielectric medium. The results from these test cases show that our parallel approach, which is built on a kernel-independent FMM, can enable highly efficient and accurate simulations and allow for considerable flexibility in a broad range of applications.