Efficient Laplace-Beltrami Solutions via Multipole Acceleration

Abstract

The Laplace-Beltrami operator an essential object in too many branches of mathematics and physics which forms a fundamental tool in solving partial differential equations in manifolds. Its applicability covers geometry processing, quantum mechanics and machine learning in non-Euclidean space. Numerical solutions of the Laplace-Beltrami equations analytically were problematic by dense matrices and high dimensional integration methods are computationally intensive. Multipole acceleration techniques, which have been designed to improve computational effectiveness in electrostatics, represent a revolutionary approach to overcoming these difficulties. This paper presents an attempt at incorporating multipole acceleration into the Laplace-Beltrami solutions to increase the efficiency and applicability of the code. Concepts of the technique are introduced and we consider the technique when applied on curved manifolds and Efficiency gained. The outcomes show that these methods have substantial benefits in terms of computational performance and they are therefore well-applying to large-scale problems.