[Paper Review] An Integral Equation Formulation of the $N$-body Dielectric Spheres Problem. Part II: Complexity Analysis

Bérenger Bramas, Muhammad Hassan+1 more|arXiv (Cornell University)|Nov 17, 2019

Electromagnetic Scattering and Analysis83 references4 citations

TL;DR

This paper presents a complexity analysis of a Galerkin boundary integral equation method for solving the N-body dielectric spheres problem, demonstrating that the number of GMRES and CG iterations required to solve the linear system remains bounded independently of N. Combined with the fast multipole method (FMM), this ensures linear scaling in computational cost, establishing the method as linear scaling in accuracy for large-scale electrostatic simulations of polarizable dielectric particles.

ABSTRACT

This article is the second in a series of two papers concerning the mathematical study of a boundary integral equation of the second kind that describes the interaction of $N$ dielectric spherical particles undergoing mutual polarisation. The first article presented the numerical analysis of the Galerkin method used to solve this boundary integral equation and derived $N$-independent convergence rates for the induced surface charges and total electrostatic energy. The current article will focus on computational aspects of the algorithm. We provide a convergence analysis of the iterative method used to solve the underlying linear system and show that the number of liner solver iterations required to obtain a solution is independent of $N$. Additionally, we present two linear scaling solution strategies for the computation of the approximate induced surface charges. Finally, we consider a series of numerical experiments designed to validate our theoretical results and explore the dependence of the numerical errors and computational cost of solving the underlying linear system on different system parameters.

Motivation & Objective

• To establish that the Galerkin boundary integral equation method for N dielectric spheres is linear scaling in cost, ensuring computational efficiency for large N.
• To prove that the number of GMRES and CG iterations required to solve the linear system is independent of the number of particles N.
• To validate theoretically derived convergence bounds through numerical experiments under varying system parameters.
• To demonstrate that the method maintains N-error stability while achieving O(N) computational complexity.
• To lay the foundation for extending the method to compute electrostatic forces and other operators beyond the Laplace equation.

Proposed method

• The method employs a Galerkin discretization of a second-kind boundary integral equation formulated for N dielectric spheres in a homogeneous medium.
• The resulting linear system is solved using iterative solvers—GMRES and, after symmetrization, the conjugate gradient (CG) method.
• The Fast Multipole Method (FMM) is used to compute matrix-vector products in O(N) operations, enabling scalable linear system solves.
• A convergence analysis of GMRES and CG is performed, proving the existence of N-independent upper bounds on the number of iterations for a given tolerance.
• The linear system is symmetrized to allow the use of CG, which improves efficiency and enables a parallel convergence analysis.
• Numerical experiments are conducted on a laptop using varying N and ℓ_max (5 and 10), with solver tolerances set to 10^−6 and 10^−9, respectively, to validate scalability.

Experimental results

Research questions

• RQ1 Does the number of GMRES iterations required to solve the linear system grow with the number of dielectric spheres N?
• RQ2 Can the conjugate gradient (CG) method be effectively applied to the symmetrized linear system arising from the Galerkin discretization, with iteration count independent of N?
• RQ3 Is the overall computational cost of the algorithm linear in N, given the use of FMM and N-independent iterative convergence?
• RQ4 How do the discretization error and FMM error interact in practice, and does the FMM error dominate the solution accuracy?
• RQ5 Can the method be extended to compute electrostatic forces between dielectric particles with similar scalability and accuracy properties?

Key findings

• The number of GMRES iterations required to achieve a given relative error tolerance is bounded independently of N, proving N-independent convergence for the iterative solver.
• The number of CG iterations required to solve the symmetrized linear system is also bounded independently of N, with a convergence bound qualitatively similar to that of GMRES.
• Numerical experiments confirm linear scaling behavior in computation time for both GMRES and CG approaches, with nearly identical performance across varying N and ℓ_max.
• The FMM error does not dominate the discretization error in the tested geometric configurations, supporting the validity of the asymptotic complexity analysis.
• The combination of N-error stable Galerkin discretization and O(N) iterative solvers using FMM establishes the method as linear scaling in accuracy.
• The results support the feasibility of extending the framework to compute electrostatic forces with similar scalability, as planned in future work.