Computing Equilibrium Points of Electrostatic Potentials

TL;DR

This paper introduces an efficient algorithm for approximating equilibrium points in electrostatic potentials, addressing challenges posed by singularities and rapid function changes, and explores the problem's computational complexity.

Contribution

It proposes a novel piecewise Taylor series approximation method for computing electrostatic equilibrium points and analyzes the problem's complexity class.

Findings

Algorithm finds approximate equilibrium points efficiently under certain conditions.

The problem is shown to be CLS-hard and in PPAD, highlighting its computational complexity.

Approximate solutions can be computed in poly-logarithmic time relative to the approximation parameter.

Abstract

We study the computation of equilibrium points of electrostatic potentials: locations in space where the electrostatic force arising from a collection of charged particles vanishes. This is a novel scenario of optimization in which solutions are guaranteed to exist due to a nonconstructive argument, but gradient descent is unreliable due to the presence of singularities. We present an algorithm based on piecewise approximation of the potential function by Taylor series. The main insight is to divide the domain into a grid with variable coarseness, where grid cells are exponentially smaller in regions where the function changes rapidly compared to regions where it changes slowly. Our algorithm finds approximate equilibrium points in time poly-logarithmic in the approximation parameter, but these points are not guaranteed to be close to exact solutions. Nevertheless, we show that such points can be computed efficiently under a mild assumption that we call "strong non-degeneracy". We complement these algorithmic results by studying a generalization of this problem and showing that it is CLS-hard and in PPAD, leaving its precise classification as an intriguing open problem.