Optimal design of optical analog solvers of linear systems

TL;DR

This paper proposes a novel physical approach using optical analogs to solve linear systems efficiently by optimally placing scattering objects, leveraging asymptotic expansions for faster computation.

Contribution

It introduces a new algorithm for optimal placement of objects in optical systems to solve linear equations, improving speed over classical methods.

Findings

Algorithm achieves faster solutions than classical BEM.

Asymptotic expansions enable efficient computation.

Method effectively solves linear systems using physical wave propagation.

Abstract

In this paper, given a linear system of equations A x = b, we are finding locations in the plane to place objects such that sending waves from the source points and gathering them at the receiving points solves that linear system of equations. The ultimate goal is to have a fast physical method for solving linear systems. The issue discussed in this paper is to apply a fast and accurate algorithm to find the optimal locations of the scattering objects. We tackle this issue by using asymptotic expansions for the solution of the underlyingpartial differential equation. This also yields a potentially faster algorithm than the classical BEM for finding solutions to the Helmholtz equation.