Annealing-based approach to solving partial differential equations

TL;DR

This paper introduces an annealing-based method for solving PDEs by transforming discretized equations into eigenvalue problems, enabling efficient eigenvector computation with high precision.

Contribution

It presents a novel annealing algorithm that efficiently computes eigenvectors for PDEs without increasing variable count, improving precision and scalability.

Findings

Eigenvector computation efficiency depends on system size and annealing time.

Simulated annealing demonstrates scalable iteration counts relative to problem size.

Performance varies with problem characteristics and annealing parameters.

Abstract

Solving partial differential equations (PDEs) using an annealing-based approach involves solving generalized eigenvalue problems. Discretizing a PDE yields a system of linear equations (SLE). Solving an SLE can be formulated as a general eigenvalue problem, which can be transformed into an optimization problem with an objective function given by a generalized Rayleigh quotient. The proposed algorithm requires iterative computations. However, it enables efficient annealing-based computation of eigenvectors to arbitrary precision without increasing the number of variables. Investigations using simulated annealing demonstrate how the number of iterations scales with system size and annealing time. Computational performance depends on system size, annealing time, and problem characteristics.