An Efficient Spectral Trust-Region Deflation Method for Multiple Solutions

TL;DR

This paper introduces a spectral trust-region deflation method that efficiently finds multiple solutions of nonlinear PDEs with high accuracy, flexibility in initial guesses, and faster convergence compared to existing methods.

Contribution

It develops a novel spectral-Galerkin discretization combined with a deflation technique and a trust-region solver to systematically find multiple solutions of nonlinear PDEs.

Findings

The method successfully finds multiple solutions with high accuracy.

It is flexible with initial guesses, even starting from the same initial point.

The approach is faster and explores solutions not previously reported.

Abstract

It is quite common that a nonlinear partial differential equation (PDE) admits multiple distinct solutions and each solution may carry a unique physical meaning. One typical approach for finding multiple solutions is to use the Newton method with different initial guesses that ideally fall into the basins of attraction confining the solutions. In this paper, we propose a fast and accurate numerical method for multiple solutions comprised of three ingredients: (i) a well-designed spectral-Galerkin discretization of the underlying PDE leading to a nonlinear algebraic system (NLAS) with multiple solutions; (ii) an effective deflation technique to eliminate a known (founded) solution from the other unknown solutions leading to deflated NLAS; and (iii) a viable nonlinear least-squares and trust-region (LSTR) method for solving the NLAS and the deflated NLAS to find the multiple solutions sequentially one by one. We demonstrate through ample examples of differential equations and comparison with relevant existing approaches that the spectral LSTR-Deflation method has the merits: (i) it is quite flexible in choosing initial values, even starting from the same initial guess for finding all multiple solutions; (ii) it guarantees high-order accuracy; and (iii) it is quite fast to locate multiple distinct solutions and explore new solutions which are not reported in literature.