Deflation techniques for finding distinct solutions of nonlinear partial differential equations

TL;DR

This paper introduces an infinite-dimensional deflation method to systematically find multiple distinct solutions of nonlinear PDEs by modifying the residual, enabling convergence to different solutions from the same initial guess.

Contribution

The paper develops a novel deflation algorithm for nonlinear PDEs that efficiently finds multiple solutions without multiple initial guesses, with effective preconditioning strategies.

Findings

The deflation method successfully finds multiple solutions in various nonlinear PDE problems.

Preconditioning keeps Krylov iteration counts stable during deflation.

The approach applies to problems in special functions, phase separation, geometry, and fluid mechanics.

Abstract

Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find starting points that lie in different basins of attraction. In this paper, we present an infinite-dimensional deflation algorithm for systematically modifying the residual of a nonlinear PDE problem to eliminate known solutions from consideration. This enables the Newton--Kantorovitch iteration to converge to several different solutions, even starting from the same initial guess. The deflated Jacobian is dense, but an efficient preconditioning strategy is devised, and the number of Krylov iterations is observed not to grow as solutions are deflated. The power of the approach is demonstrated on several problems from special functions, phase separation, differential geometry and fluid mechanics that permit distinct solutions.