Deflation techniques for distinct solutions of nonlinear PDEs
Nonlinear problems can permit several distinct solutions. For example, most optimisation problems arising in practice are nonconvex and permit multiple local minima. This leads to the question: if a problem has more than one solution, how can we compute them? In this talk, I present an algorithm for this purpose, called deflation. Given the residual of a nonlinear PDE, and one solution of it, deflation constructs a new problem with all of the solutions of the original problem, except for the one being deflated. This allows Newton's method to converge to different solutions, even starting from the same initial guess. An efficient preconditioning strategy is devised, and the number of Krylov iterations is observed not to grow as solutions are deflated; deflation scales to problems with billions of degrees of freedom. The technique is then applied to computing distinct solutions of nonlinear PDEs, tracing bifurcation diagrams, and to computing multiple local minima of PDE-constrained optimisation problems.