# Deflation techniques for distinct solutions of nonlinear PDEs

**Patrick Farrell
**

Mathematical Institute

University of Oxford

Oxford, United Kingdom

### Abstract:

Nonlinear problems can permit several distinct solutions. Familiar examples
include eigenvalue problems and multiple local minima in nonconvex optimisation
problems. This naturally leads to the question: if a nonlinear equation 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.