# CBC Talk on Deflation techniques for distinct solutions of nonlinear PDEs - October 1, 2015

Total number of participants: 13

Total number of guests outside of CBC: 2

Number of different nationalities represented: 5

Total number of speakers: 1

Total number of talks: 1

## 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.

### About Patrick Farrell

My research lies at the junction of mathematics, physics, engineering, and computation. In my thesis and early postdoctoral work, my research focussed on adaptive mesh discretisations: changing the computational mesh in some way to make orders of magnitude efficiency gains in solving complex models. In this work, I solved a problem of computational geometry that had been open in the literature for over twenty years. In my later postdoctoral work, I have focussed on the problem of automatically deriving adjoint models. Adjoints are absolutely essential for many important problems, such as weather prediction, optimising the shape of wings, and quantifying the accuracy of nuclear simulations. I have developed an entirely new approach to deriving adjoint models, which yields dramatic gains in automation, robustness and efficiency -- in some cases, from years to days. With these methods in hand, it is now possible to contemplate the automated solution of optimisation problems constrained by the laws of physics. Such applications are of huge interest and importance across all of engineering and the quantitative sciences.

Center for Biomedical Computing (CBC) aims to develop and apply novel simulation technologies to reach new understanding of complex physical processes affecting human health. We target selected medical problems where insight from mathematical modeling can contribute to changing clinical practice.