MOX Seminar Series

Computing disconnected bifurcation diagrams of partial differential equations

Speaker: Patrick Farrell
Affiliation: Mathematical Institute, Oxford
When: Thursday 26th April 2018
Time: 14:00:00
Abstract: Computing the distinct solutions $u$ of an equation $f(u, \lambda) = 0$ as a parameter $\lambda \in \mathbb{R}$ is varied is a central task in applied mathematics and engineering. The solutions are captured in a bifurcation diagram, plotting (some functional of) $u$ as a function of $\lambda$. In this talk I will present a new algorithm, deflated continuation, for this task. Deflated continuation has three advantages. First, it is capable of computing disconnected bifurcation diagrams; previous algorithms only aimed to compute that part of the bifurcation diagram continuously connected to the initial data. Second, its implementation is extremely simple: it only requires a minor modification to any existing Newton-based solver. Third, it can scale to very large discretisations if a good preconditioner is available. Among other problems, we will apply this to a famous singularly perturbed ODE, Carrier's problem. The computations reveal a striking and beautiful bifurcation diagram, with an infinite sequence of alternating pitchfork and fold bifurcations as the singular perturbation parameter tends to zero. The analysis yields a novel and complete taxonomy of the solutions to the problem. We will also apply it to discover previously unknown solutions to equations arising in liquid crystals and quantum mechanics. Contact:
Note: I am an Associate Professor of Numerical Analysis and Scientific Computing and EPSRC Early Career Research Fellow in the Numerical Analysis group of the University of Oxford, and a Tutorial Fellow at Oriel College, Oxford. I work on the numerical solution of partial differential equations, with a particular focus on bifurcation analysis of nonlinear equations, the automated derivation and application of adjoint models, and the interaction between computational geometry and numerical simulation. I apply the numerical techniques I develop to various applications, including tidal turbines for renewable energy, bidomain cardiac electrophysiology, radiation transport, and glaciology. I mainly code in Python, contribute regularly to the FEniCS and PETSc software projects, and lead the development of dolfin-adjoint.