### 19 APRIL 2017 | SEMINAR

#### Abstract

Computing the solutions $u$ of an equation $f(u, \lambda) = 0$ as the parameter $\lambda$ is varied is a central task in applied mathematics and engineering. In this talk I will present a new algorithm, deflated continuation, for this task.

Deflated continuation has three main 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, and demonstrates that a claim of Bender & Orszag (1999) is incorrect. We will also use the algorithm to calculate distinct local minimisers of a topology optimisation problem via the combination of deflated continuation and a semismooth Newton method.

#### Personal bio of Prof Patrick Farrell

Employment: Associate Professor in Numerical Analysis and Scientific Computing Mathematical Institute , University of Oxford and Tutorial Fellow in Applied Mathematics – Oriel College, University of Oxford

Qualifications: PhD in Computational Physics – Imperial College London – Thesis title: Galerkin projection of discrete fields via supermesh construction

Prizes: Association of Computational Mechanics in Engineering 2010; Finalist and UK Representative, European Community on Computational Methods in Applied Sciences Award; Fox Prize, 2015, Wilkinson Prize 2015

## Details

Date: 19 APRIL 2017
Time: 2:00 pm