Scalable bifurcation analysis of nonlinear partial differential equations

Time: 16:50 - 17:10  Computing the solutions of an equation as a parameter is varied is a central task in applied mathematics and engineering. In this work I will present a new algorithm, called deflated continuation, for... [ view full abstract ]

Time: 16:50 - 17:10  

Computing the solutions of an equation as a parameter is varied is a central task in applied mathematics and engineering. In this work I will present a new algorithm, called deflated continuation, for this task, and describe a massively parallel implementation of it built on top of FEniCS.

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 our software 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.