Computing the solutions of a nonlinear equation as a parameter 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 two main advantages over previous approaches. 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, and does not require solving any new auxiliary problems. As a consequence, it can scale to very large discretisations if a good preconditioner is available.
We will demonstrate the utility of the new algorithm by using it to discover previously unknown solutions to several problems of physical interest.