The purpose of this project is the development of new numerical techniques for PDEs driven by
(deterministic or random) rough paths. Clearly, the numerical analysis for stochastic PDEs –
typically driven by finite or infinite dimensional Brownian motion or white noise – is a
well-established field with a long history. On the other hand, little is known about the numerical
treatment of PDEs driven by genuine rough paths, beyond the Brownian setting. In fact, even in the
case of now well-understood rough ordinary differential equations there are still open questions.
However, some of the techniques proposed for the theoretical analysis of rough PDEs can lead to
viable approaches to their computational solution, as well. We have identified two distinct promising
techniques that we are going to explore in this project: The first is based on the application of
semi-groups to rough PDEs and their approximation by Galerkin finite element methods. The
second is based on Feynman–Kac representations and the numerical approximation of the underlying
stochastic differential equation, combined with spatial regression. Both techniques have been used in
the past to give sense to certain classes of rough PDEs, see Deya–Gubinelli–Tindel (2012) or the
recent book by Friz–Hairer (2014, Chapter 12), respectively. In this project we evolve the two
approaches into new implementable numerical methods and focus on questions like error analysis and
computational complexity.
Post-Docs: M. Redmann (WIAS), Y. Wu (TU Berlin; jointly with P4) |
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