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P3: Numerical analysis of rough PDEs
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.
Principal investigators: C. Bayer  (WIAS), R. Kruse  (TU Berlin), J. Schoenmakers  (WIAS)