Structure-preserving numerical methods for stochastic differential equations

In this talk, geometric numerical integration meets stochastic numerics. In particular, the attention is focused on recent advances regarding the numerical preservation of qualitative/quantitative aspects and invariance laws characterizing the dynamics of various stochastic ordinary and partial differential equations. The talk moves towards the following two tracks:

-geometric numerical integration of stochastic Hamiltonian problems. For these problems, two different scenarios are clarified: if the noise is driven in the Ito sense, the expected Hamiltonian function exhibits a linear drift in time; in the Stratonovich case, the Hamiltonian is pathwise preserved. In both cases, the talk aims to highlight the attitude of selected numerical methods in preserving the aforementioned behaviors. A long-term investigation via backward error analysis is also presented;

-structure-preserving numerics of dissipative problems. The investigation moves towards the numerical conservation of mean-square contractivity in the time integration of dissipative problems, via stochastic theta-methods. The analysis shows that numerical contractivity in mean-square sense is hidden within proper stepsize restrictions.

A general guideline is given by the assessment of a bridge between numerical modelling of stochastic problems and that for the underlying deterministic ones. This talk is based on the joint research in collaboration with Chuchu Chen (Chinese Academy of Sciences), David Cohen (Chalmers University of Technology & University of Gothenburg), Stefano Di Giovacchino (University of L’Aquila), and Annika Lang (Chalmers University of Technology & University of Gothenburg).

Contatti:
luca.formaggia@polimi.it