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# Weak convergence of numerical methods for SPDEs with applications to neurosciences

*This research is carried out in the framework of Matheon supported by Einstein Foundation Berlin (MATHEON Project CH-TU25)*

Project head: Raphael Kruse

Staff: Rico Weiske

Cooperations:

Andrea Barth (U Stuttgart), Wolf-Jürgen Beyn (U Bielefeld), Etienne Emmrich (TU Berlin), Elena Isaak (U Bielefeld), Stig Larsson (Chalmers), Wilhelm Stannat (TU Berlin), Michael Scheutzow (TU Berlin)

Stochastic partial differential equations (SPDEs) have been one of the most active areas of research within probability theory since the early 1970s. Since then they have attracted the attention of many experts from different disciplines within mathematics and physics such as interacting particle systems, fluid dynamics, nonlinear filtering or statistical physics, just to name a few. Numerical methods for SPDEs have also been studied more extensively during the last two decades. They have proven themselves to be valuable in bridging the SPDE theory to a rich variety of applications, for example, in bond market models and environmental pollution models, in segmentation methods in image processing or in models which are concerned with the fiber lay-down dynamics in the production process of nonwovens. In this project we focus on the numerical approximation of models arising in the modelling of an axon in neurosciences.

For instance, we consider stochastic versions of the Hodgkin-Huxley equations, FitzHugh-Nagumo equations, or the Nagumo equation. The considered numerical methods combine discretization strategies such as

- Galerkin finite element methods
- Temporal time stepping methods (Euler-Maruyama, Milstein schemes)
- Multilevel Monte Carlo methods
- Noise discretization.

The goals of the project involve the development of computationally tractable fully discrete numerical schemes for the simulation of stochastic reaction diffusion equations and their mathematical justification.

## Publications

R. Kruse and M. Scheutzow

A discrete stochastic Gronwall lemma

Math. Comp. Simulation, Vol. **143**, (2018), pp. 149-157

DOI: 10.1016/j.matcom.2016.07.002

ArXiv: 1601.07503

W.-J. Beyn, E. Isaak and R. Kruse

Stochastic C-stability and B-consistency of explicit and implicit Milstein-type schemes *J. Sci. Comput., Vol. 70 (3), (2017), pp. 1042-1077*

DOI: 10.1007/s10915-016-0290-x

ArXiv: 1512.06905

A. Andersson and R. Kruse

Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition*BIT Numer. Math., Vol. 57 (1), (2017), pp. 21-53*

DOI: 10.1007/s10543-016-0624-y

ArXiv: 1509.00609

A. Andersson, R. Kruse and S. Larsson

Duality in refined Sobolev-Malliavin spaces and weak approximations of SPDE

Stoch. Partial Differ. Equ. Anal. Comput. (2015) (Online First)

DOI: 10.1007/s40072-015-0065-7

Arxiv: 1312.5893

W-J. Beyn, E. Isaak and R. Kruse

Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes

J. Sci. Comput. (2015) (Online first)

DOI: 10.1007/s10915-015-0114-4

Arxiv: 1411.6961