Berlin 2024 – scientific programme

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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik

MP 7: Poster (joint session MP/QI)

MP 7.10: Poster

Tuesday, March 19, 2024, 11:00–13:00, Poster B

Numerical simulations of stochastic optimal control model for navigation of finite size microswimmers — •Malte Thumann — Institut für Numerische und Angewandte Mathematik, Universität Göttingen

Using stochastic optimal control theory, we study the optimal navigation of finite size microswimmers in the presence of a fluid flow and thermal fluctuations in two-dimensional space. The resulting Hamilton-Jacobi-Bellmann (HJB) equation is a nonlinear convection-diffusion type partial differential equation (PDE) that describes the optimal torque an active swimmer must satisfy to navigate towards a desired target. This equation is numerically solvable in a three-dimensional phase space (position and orientation) for a given set of initial conditions. We discretise the HJB equation in a finite element framework known as the discontinuous Galerkin method, which operates over a trial space of functions that are only piecewise continuous. This allows for a more stable and flexible discretisation scheme, in particular to cope with the challenging task of implementing singular boundary conditions arising from the stochastic optimal control approach. Using the optimal torque solution, we perform stochastic simulations to determine the optimal mean microswimmer path. Our work emphasises that finite element methods are a suitable discretisation technique to handle PDEs arising in theoretical biophysics in a non-trivial setting with complex geometries, singularities, or higher order local approximations.

Keywords: Stochastic optimal control theory; Finite element method; Discontinuous Galerkin method; Navigation of active particles; Stochastic simulations