| Popis: |
In this thesis, we consider numerical aspects concerning the simulation and strong approximation of solutions to McKean--Vlasov SDEs and interacting particle systems under non-standard assumptions, as well as numerical schemes for a broad class of mean-field control problems. First, we introduce novel adaptive time-stepping schemes and tamed Milstein schemes with the aim to achieve stable and strongly convergent discrete-time approximations to McKean--Vlasov equations and associated particle systems, where the drift and diffusion coefficients are allowed to grow super-linearly. The performance of the proposed time-stepping schemes is intensively investigated for various examples, including important models from neuroscience. Next, we study numerical schemes, as well as strong well-posedness, for a class of one-dimensional McKean--Vlasov SDEs with drifts that have a finite number of spatial discontinuities. We propose a transformation technique in combination with a fixed-point argument to show strong existence and uniqueness results. Further, we discuss different particle approximation methods and time-stepping schemes for such McKean--Vlasov equations and prove strong convergence results in the number of particles and time-steps, and provide numerical tests for models appearing in neuroscience and systemic risk. Finally, we develop a novel algorithm to obtain approximations of optimal feedback controls for control problems where the controlled state process is of McKean--Vlasov type. Here, we will employ deep-learning methods to solve these problems relying on a suitable formulation based on partial differential equations. An intensive numerical study of the proposed method is performed for high-dimensional mean-field Cucker--Smale models. |
