Abstract
This thesis is concerned with the approximation of various problems related to the stochastic Landau-Lifshitz-Gilbert equation (SLLG), which models the dynamics of a ferromagnetic body at elevated temperatures. The SLLG is a nonlinear stochastic partial differential equation which possesses an inherent non-convex side constraint. Firstly, the time discretization of the stochastic partial differential equation is addressed, where we study the convergence behavior of a structure-preserving discretization. Secondly, the approximation and simulation of the stochastic optimal control problem subject to the SLLG is studied by means of the necessary first order optimality conditions. The thesis is split into three parts. In the first part we focus on the time discretization of the SLLG. We show convergence in probability with rate of order 1/2 for a time discretized scheme which is based on the midpoint rule and preserves the sphere constraint. Main difficulties were the analytical and numerical treatment of the nonlinear and stochastic terms. Computational studies carried out in this part support this convergence rate. The second and the third part are contributed to the stochastic optimal control problem. In the second part, we prove strong convergence with optimal rates for a spatial discretization of the forward-backward stochastic heat equation which describes the stochastic optimal control problem subject to the stochastic heat equation. As an intermediate step, we show optimal rates for a spatial discretization of the backward stochastic heat equation. A full discretization which is based on the implicit Euler method for a temporal discretization and a least squares Monte-Carlo method is then proposed. Next to an iterative solution strategy which is based on a well-known Picard-type algorithm, the new stochastic gradient method turns out to be much more flexible. Concluding computational experiments compare the efficiency of different discretization approaches. The third part combines the methodology of the second part with the SLLG. Here, we control the dynamics of a fixed number of ferromagnetic spins at elevated temperatures by minimizing a quadratic functional subject to the SLLG. Existence of a minimum of the stochastic optimal control problem with control constraints is shown. The related first order optimality conditions consist of a coupled forward-backward SDE system, which is numerically solved by a structure-inheriting discretization, the least squares Monte-Carlo method to approximate related conditional expectations, and the stochastic gradient method. Computational experiments are reported which motivate optimal controls in the case of interacting anisotropy, stray field, exchange energies, and acting noise.
Author and committee
dc:creator, dc:contributor.*- Author
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- Dunst, Thomas
Identifiers
dc:identifier.*- Identifier
- hdl:10900/71846