We prove strong convergence with optimal rates for a spatial discretization of the backward stochastic heat equation, and the forward-backward stochastic heat equation from stochastic optimal control. A full discretization based on the implicit Euler method for a temporal discretization, and a least squares Monte-Carlo method, in combination with the new stochastic gradient method are then proposed, and simulation results are reported.

We consider parabolic stochastic partial differential equations driven by space-time Lévy noise. Different discretization methods to accurately simulate jumps are proposed and analyzed in the context of an implicit time discretization. Computational studies based on a finite element discretization are provided to illustrate combined truncation and time-discretization effects.

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Abstract. We consider an optimal control problem subject to the Landau-Lifshitz-Gilbert equation which describes the evolution of magnetizations in S 2 . The problem is motivated in order to control switching processes of ferromagnets. Existence of an optimum and the first order necessary optimality system are derived. We show (up to subsequences) convergence of state, adjoint and control variables of a time discretization (semi-implicit Euler method) for vanishing time step size. A main step here is to verify corresponding stability properties for the semi-discrete state, which is nontrivial since the iterates take values which only approximate S 2 . We use a perturbation argument within a variational discretization in order to show error bounds for the semi-discrete state variables, from which we may then infer uniform bounds for the semi-discrete state and also adjoint variables. Numerical studies underline these results and compare this discretization with a further variant which bases on a projection strategy for the state equation to enhance iterates to better approximate S 2 .