Andreas Prohl

Xiaobing Feng, Yukun Li, Andreas Prohl

This paper develops and analyzes a semi-discrete and a fully discrete finite element method for a one-dimensional quasilinear parabolic stochastic partial differential equation (SPDE) which describes the stochastic mean curvature flow for planar curves of graphs. To circumvent the difficulty caused by the low spatial regularity of the SPDE solution, a regularization procedure is first proposed to approximate the SPDE, and an error estimate for the regularized problem is derived. A semi-discrete finite element method, and a space-time fully discrete method are then proposed to approximate the solution of the regularized SPDE problem. Strong convergence with rates are established for both, semi- and fully discrete methods. Computational experiments are provided to study the interplay of the geometric evolution and gradient type-noises.

John W. Barrett, Xiaobing Feng, Andreas Prohl

Motivated by emerging applications from imaging processing, the heat flow of a generalized $p$-harmonic map into spheres is studied for the whole spectrum, $1\leq p<\infty$, in a unified framework. The existence of global weak solutions is established for the flow using the energy method together with a regularization and a penalization technique. In particular, a $BV$-solution concept is introduced and the existence of such a solution is proved for the 1-harmonic map heat flow. The main idea used to develop such a theory is to exploit the properties of measures of the forms $\cA\cdot\nab\bv$ and $\cA\wedge\nab\bv$; which pair a divergence-$L^1$, or a divergence-measure, tensor field $\cA$, and a $BV$-vector field $\bv$. Based on these analytical results, a practical fully discrete finite element method is then proposed for approximating weak solutions of the $p$-harmonic map heat flow, and the convergence of the proposed numerical method is also established.

Lubomir Banas, Zdzislaw Brzezniak, Martin Ondrejat et al.

We give a sufficient condition on nonlinearities of an SDE on a compact connected Riemannian manifold $M$ which implies that laws of all solutions converge weakly to the normalized Riemannian volume measure on $M$. This result is further applied to characterize invariant and ergodic measures for various SDEs on manifolds.

Max Jensen, Ananta Majee, Andreas Prohl et al.

We use optimal control via a distributed exterior field to steer the dynamics of an ensemble of N interacting ferromagnetic particles which are immersed into a heat bath by minimizing a quadratic functional. By using dynamic programing principle, we show the existence of a unique strong solution of the optimal control problem. By the Hopf-Cole transformation, the related Hamilton-Jacobi-Bellman equation from dynamic programming principle may be re-cast into a linear PDE on the manifold M = (S^2)^N, whose classical solution may be represented via Feynman-Kac formula. We use this probabilistic representation for Monte-Carlo simulations to illustrate optimal switching dynamics.

Chuchu Chen, Jialin Hong, Andreas Prohl

We propose a $θ$-scheme to discretize the $d$-dimensional stochastic cubic Schrödinger equation in Stratono\-vich sense. A uniform bound for the Hamiltonian of the discrete problem is obtained, which is a crucial property to verify the convergence in probability towards a mild solution. Furthermore, based on the uniform bounds of iterates in ${\mathbb H}^2(\mathcal{O})$ for $\mathcal{O}\subset\mathbb{R}^{1}$, the optimal convergence order 1 in strong local sense is obtained.