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4.0

Jun 30, 2018
06/18

by
Markus Klein; Andreas Prohl

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We consider an optimal control problem subject to the thin-film equation which is deduced from the Navier--Stokes equation. The PDE constraint lacks well-posedness for general right-hand sides due to possible degeneracies; state constraints are used to circumvent this problematic issue and to ensure well-posedness, and the rigorous derivation of necessary optimality conditions for the optimal control problem is performed. A multi-parameter regularization is considered which addresses both, the...

Topics: Mathematics, Analysis of PDEs, Optimization and Control

Source: http://arxiv.org/abs/1410.2077

4
4.0

Jun 30, 2018
06/18

by
Chuchu Chen; Jialin Hong; Andreas Prohl

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We propose a $\theta$-scheme to discretize the $d$-dimensional stochastic cubic Schr\"odinger 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.

Topics: Mathematics, Numerical Analysis

Source: http://arxiv.org/abs/1410.6231

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44

Sep 23, 2013
09/13

by
Xiaobing Feng; Yukun Li; Andreas Prohl

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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...

Source: http://arxiv.org/abs/1303.5930v1

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61

Sep 21, 2013
09/13

by
John W. Barrett; Xiaobing Feng; Andreas Prohl

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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

Source: http://arxiv.org/abs/0712.2528v1

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38

Sep 18, 2013
09/13

by
Lubomir Banas; Zdzislaw Brzezniak; Martin Ondrejat; Andreas Prohl

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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.

Source: http://arxiv.org/abs/1102.0728v1

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3.0

Jun 30, 2018
06/18

by
L'ubomír Banas; Zdzisław Brzeźniak; Misha Neklyudov; Martin Ondreját; Andreas Prohl

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We study ergodic properties of stochastic geometric wave equations on a particular model with the target being the 2D sphere while considering the space variable-independent solutions only. This simplification leads to a degenerate stochastic equation in the tangent bundle of the 2D sphere. Studying this equation, we prove existence and non-uniqueness of invariant probability measures for the original problem and we obtain also results on attractivity towards an invariant measure. We also...

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1403.6811