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Dive into the research topics where Annika Lang is active.

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Featured researches published by Annika Lang.


Stochastics An International Journal of Probability and Stochastic Processes | 2012

Simulation of stochastic partial differential equations using finite element methods

Andrea Barth; Annika Lang

These notes describe numerical issues that may arise when implementing a simulation method for a stochastic partial differential equation (SPDE). It is shown that an additional approximation of the noise does not necessarily affect the order of convergence of a discretization method for a SPDE driven by Lévy noise. Furthermore, finite element methods are explicitly given and simulations are done. In statistical tests, it is shown that the simulations obey the theoretical orders of convergence.


International Journal of Computer Mathematics | 2012

Multilevel Monte Carlo method with applications to stochastic partial differential equations

Andrea Barth; Annika Lang

In this work, the approximation of Hilbert-space-valued random variables is combined with the approximation of the expectation by a multilevel Monte Carlo (MLMC) method. The number of samples on the different levels of the multilevel approximation are chosen such that the errors are balanced. The overall work then decreases in the optimal case to O(h −2) if h is the error of the approximation. The MLMC method is applied to functions of solutions of parabolic and hyperbolic stochastic partial differential equations as needed, for example, for option pricing. Simulations complete the paper.


Computational Statistics & Data Analysis | 2012

A new similarity measure for nonlocal filtering in the presence of multiplicative noise

Tanja Teuber; Annika Lang

A new similarity measure and nonlocal filters for images corrupted by multiplicative noise are presented. The considered filters are generalizations of the nonlocal means filter of Buades et al., which is known to be well suited for removing additive Gaussian noise. To adapt this filter to different noise models, the involved patch comparison has first of all to be performed by a suitable noise dependent similarity measure. For this purpose, a recently proposed probabilistic measure for general noise models by Deledalle et al. is studied. This measure is analyzed in the context of conditional density functions and its properties are examined for data corrupted by additive and multiplicative noise. Since it turns out to have unfavorable properties for multiplicative noise, a new similarity measure is deduced consisting of a probability density function specially chosen for this type of noise. The properties of this new measure are studied theoretically as well as by numerical experiments. To finally obtain nonlocal filters, a weighted maximum likelihood estimation framework is applied, which also incorporates the noise statistics. Moreover, the weights occurring in these filters are defined using the new similarity measure and different adaptations are proposed to further improve the results. Finally, restoration results for images corrupted by multiplicative Gamma and Rayleigh noise are presented to demonstrate the very good performance of these nonlocal filters.


international conference on scale space and variational methods in computer vision | 2011

Nonlocal filters for removing multiplicative noise

Tanja Teuber; Annika Lang

In this paper, we propose nonlocal filters for removing multiplicative noise in images. The considered filters are deduced in a weighted maximum likelihood estimation framework and the occurring weights are defined by a new similarity measure for comparing data corrupted by multiplicative noise. For the deduction of this measure we analyze a probabilistic measure recently proposed for general noise models by Deledalle et al. and study its properties in the presence of additive and multiplicative noise. Since it turns out to have unfavorable properties facing multiplicative noise we propose a new similarity measure consisting of a density specially chosen for this type of noise. The properties of our new measure are examined theoretically as well as by numerical experiments. Afterwards, it is applied to define the weights of our nonlocal filters and different adaptations are proposed to further improve the results. Throughout the paper, our findings are exemplified for multiplicative Gamma noise. Finally, restoration results are presented to demonstrate the good properties of our new filters.


Stochastics An International Journal of Probability and Stochastic Processes | 2010

Almost sure convergence of a semidiscrete Milstein scheme for SPDEs of Zakai type

Annika Lang; Pao Liu Chow; Jürgen Potthoff

A semidiscrete Milstein scheme for stochastic partial differential equations of Zakai type on a bounded domain of is derived. It is shown that the order of convergence of this scheme is 1 for convergence in mean square sense. For almost sure convergence, the order of convergence is proved to be for any .


international conference on conceptual structures | 2010

Mean square convergence of a semidiscrete scheme for SPDEs of Zakai type driven by square integrable martingales

Annika Lang

In this short note, a direct proof of L2 convergence of an Euler-Maruyama approximation of a Zakai equation driven by a square integrable martingale is shown. The order of convergence is as known for real-valued stochastic differential equations and for less general driving noises O(√Δt) for a time discretization step size Δt.


Potential Analysis | 2014

Kolmogorov-Chentsov Theorem and Differentiability of Random Fields on Manifolds

Roman Andreev; Annika Lang

A version of the Kolmogorov–Chentsov theorem on sample differentiability and Hölder continuity of random fields on domains of cone type is proved, and the result is generalized to manifolds.


Journal of Computational and Applied Mathematics | 2010

A Lax equivalence theorem for stochastic differential equations

Annika Lang

In this paper, a stochastic mean square version of Laxs equivalence theorem for Hilbert space valued stochastic differential equations with additive and multiplicative noise is proved. Definitions for consistency, stability, and convergence in mean square of an approximation of a stochastic differential equation are given and it is shown that these notions imply similar results as those known for approximations of deterministic partial differential equations. Examples show that the assumptions made are met by standard approximations.


arXiv: Probability | 2013

Covariance structure of parabolic stochastic partial differential equations

Annika Lang; Stig Larsson; Christoph Schwab

In this paper parabolic random partial differential equations and parabolic stochastic partial differential equations driven by a Wiener process are considered. A deterministic, tensorized evolution equation for the second moment and the covariance of the solutions of the parabolic stochastic partial differential equations is derived. Well-posedness of a space–time weak variational formulation of this tensorized equation is established.


Stochastics An International Journal of Probability and Stochastic Processes | 2012

Erratum to Almost sure convergence of a semi-discrete Milstein scheme for SPDEs of Zakai type

Annika Lang; Pao Liu Chow; Jürgen Potthoff

References [1] A. Barth and A. Lang, L and almost sure convergence of a Milstein scheme for stochastic partial differential equation, (2009), SAM report 2011-15. [2] A. Barth and A. Lang, Milstein approximation for advection-diffusion equations driven multiplicative noncontinuous martingale noises, (2011), SAM report 2011-36. [3] A. Jentzen, private communication. [4] A. Jentzen and M. Röckner, A break of the complexity of the numerical approximation of nonlinear SPDEs with multiplicative noise, (2010), arXiv: math.NA.1001.2751. [5] A. Lang, P.L. Chow, and J. Potthoff, Almost sure convergence of a semidiscrete Milstein scheme for SPDEs of Zakai type, Stochastics 82 (2010), pp. 315–326.

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Andrea Barth

University of Stuttgart

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Andreas Petersson

Chalmers University of Technology

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Stig Larsson

Chalmers University of Technology

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Roman Andreev

Austrian Academy of Sciences

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Kristin Kirchner

Chalmers University of Technology

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