@article{642, abstract = {Cauchy problems with SPDEs on the whole space are localized to Cauchy problems on a ball of radius R. This localization reduces various kinds of spatial approximation schemes to finite dimensional problems. The error is shown to be exponentially small. As an application, a numerical scheme is presented which combines the localization and the space and time discretization, and thus is fully implementable.}, author = {Gerencser, Mate and Gyöngy, István}, issn = {00255718}, journal = {Mathematics of Computation}, number = {307}, pages = {2373 -- 2397}, publisher = {American Mathematical Society}, title = {{Localization errors in solving stochastic partial differential equations in the whole space}}, doi = {10.1090/mcom/3201}, volume = {86}, year = {2017}, } @inbook{649, abstract = {We give a short overview on a recently developed notion of Ricci curvature for discrete spaces. This notion relies on geodesic convexity properties of the relative entropy along geodesics in the space of probability densities, for a metric which is similar to (but different from) the 2-Wasserstein metric. The theory can be considered as a discrete counterpart to the theory of Ricci curvature for geodesic measure spaces developed by Lott–Sturm–Villani.}, author = {Maas, Jan}, booktitle = {Modern Approaches to Discrete Curvature}, editor = {Najman, Laurent and Romon, Pascal}, isbn = {978-3-319-58001-2}, issn = {978-3-319-58002-9}, pages = {159 -- 174}, publisher = {Springer}, title = {{Entropic Ricci curvature for discrete spaces}}, doi = {10.1007/978-3-319-58002-9_5}, volume = {2184}, year = {2017}, } @article{956, abstract = {We study a class of ergodic quantum Markov semigroups on finite-dimensional unital C⁎-algebras. These semigroups have a unique stationary state σ, and we are concerned with those that satisfy a quantum detailed balance condition with respect to σ. We show that the evolution on the set of states that is given by such a quantum Markov semigroup is gradient flow for the relative entropy with respect to σ in a particular Riemannian metric on the set of states. This metric is a non-commutative analog of the 2-Wasserstein metric, and in several interesting cases we are able to show, in analogy with work of Otto on gradient flows with respect to the classical 2-Wasserstein metric, that the relative entropy is strictly and uniformly convex with respect to the Riemannian metric introduced here. As a consequence, we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum Markov semigroups with detailed balance.}, author = {Carlen, Eric and Maas, Jan}, issn = {00221236}, journal = {Journal of Functional Analysis}, number = {5}, pages = {1810 -- 1869}, publisher = {Academic Press}, title = {{Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance}}, doi = {10.1016/j.jfa.2017.05.003}, volume = {273}, year = {2017}, } @inproceedings{989, abstract = {We present a generalized optimal transport model in which the mass-preserving constraint for the L2-Wasserstein distance is relaxed by introducing a source term in the continuity equation. The source term is also incorporated in the path energy by means of its squared L2-norm in time of a functional with linear growth in space. This extension of the original transport model enables local density modulations, which is a desirable feature in applications such as image warping and blending. A key advantage of the use of a functional with linear growth in space is that it allows for singular sources and sinks, which can be supported on points or lines. On a technical level, the L2-norm in time ensures a disintegration of the source in time, which we use to obtain the well-posedness of the model and the existence of geodesic paths. The numerical discretization is based on the proximal splitting approach [18] and selected numerical test cases show the potential of the proposed approach. Furthermore, the approach is applied to the warping and blending of textures.}, author = {Maas, Jan and Rumpf, Martin and Simon, Stefan}, editor = {Lauze, François and Dong, Yiqiu and Bjorholm Dahl, Anders}, issn = {03029743}, location = {Kolding, Denmark}, pages = {563 -- 577}, publisher = {Springer}, title = {{Transport based image morphing with intensity modulation}}, doi = {10.1007/978-3-319-58771-4_45}, volume = {10302}, year = {2017}, } @article{447, abstract = {We consider last passage percolation (LPP) models with exponentially distributed random variables, which are linked to the totally asymmetric simple exclusion process (TASEP). The competition interface for LPP was introduced and studied in Ferrari and Pimentel (2005a) for cases where the corresponding exclusion process had a rarefaction fan. Here we consider situations with a shock and determine the law of the fluctuations of the competition interface around its deter- ministic law of large number position. We also study the multipoint distribution of the LPP around the shock, extending our one-point result of Ferrari and Nejjar (2015).}, author = {Ferrari, Patrik and Nejjar, Peter}, journal = {Revista Latino-Americana de Probabilidade e Estatística}, pages = {299 -- 325}, publisher = {Instituto Nacional de Matematica Pura e Aplicada}, title = {{Fluctuations of the competition interface in presence of shocks}}, doi = {10.30757/ALEA.v14-17}, volume = {9}, year = {2017}, } @article{1448, abstract = {We develop a new and systematic method for proving entropic Ricci curvature lower bounds for Markov chains on discrete sets. Using different methods, such bounds have recently been obtained in several examples (e.g., 1-dimensional birth and death chains, product chains, Bernoulli–Laplace models, and random transposition models). However, a general method to obtain discrete Ricci bounds had been lacking. Our method covers all of the examples above. In addition we obtain new Ricci curvature bounds for zero-range processes on the complete graph. The method is inspired by recent work of Caputo, Dai Pra and Posta on discrete functional inequalities.}, author = {Fathi, Max and Maas, Jan}, journal = {The Annals of Applied Probability}, number = {3}, pages = {1774 -- 1806}, publisher = {Institute of Mathematical Statistics}, title = {{Entropic Ricci curvature bounds for discrete interacting systems}}, doi = {10.1214/15-AAP1133}, volume = {26}, year = {2016}, } @article{1261, abstract = {We consider a non-standard finite-volume discretization of a strongly non-linear fourth order diffusion equation on the d-dimensional cube, for arbitrary . The scheme preserves two important structural properties of the equation: the first is the interpretation as a gradient flow in a mass transportation metric, and the second is an intimate relation to a linear Fokker-Planck equation. Thanks to these structural properties, the scheme possesses two discrete Lyapunov functionals. These functionals approximate the entropy and the Fisher information, respectively, and their dissipation rates converge to the optimal ones in the discrete-to-continuous limit. Using the dissipation, we derive estimates on the long-time asymptotics of the discrete solutions. Finally, we present results from numerical experiments which indicate that our discretization is able to capture significant features of the complex original dynamics, even with a rather coarse spatial resolution.}, author = {Maas, Jan and Matthes, Daniel}, journal = {Nonlinearity}, number = {7}, pages = {1992 -- 2023}, publisher = {IOP Publishing Ltd.}, title = {{Long-time behavior of a finite volume discretization for a fourth order diffusion equation}}, doi = {10.1088/0951-7715/29/7/1992}, volume = {29}, year = {2016}, } @article{1635, abstract = {We calculate a Ricci curvature lower bound for some classical examples of random walks, namely, a chain on a slice of the n-dimensional discrete cube (the so-called Bernoulli-Laplace model) and the random transposition shuffle of the symmetric group of permutations on n letters.}, author = {Erbar, Matthias and Maas, Jan and Tetali, Prasad}, journal = {Annales de la faculté des sciences de Toulouse}, number = {4}, pages = {781 -- 800}, publisher = {Faculté des sciences de Toulouse}, title = {{Discrete Ricci curvature bounds for Bernoulli-Laplace and random transposition models}}, doi = {10.5802/afst.1464}, volume = {24}, year = {2015}, } @article{1639, abstract = {In this paper the optimal transport and the metamorphosis perspectives are combined. For a pair of given input images geodesic paths in the space of images are defined as minimizers of a resulting path energy. To this end, the underlying Riemannian metric measures the rate of transport cost and the rate of viscous dissipation. Furthermore, the model is capable to deal with strongly varying image contrast and explicitly allows for sources and sinks in the transport equations which are incorporated in the metric related to the metamorphosis approach by Trouvé and Younes. In the non-viscous case with source term existence of geodesic paths is proven in the space of measures. The proposed model is explored on the range from merely optimal transport to strongly dissipative dynamics. For this model a robust and effective variational time discretization of geodesic paths is proposed. This requires to minimize a discrete path energy consisting of a sum of consecutive image matching functionals. These functionals are defined on corresponding pairs of intensity functions and on associated pairwise matching deformations. Existence of time discrete geodesics is demonstrated. Furthermore, a finite element implementation is proposed and applied to instructive test cases and to real images. In the non-viscous case this is compared to the algorithm proposed by Benamou and Brenier including a discretization of the source term. Finally, the model is generalized to define discrete weighted barycentres with applications to textures and objects.}, author = {Maas, Jan and Rumpf, Martin and Schönlieb, Carola and Simon, Stefan}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis}, number = {6}, pages = {1745 -- 1769}, publisher = {EDP Sciences}, title = {{A generalized model for optimal transport of images including dissipation and density modulation}}, doi = {10.1051/m2an/2015043}, volume = {49}, year = {2015}, } @article{1517, abstract = {We study the large deviation rate functional for the empirical distribution of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer that this functional is asymptotically equivalent (in the sense of Γ-convergence) to the Jordan-Kinderlehrer-Otto functional arising in the Wasserstein gradient flow structure of the Fokker-Planck equation. In higher dimensions, part of this statement (the lower bound) has been recently proved by Duong, Laschos and Renger, but the upper bound remained open, since the proof of Duong et al relies on regularity properties of optimal transport maps that are restricted to one dimension. In this note we present a new proof of the upper bound, thereby generalising the result of Adams et al to arbitrary dimensions. }, author = {Erbar, Matthias and Maas, Jan and Renger, Michiel}, journal = {Electronic Communications in Probability}, publisher = {Institute of Mathematical Statistics}, title = {{From large deviations to Wasserstein gradient flows in multiple dimensions}}, doi = {10.1214/ECP.v20-4315}, volume = {20}, year = {2015}, }