@article{12959, abstract = {This paper deals with the large-scale behaviour of dynamical optimal transport on Zd -periodic graphs with general lower semicontinuous and convex energy densities. Our main contribution is a homogenisation result that describes the effective behaviour of the discrete problems in terms of a continuous optimal transport problem. The effective energy density can be explicitly expressed in terms of a cell formula, which is a finite-dimensional convex programming problem that depends non-trivially on the local geometry of the discrete graph and the discrete energy density. Our homogenisation result is derived from a Γ -convergence result for action functionals on curves of measures, which we prove under very mild growth conditions on the energy density. We investigate the cell formula in several cases of interest, including finite-volume discretisations of the Wasserstein distance, where non-trivial limiting behaviour occurs.}, author = {Gladbach, Peter and Kopfer, Eva and Maas, Jan and Portinale, Lorenzo}, issn = {1432-0835}, journal = {Calculus of Variations and Partial Differential Equations}, number = {5}, publisher = {Springer Nature}, title = {{Homogenisation of dynamical optimal transport on periodic graphs}}, doi = {10.1007/s00526-023-02472-z}, volume = {62}, year = {2023}, } @article{11700, abstract = {This paper contains two contributions in the study of optimal transport on metric graphs. Firstly, we prove a Benamou–Brenier formula for the Wasserstein distance, which establishes the equivalence of static and dynamical optimal transport. Secondly, in the spirit of Jordan–Kinderlehrer–Otto, we show that McKean–Vlasov equations can be formulated as gradient flow of the free energy in the Wasserstein space of probability measures. The proofs of these results are based on careful regularisation arguments to circumvent some of the difficulties arising in metric graphs, namely, branching of geodesics and the failure of semi-convexity of entropy functionals in the Wasserstein space.}, author = {Erbar, Matthias and Forkert, Dominik L and Maas, Jan and Mugnolo, Delio}, issn = {1556-181X}, journal = {Networks and Heterogeneous Media}, number = {5}, pages = {687--717}, publisher = {American Institute of Mathematical Sciences}, title = {{Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph}}, doi = {10.3934/nhm.2022023}, volume = {17}, year = {2022}, } @article{11739, abstract = {We consider finite-volume approximations of Fokker--Planck equations on bounded convex domains in $\mathbb{R}^d$ and study the corresponding gradient flow structures. We reprove the convergence of the discrete to continuous Fokker--Planck equation via the method of evolutionary $\Gamma$-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalizing the one-dimensional result obtained by Disser and Liero. The proof is of variational nature and relies on a Mosco convergence result for functionals in the discrete-to-continuum limit that is of independent interest. Our results apply to arbitrary regular meshes, even though the associated discrete transport distances may fail to converge to the Wasserstein distance in this generality.}, author = {Forkert, Dominik L and Maas, Jan and Portinale, Lorenzo}, issn = {1095-7154}, journal = {SIAM Journal on Mathematical Analysis}, keywords = {Fokker--Planck equation, gradient flow, evolutionary $\Gamma$-convergence}, number = {4}, pages = {4297--4333}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Evolutionary $\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions}}, doi = {10.1137/21M1410968}, volume = {54}, year = {2022}, } @article{10023, abstract = {We study the temporal dissipation of variance and relative entropy for ergodic Markov Chains in continuous time, and compute explicitly the corresponding dissipation rates. These are identified, as is well known, in the case of the variance in terms of an appropriate Hilbertian norm; and in the case of the relative entropy, in terms of a Dirichlet form which morphs into a version of the familiar Fisher information under conditions of detailed balance. Here we obtain trajectorial versions of these results, valid along almost every path of the random motion and most transparent in the backwards direction of time. Martingale arguments and time reversal play crucial roles, as in the recent work of Karatzas, Schachermayer and Tschiderer for conservative diffusions. Extensions are developed to general “convex divergences” and to countable state-spaces. The steepest descent and gradient flow properties for the variance, the relative entropy, and appropriate generalizations, are studied along with their respective geometries under conditions of detailed balance, leading to a very direct proof for the HWI inequality of Otto and Villani in the present context.}, author = {Karatzas, Ioannis and Maas, Jan and Schachermayer, Walter}, issn = {1526-7555}, journal = {Communications in Information and Systems}, keywords = {Markov Chain, relative entropy, time reversal, steepest descent, gradient flow}, number = {4}, pages = {481--536}, publisher = {International Press}, title = {{Trajectorial dissipation and gradient flow for the relative entropy in Markov chains}}, doi = {10.4310/CIS.2021.v21.n4.a1}, volume = {21}, year = {2021}, } @article{8758, abstract = {We consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reaction-rate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailed-balance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradient-flow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailed-balance steady state. The limit of large volumes is studied in the sense of evolutionary Γ-convergence of gradient flows. Moreover, we use the gradient structures to derive hybrid models for coupling different modeling levels.}, author = {Maas, Jan and Mielke, Alexander}, issn = {15729613}, journal = {Journal of Statistical Physics}, number = {6}, pages = {2257--2303}, publisher = {Springer Nature}, title = {{Modeling of chemical reaction systems with detailed balance using gradient structures}}, doi = {10.1007/s10955-020-02663-4}, volume = {181}, year = {2020}, } @article{71, abstract = {We consider dynamical transport metrics for probability measures on discretisations of a bounded convex domain in ℝd. These metrics are natural discrete counterparts to the Kantorovich metric 𝕎2, defined using a Benamou-Brenier type formula. Under mild assumptions we prove an asymptotic upper bound for the discrete transport metric Wt in terms of 𝕎2, as the size of the mesh T tends to 0. However, we show that the corresponding lower bound may fail in general, even on certain one-dimensional and symmetric two-dimensional meshes. In addition, we show that the asymptotic lower bound holds under an isotropy assumption on the mesh, which turns out to be essentially necessary. This assumption is satisfied, e.g., for tilings by convex regular polygons, and it implies Gromov-Hausdorff convergence of the transport metric.}, author = {Gladbach, Peter and Kopfer, Eva and Maas, Jan}, issn = {10957154}, journal = {SIAM Journal on Mathematical Analysis}, number = {3}, pages = {2759--2802}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Scaling limits of discrete optimal transport}}, doi = {10.1137/19M1243440}, volume = {52}, year = {2020}, } @article{7573, abstract = {This paper deals with dynamical optimal transport metrics defined by spatial discretisation of the Benamou–Benamou formula for the Kantorovich metric . Such metrics appear naturally in discretisations of -gradient flow formulations for dissipative PDE. However, it has recently been shown that these metrics do not in general converge to , unless strong geometric constraints are imposed on the discrete mesh. In this paper we prove that, in a 1-dimensional periodic setting, discrete transport metrics converge to a limiting transport metric with a non-trivial effective mobility. This mobility depends sensitively on the geometry of the mesh and on the non-local mobility at the discrete level. Our result quantifies to what extent discrete transport can make use of microstructure in the mesh to reduce the cost of transport.}, author = {Gladbach, Peter and Kopfer, Eva and Maas, Jan and Portinale, Lorenzo}, issn = {00217824}, journal = {Journal de Mathematiques Pures et Appliquees}, number = {7}, pages = {204--234}, publisher = {Elsevier}, title = {{Homogenisation of one-dimensional discrete optimal transport}}, doi = {10.1016/j.matpur.2020.02.008}, volume = {139}, year = {2020}, } @article{6358, abstract = {We study dynamical optimal transport metrics between density matricesassociated to symmetric Dirichlet forms on finite-dimensional C∗-algebras. Our settingcovers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein–Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, andspectral gap estimates.}, author = {Carlen, Eric A. and Maas, Jan}, issn = {15729613}, journal = {Journal of Statistical Physics}, number = {2}, pages = {319--378}, publisher = {Springer Nature}, title = {{Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems}}, doi = {10.1007/s10955-019-02434-w}, volume = {178}, year = {2020}, } @unpublished{10022, abstract = {We consider finite-volume approximations of Fokker-Planck equations on bounded convex domains in R^d and study the corresponding gradient flow structures. We reprove the convergence of the discrete to continuous Fokker-Planck equation via the method of Evolutionary Γ-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalising the one-dimensional result obtained by Disser and Liero. The proof is of variational nature and relies on a Mosco convergence result for functionals in the discrete-to-continuum limit that is of independent interest. Our results apply to arbitrary regular meshes, even though the associated discrete transport distances may fail to converge to the Wasserstein distance in this generality.}, author = {Forkert, Dominik L and Maas, Jan and Portinale, Lorenzo}, booktitle = {arXiv}, pages = {33}, title = {{Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions}}, year = {2020}, } @article{73, abstract = {We consider the space of probability measures on a discrete set X, endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset Y⊆X, it is natural to ask whether they can be connected by a constant speed geodesic with support in Y at all times. Our main result answers this question affirmatively, under a suitable geometric condition on Y introduced in this paper. The proof relies on an extension result for subsolutions to discrete Hamilton-Jacobi equations, which is of independent interest.}, author = {Erbar, Matthias and Maas, Jan and Wirth, Melchior}, issn = {09442669}, journal = {Calculus of Variations and Partial Differential Equations}, number = {1}, publisher = {Springer}, title = {{On the geometry of geodesics in discrete optimal transport}}, doi = {10.1007/s00526-018-1456-1}, volume = {58}, year = {2019}, } @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{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}, } @article{2131, abstract = {We study approximations to a class of vector-valued equations of Burgers type driven by a multiplicative space-time white noise. A solution theory for this class of equations has been developed recently in Probability Theory Related Fields by Hairer and Weber. The key idea was to use the theory of controlled rough paths to give definitions of weak/mild solutions and to set up a Picard iteration argument. In this article the limiting behavior of a rather large class of (spatial) approximations to these equations is studied. These approximations are shown to converge and convergence rates are given, but the limit may depend on the particular choice of approximation. This effect is a spatial analogue to the Itô-Stratonovich correction in the theory of stochastic ordinary differential equations, where it is well known that different approximation schemes may converge to different solutions.}, author = {Hairer, Martin M and Jan Maas and Weber, Hendrik}, journal = {Communications on Pure and Applied Mathematics}, number = {5}, pages = {776 -- 870}, publisher = {Wiley-Blackwell}, title = {{Approximating Rough Stochastic PDEs}}, doi = {10.1002/cpa.21495}, volume = {67}, year = {2014}, } @article{2132, abstract = {We consider discrete porous medium equations of the form ∂tρt=Δϕ(ρt), where Δ is the generator of a reversible continuous time Markov chain on a finite set χ, and ϕ is an increasing function. We show that these equations arise as gradient flows of certain entropy functionals with respect to suitable non-local transportation metrics. This may be seen as a discrete analogue of the Wasserstein gradient flow structure for porous medium equations in ℝn discovered by Otto. We present a one-dimensional counterexample to geodesic convexity and discuss Gromov-Hausdorff convergence to the Wasserstein metric.}, author = {Erbar, Matthias and Jan Maas}, journal = {Discrete and Continuous Dynamical Systems- Series A}, number = {4}, pages = {1355 -- 1374}, publisher = {Southwest Missouri State University}, title = {{Gradient flow structures for discrete porous medium equations}}, doi = {10.3934/dcds.2014.34.1355 }, volume = {34}, year = {2014}, }