Dynamical Optimal Transport on Discrete Surfaces

We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finite-dimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspective, our technique provides a smooth interpolation between distributions on discrete surfaces with less diffusion than state-of-the-art algorithms involving entropic regularization. Beyond interpolation, we show how our discrete notion of optimal transport extends to other tasks, such as distribution-valued Dirichlet problems and time integration of gradient flows

Fear or freedom? Visually impaired students’ ambivalent perspectives on physical education

With a growing interest in sport, fitness, and a healthy lifestyle, bodily practices are increasing in importance in our society. In the school context, physical education (PE) is the subject where these practices play a central role. But, the German language discourse shows in an exemplary manner that inherent body-related social normality requirements are articulated in didactic traditions and curricular requirements, and that these normality requirements have exclusionary potential for those students who do not fit into the norms. Against this background, this article seeks to understand children with visual impairments’ (CWVI’s) individual constructions of PE in a school specialized for CWVI in Germany. This interview study with eight CWVI focused on individual opportunities and challenges concerning central aspects in PE. The findings show that the CWVI draw ambivalent perspectives on PE that range from existential fears (e.g., fears of heights) to feeling free in working off energy. These aspects especially gain importance in connection to the body, when the general wish to learn and experience with the body seems to be disturbed by normality requirements – like doing certain movements in a pre-defined way – which lead to existential challenges for the CWVI. Further, the relationship between blind and visually impaired students in PE seems ambivalent. Within this special school setting, the segregation according to the external differentiation in “handicapped” and “non-handicapped” somehow leads to a kind of subsegregation at the blind and visually impaired school.Peer Reviewe

Stochastic Wasserstein Barycenters

We present a stochastic algorithm to compute the barycenter of a set of probability distributions under the Wasserstein metric from optimal transport. Unlike previous approaches, our method extends to continuous input distributions and allows the support of the barycenter to be adjusted in each iteration. We tackle the problem without regularization, allowing us to recover a sharp output whose support is contained within the support of the true barycenter. We give examples where our algorithm recovers a more meaningful barycenter than previous work. Our method is versatile and can be extended to applications such as generating super samples from a given distribution and recovering blue noise approximations.Comment: ICML 201

Natural parameter conditions for singular perturbations of chemical and biochemical reaction networks

We consider reaction networks that admit a singular perturbation reduction in a certain parameter range. The focus of this paper is on deriving "small parameters" (briefly for small perturbation parameters), to gauge the accuracy of the reduction, in a manner that is consistent, amenable to computation and permits an interpretation in chemical or biochemical terms. Our work is based on local timescale estimates via ratios of the real parts of eigenvalues of the Jacobian near critical manifolds; this approach is familiar from computational singular perturbation theory. While parameters derived by this method cannot provide universal estimates for the accuracy of a reduction, they represent a critical first step toward this end. Working directly with eigenvalues is generally unfeasible, and at best cumbersome. Therefore we focus on the coefficients of the characteristic polynomial to derive parameters, and relate them to timescales. Thus we obtain distinguished parameters for systems of arbitrary dimension, with particular emphasis on reduction to dimension one. As a first application, we discuss the Michaelis--Menten reaction mechanism system in various settings, with new and perhaps surprising results. We proceed to investigate more complex enzyme catalyzed reaction mechanisms (uncompetitive, competitive inhibition and cooperativity) of dimension three, with reductions to dimension one and two. The distinguished parameters we derive for these three-dimensional systems are new; in fact no rigorous derivation of small parameters seems to exist in the literature so far. Numerical simulations are included to illustrate the efficacy of the parameters obtained, but also to show that certain limitations must be observed.Comment: 57 pages, 17 figure

Large-D Expansion from Variational Perturbation Theory

We derive recursively the perturbation series for the ground-state energy of the D-dimensional anharmonic oscillator and resum it using variational perturbation theory (VPT). From the exponentially fast converging approximants, we extract the coefficients of the large-D expansion to higher orders. The calculation effort is much smaller than in the standard field-theoretic approach based on the Hubbard-Stratonovich transformation.Comment: Author Information under http://hbar.wustl.edu/~sbrandt and http://www.theo-phys.uni-essen.de/tp/ags/pelster_di

Dynamical optimal transport on discrete surfaces

We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finitedimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspective, our technique provides a smooth interpolation between distributions on discrete surfaces with less diffusion than state-of-the-art algorithms involving entropic regularization. Beyond interpolation, we show how our discrete notion of optimal transport extends to other tasks, such as distribution-valued Dirichlet problems and time integration of gradient flows