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

Discrete probability density estimation using multirate DSP models

We propose a model based approach for estimation of probability mass functions for discrete random variables. The model is based on tools from multirate signal processing. Similar in principle to the kernel based methods, the approach takes advantage of well-known results from multirate signal processing theory. Similarities to and differences from wavelet based approaches is also indicated where appropriate. In the final form, the probability estimates are obtained by filtering the square root of the histogram through a multirate system whose components are biorthogonal partners of each other

Extreme events in discrete nonlinear lattices

We perform statistical analysis on discrete nonlinear waves generated though modulational instability in the context of the Salerno model that interpolates between the intergable Ablowitz-Ladik (AL) equation and the nonintegrable discrete nonlinear Schrodinger (DNLS) equation. We focus on extreme events in the form of discrete rogue or freak waves that may arise as a result of rapid coalescence of discrete breathers or other nonlinear interaction processes. We find power law dependence in the wave amplitude distribution accompanied by an enhanced probability for freak events close to the integrable limit of the equation. A characteristic peak in the extreme event probability appears that is attributed to the onset of interaction of the discrete solitons of the AL equation and the accompanied transition from the local to the global stochasticity monitored through the positive Lyapunov exponent of a nonlinear map.Comment: 5 pages, 4 figures; reference added, figure 2 correcte