Couplings, metrics and contraction rates for Langevin diffusions Andreas Eberle, University of Bonn, Germany
March 23, 2018
11:20 am-12:20 pm
Armitage 121

Carefully constructed Markovian couplings and specifically designed Kantorovich metrics can be used to derive relatively precise bounds on the distance between the laws of two Langevin processes. In the case of two overdamped Langevin diffusions with the same drift, the processes are coupled by reflection, and the metric is an L1 Wasserstein distance based on an appropriately chosen concave distance function. If the processes have different drifts, then the reflection coupling can be replaced by a „sticky coupling“ where the distance between the two copies is bounded from above by a one-dimensional diffusion process with a sticky boundary at 0. This new type of coupling leads to long-time stable bounds on the total variation distance between the two laws. Similarly, two kinetic Langevin processes can be coupled using a particular combination of a reflection and a synchronous coupling that is sticky on a hyperplane. Again, the coupling distance is contractive on average w.r.t. an appropriately designed Wasserstein distance. This can be applied to derive new bounds of kinetic order for convergence to equilibrium at the borderline between the overdamped and the underdamped regime. Similar approaches are also useful when studying mean-field interacting particle systems, McKean-Vlasov diffusions, or diffusions on infinite dimensional state spaces. (joint work with Arnaud Guillin and Raphael Zimmer).

Lecture with Stephen McKeown, Princeton University
April 6, 2018
3 pm
319 Cooper Street, Room 110

Abstract coming soon.

Lecture with Sönmez Şahutoğlu, University of Toledo April 20, 2018
3 pm
319 Cooper Street, Room 110

Abstract coming soon.

Past Seminars

“Metastability of the Nonlinear Wave Equation” Katherine Newhall, UNC
February 23, 2018
11:20 am- 12:20 pm
Armitage 121

I will discuss the long-time dynamics of infinite energy solutions to a wave equation with nonlinear forcing. Of particular interest is when these solutions display metastability in the sense that they spend long periods of time in disjoint regions of phase-space and only rarely transition between them. This phenomenon is quantified by calculating exactly via Transition State Theory (TST) the mean frequency at which the solutions of the nonlinear wave equation with initial conditions drawn from its invariant measure cross a dividing surface lying in between the metastable sets. Numerical results suggest a regime for which the dynamics are not fundamentally different from that observed in the stochastic counterpart in which random noise and damping terms are added to the equation, as well as a regime for which successive transitions between the metastable sets are correlated and the coarse-graining to a Markov chain fails.

“Pricing Uber’s Marketplace”
James Davis, Senior Data Scientist at Uber & a 2010 graduate of Rutgers–Camden Mathematics
February 16, 2018
11:20 am- 12:20 pm
Penn 401 (Note special location)

Dynamic pricing at Uber is a key part of what makes the services so reliable. We will see how dynamic pricing works in an up-close way. First, we will walk through pricing during a typical day in Philadelphia, Then, after we explain the key concepts behind dynamic pricing, we will show these concepts in action after the Eagles Superbowl win. We will show just how the Uber market evolved and how pricing responded. If there is enough time we will discuss some big upcoming challenges for Uber’s pricing system.

“Kinetics of Particles with Short-range Interactions” Miranda Holmes-Cerfon, NYU Courant
February 9, 2018
11:20 am- 12:20 pm
Armitage 121

Particles in soft-matter systems (such as colloids) tend to have very short-range interactions, so traditional theories, that assume the energy landscape is smooth enough, will struggle to capture their dynamics. We propose a new framework to look at such particles, based on taking the limit as the range of the interaction goes to zero. In this limit, the energy landscape is a set of geometrical manifolds plus a single control parameter, while the dynamics on top of the manifolds are given by a hierarchy of Fokker-Planck equations coupled by “sticky” boundary conditions. We show how to compute dynamical quantities such as transition rates between clusters of hard spheres, and then show this agrees quantitatively with experiments. The framework may also be used to more efficiently ask questions about programmable self-assembly.

“Stochastic Simulation of Multiscale Reaction-Diffusion Models via First Exit Times” Lina Meinecke, Department of Mathematics at UC Irvine
January 26, 2018
11:20 am- 12:20 pm
Armitage 121

Mathematical models are important tools in systems biology, since the regulatory networks in biological cells are too complicated to understand by biological experiments alone. However, analytical solutions can be derived only for the simplest models and hence numerical simulations are often necessary.
This talk focuses on the mesoscopic simulation level, which captures both, space dependent behavior by diffusion and the inherent stochasticity of cellular systems. Space is partitioned into compartments by a mesh and the number of molecules of each species in each compartment gives the state of the system. We first examine how to compute the jump coefficients for a discrete stochastic jump process on unstructured meshes from a first exit time approach guaranteeing the correct speed of diffusion. Furthermore, we analyze different methods leading to non-negative coefficients by backward analysis and derive a new method, minimizing both the error in the diffusion coefficient and in the particle distribution.
The second part investigates macromolecular crowding effects. A high percentage of the cytosol and membranes of cells are occupied by molecules. This impedes the diffusive motion and also affects the reaction rates. Most algorithms for cell simulations are either derived for a dilute medium or become computationally very expensive when applied to a crowded environment. Therefore, we develop a multiscale approach, which takes the microscopic positions of the molecules into account, while still allowing for efficient stochastic simulations on the mesoscopic level. Finally, we compare on- and off-lattice models on the microscopic level when applied to a crowded environment.

“Symplectic Runge-Kutta schemes for adjoint equations, automatic differentiation, optimal control and more” Jesús María Sanz-Serna, Universidad Carlos III de Madrid, Spain
November 17, 2017
11:20 am- 12:20 pm
Armitage 123

Symplectic Runge-Kutta (RK) schemes were introduced in the 1980’s to integrate numerically Hamiltonian systems of differential equations. As it turns out, unbeknownst to the user, symplectic RK schemes implicitly appear in a number of applications, including automatic differentiation, optimal control, etc. The talk explains in very general terms the links between symplectic integration and those application areas.

“The Ins and Outs of the Diffusion Approximation”
Charles Epstein, Department of Mathematics at UPENN
October 17, 2017
12:45 pm- 1:45 pm
Science Building Lecture Hall

Population Genetics is the study of how the distribution of different types of individuals evolves in a reproducing population. The standard models incorporate effects of random mating, mutation, selection, and migration. In their simplest form these models are discrete Markov chains that model a finite population. These discrete models are very difficult to analyze and compute with, and so methods were developed to replace the discrete models with continuum models allowing for the usage of calculus. We describe both the discrete and continuous models and how they are related, as well as recent work on the analysis and numerics of the continuous models.

“Notes on the Minkowski Measure, the Minkowski Symmetral, and the Banach-Mazur Distance”
Xing Huang, Ph.D. student from China
September 29, 2017
11:20 am- 12:20 pm
Armitage 123

In this talk we derive some basic inequalities connecting the Minkowski measure of symmetry, the Minkowski symmetral and the Banach-Mazur distance. We then explore the geometric contents of these inequalities and shed light on the structure of the quotient of the space of convex bodies modulo affine transformations.