Abstract. Recent developments in statistical transport modeling, ranging from rarefied gas dynamics, collisional plasmas and electron transport in nanostructures, to self-organized or social interacting dynamics, share a common description
based in a Markovian framework of birth and death processes. Under the regime of molecular chaos propagation,
their evolution is given by kinetic equations of non-linear collisional (integral) Boltzmann type. We will present an
overview of analytical issues and novel numerical methods for these equations that preserve the expected conserved
properties of the described phenomena, while enabling rigorous stability, convergence and error analysis.
Abstract. Radar imaging is a technology that has been developed, very successfully, within the engineering community during the last 50 years. Radar systems on satellites now make beautiful images of regions of our earth and of other planets such as Venus. One of the key components of this impressive technology is mathematics, and many of the open problems are mathematical ones. This lecture will explain, from first principles, some of the basics of radar and the mathematics involved in producing high-resolution radar images.
Abstract. The adiabatic theorem gives conditions that guarantee that a system defined by Schrödinger's equation remains in its ground state when started in its ground state and evolved slowly. Realistically, such systems are subject to perturbations in the initial condition, systematic time-dependent perturbations in the Hamiltonian, coupling to low-energy quantum systems, and decoherent time-dependent perturbations in the Hamiltonian. Using Wilkinson-style perturbation analysis, we derive bounds on the effects of these perturbations. This is joint work with Michael J. O'Hara.
Abstract. Guided by a geometric understanding developed in earlier works of Wang and Young, we carry out some numerical studies of shear-induced chaos. The settings considered include periodic kicking of limit cycles, random kicks at Poisson times, and continuous-time driving by white noise. The forcing of a quasi-periodic model describing two coupled oscillators is also investigated. In all cases, positive Lyapunov exponents are found in suitable parameter ranges when the forcing is suitably directed.
Abstract. Many current questions in applied analysis are motivated by issues in the physical sciences or engineering. They require state-of-the-art techniques, new ideas, and the introduction of innovative tools in the calculus of variations, differential equations, and geometric measure theory. This talk will focus on the variational approach as it is used to treat problems on foams, imaging, micromagnetics, and thin structures.
Abstract. This is a story that involves many strands and many players, and like all such stories, it can be told in many ways. I will start by picking up a strand involving superfast wavelet transforms. As the story unfolds, other strands and players will be brought in, to discuss superfast as well as other effective ways in which to compute sparse transforms from what might, at first sight, seem incomplete data. It is a story that features many young researchers, and in which the speaker herself plays an incidental role only.
Abstract. The palpation exam, where the doctor presses against the skin to locate abnormal tissue, senses tissue stiffness changes. We create images of shear stiffness changes using data from Mathias Fink's transient elastography experiment. That experiment yields time and space dependent interior displacements of a propagating elastic wave using an ultrafast ultrasound based imaging technique. In this talk, we present well-posedness results, algorithms and images utilizing positions of the propagating front of the wave.
Abstract. In microscopic systems formed by living cells, small numbers of reactant molecules can result in dynamical behavior that is discrete and stochastic rather than continuous and deterministic. In simulating and analyzing such behavior it is essential to employ methods that directly take into account the underlying discrete stochastic nature of the molecular events. This leads to an accurate description of the system that in many important cases is impossible to obtain through deterministic continuous modeling (e.g. ODEs). Gillespie’s Stochastic Simulation Algorithm (SSA) has been widely used to treat these problems. However as a procedure that simulates every reaction event, it is prohibitively inefficient for most realistic problems.
We report on our progress in developing a multiscale computational framework for the numerical simulation of chemically reacting systems, where each reaction will be treated at the appropriate scale. The framework is based on a sequence of approximations ranging from SSA at the smallest scale, through a ``birth-death’’ Markov process approximation, Gillespie’s recently-developed tau-leaping approximation, a continuous stochastic differential equation (SDE) approximation, and finally to the familiar reaction rate equations (ODEs) at the coarsest scales.