Wednesdays at 4:15 PM in 380C.
Fourier phasing is the problem of retrieving Fourier phase information from Fourier intensity data. The standard Fourier phase retrieval (without a mask) is known to have many solutions which cause the standard phasing algorithms to stagnate and produce wrong or inaccurate solutions. In this talk Fourier phase retrieval is carried out with the introduction of a randomly fabricated mask in measurement and reconstruction. Highly probable uniqueness of solution, up to a global phase, was previously proved with exact knowledge of the mask. Here the uniqueness result is extended to the case where only rough information about the mask’s phases is assumed. The exponential probability bound for uniqueness is given in terms of the uncertainty-to-diversity ratio of the unknown mask. New phasing algorithms alternating between the object update and the mask update are systematically tested and demonstrated to have the capability of recovering both the object and the mask (within the object support) simultaneously, consistent with the uniqueness result. Phasing with a phase-uncertain mask is shown to be robust with respect to the correlation in the mask as well as the Gaussian and Poisson noises.
It is now well known that the Green's function between two receivers can be estimated from the cross correlation of the signals emitted by ambient noise sources and recorded by the receivers. It is also well known that a reflector can be imaged by migration of the cross correlation matrix of the signals emitted by ambient noise sources and recorded by a passive receiver array. In this talk we extend these results to situations in which the sensors are moving. We show that it is possible to use moving sensors to generate large synthetic apertures. We also exhibit a surprising super-resolution phenomenon when the sensor velocities become of the order of the speed of propagation.
This talk surveys some recent results related to the derivation of the cubic nonlinear Schroedinger equation in R^3, and the rel\ ated Cauchy problem for Gross-Pitaevskii (GP) hierarchies. Moreover, a new approach to the analysis of GP hierarchies using the \ quantum de Finetti theorem is presented, based on which we obtain a new proof of unconditional uniqueness of solutions, as well \ as a proof of scattering in the defocusing case (joint work with C. Hainzl, N. Pavlovic and R. Seiringer).
We consider a diffusion on a potential landscape which is given by a smooth Hamiltonian in the regime of small noise. We give a new proof of the Eyring-Kramers formula for the spectral gap of the associated generator of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald, Otto, Villani, and Westdickenberg and of the mean-difference estimate introduced by Chafai and Malrieu. The Eyring-Kramers formula follows as a simple corollary from two main ingredients : The first one shows that the Gibbs measures restricted to a domain of attraction has a "good" Poincaré constant mimicking the fast convergence of the diffusion to metastable states. The second ingredient is the estimation of the mean-difference by a new weighted transportation distance. It contains the main contribution of the spectral gap, resulting from exponential long waiting times of jumps between metastable states of the diffusion. This new approach also allows to derive sharp estimates on the log-Sobolev constant.
The study of shock waves propagating in biological tissue and bone has several biomedical applications. In lithotripsy, focused shock waves are used to pulverize kidney stones without surgery, while in Extracorporeal Shock Wave Therapy (ESWT), focused shock waves of smaller amplitude are used to stimulate healing and bone growth. On the negative side, blast-induced traumatic brain injury (TBI) affects countless veterans and civilians who have survived nearby explosions. In this talk I will describe some of these applications and efforts to obtain a better understanding of the affect of wave propagation on biological media. High-resolution wave propagation algorithms can robustly handle interfaces between different materials, while methods recently developed for poroelasticity may be valuable in studying wave propagation in bone.
The problem of model reduction for complex systems is an active area of research. In this talk I want to discuss how two concepts inspired by physics, namely scale dependence and renormalization, can be used to facilitate the construction of accurate reduced models for complex problems. In particular, I will be presenting the application of these concepts to the problem of detecting and tracking singularities of time-dependent partial differential equations. Results for the inviscid Burgers, the critical nonlinear Schrödinger and the incompressible Euler equations will be used to illustrate the constructions.
Historically, the theory of interest rates derivatives has served as a source for the commodity markets. As a first step this makes sense since term structures in both worlds share some of their properties. However, looking more closely, the economics driving the dynamics of yield curves and commodity forward curves have their own particularities. We will briefly discuss some of those particularities in the case of commodity markets and try to make sense of the economics that are relevant to their dynamics.
We design an asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions --- a leading-order elastic collision together with a lower-order interparticle collision. When the mean free path is small, numerically solving this equation is prohibitively expensive due to the stiff collision terms. Furthermore, since the equilibrium solution is a (zero-momentum) Fermi-Dirac distribution resulting from joint action of both collisions, the simple BGK penalization designed for the one-scale collision fails to capture the correct energy-transport limit. We present two methods to solve this problem: one is to set up a threshold on the penalization of the elastic collision such that the evolution of the solution resembles a Hilbert expansion at the continuous level; the other is based on operator splitting wherein the collisions at different scales are treated separately. Formal asymptotic analysis and numerical results confirm the efficiency and accuracy of the proposed scheme. This is joint work with Shi Jin and Li Wang.
We propose a system to describe the influence of a "road" with fast diffusion on biological invasions. Outside of the road a classical Fisher-KPP propagation with a different diffusion takes place. It is found that, if the ratio between the two diffusivities is above an explicit threshold, the road enhances the asymptotic speed of propagation in a cone of directions around the road. Outside this cone the speed of propagation coincides with the classical Fisher-KPP invasion speed. A description of the asymptotic shape of expansion is then derived.
I will describe recent results in stochastic homogenization for divergence-form, uniformly elliptic equations. Included are nonlinear equations arising as the first variation of uniformly convex energy functionals. The random field of coefficients is typically assumed to satisfy a finite range of dependence condition. Dal Maso and Modica proved a qualitative homogenization theorem covering this case. We are concerned with developing a quantitative theory-- that is, understanding the precise size and nature of the fluctuations of the solutions (and their energy density) from the homogenized limit. Here new ideas are needed, since the qualitative proof of Dal Maso and Modica was based on an abstract ergodic theorem which does not easily quantify. In joint work with Charles Smart, we introduce a method which leads to eventually to optimal quantitative estimates. Specializing to the case in which the Euler-Lagrange equation is linear, we get a new proof of some recent results of Gloria, Neukamm and Otto as well as Marahrens and Otto. Some of the top-level ideas are parallel to those we have recently developed for non-divergence form equations, which I may also briefly review.
Ensembles of solutions to partial differential equations underly mathematical theories of turbulence and phase transitions. However, our understanding of such ensembles for even well-understood PDE is quite modest. For example, there is no satisfactory theory of a soliton gas for NLS or KdV despite a detailed understanding of these PDE that ranges from the classical (explicit N-solitons) to the modern (optimal well-posedness theorems). Perhaps the simplest nonlinear PDE are scalar conservation laws. I will present a fairly complete conceptual picture for the evolution of certain random data by these PDE that has emerged over the past few years. The punchline is that the evolution of the law of the solution is determined by a completely integrable system of kinetic equations that describes the clustering of shocks. This approach allows the introduction of powerful methods from integrable systems, and suggests new lines of attack on related problems.
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