New Frontiers in Mathematical Physics
January 19th, 2016
University of Tsukuba / 筑波大学
13:30-14:30 Matti Lassas (University of Helsinki)
An Inverse Problems for a Non-linear Wave Equation and Inverse Problems in General Relativity
Abstract: We consider inverse problem for a non-linear wave equation with a time-depending metric tensor on manifolds. In addition, we study the question, do the observation of the solutions of coupled Einstein equations and matter field equations in an open subset $U$ of the space-time $M$ corresponding to sources supported in $U$ determine the properties of the metric in a maximal domain where waves can propagate from $U$ and return back to $U$. To study these problems we define the concept of light observation sets and show that these sets determine the conformal class of the metric. The results have been done in collaboration with Yaroslav Kurylev and Gunther Uhlmann. In addition to the above results we discuss the problems of geodesic tomography encountered in study of cosmic microwave background. These results are done in collaboration with Lauri Oksanen, Plamen Stefanov, and Gunther Uhlmann.
14:45-15:45 Sumio Yamada (Gakushuin University)
On Riemannian Geometry of the Einstein Equation
Abstract: In this talk, we pose the Einstein/Einstein-Maxwell equation as a Cauchy problem, and look at the moduli space of the Cauchy data, each of which consists of a three dimensional Riemannian manifold and a deformation tensor. Needless to say, the structure of the moduli space is elusive, but we use the so-call Penrose-type inequality to characterize the space. In doing so, the known exact solutions to the Einstein equations play an important role, which is closely related to the Cosmic Censorship first proposed by R. Penrose in the 1970's. This is a collaborative work with Marcus Khuri and Gilbert Weinstein.
16:15-17:15 Tapio Helin (University of Helsinki)
Maximum a posteriori estimates in Bayesian inversion
Abstract: A demanding challenge in Bayesian inversion is to efficiently characterize the posterior distribution. This task is problematic especially in high-dimensional non-Gaussian problems, where the structure of the posterior can be very chaotic and difficult to analyse. Current inverse problem literature often approaches the problem by considering suitable point estimators for the task. Here we discuss the maximum a posteriori (MAP) estimate and its definition for infinite-dimensional problems. Moreover, we consider how Bregman distance can be used to characterize the MAP estimate. This is joint work with Martin Burger.
This meeting is Session 5 of the 2nd CiRfSE Workshop.
この研究集会は 第2回 CiRfSE ワークショップ のセッション 5 です。