Title: Multiscale inverse problems for partial differential equations and applications to sonar imaging
Abstract
A common objective in many data-driven sectors is to accurately describe intrinsic features of a complex process. This is a typical inverse problem for finding parameters in a model from given data, for example determining coefficients in partial differential equations (PDEs) from solution data. Inverse problems for PDEs pose daunting theoretical and computational challenges. For example, the classical inverse conductivity problem posed by Calderon is severely ill-posed, even in the case of smooth, isotropic coefficients. The situation is worse when modeling heterogeneous materials such as composites, lung airways and vasculature, and sedimentary layers in the Earth’s crust. For a variety of reasons, including the high cost of simulations and uncertainty in the measurements, the models are often simplified by a smoothing or homogenization process. Using the analysis of inverse conductivity problems, our results identify key parameters in highly oscillatory coefficients that withstand the loss of information due to homogenization. Multiscale methods for numerical homogenization are then used to efficiently predict the forward model while recovering microscale parameters. Ideas presented in this strategy can also be applied to solving inverse problems in ocean acoustics that aim to characterize properties of the ocean floor using sonar data. Here, forward solvers incorporate simulations of Helmholtz equations on a wide range of spatial scales, allowing for detailed recovery of seafloor parameters including the material type and roughness. In order to lower the computational cost of large-scale simulations, we take advantage of a library of representative acoustic responses from various seafloor configurations.
Title: Mapping class group action on character varieties and the ergodicity
Abstract: Character varieties of a surface are central objects in several beaches of math-
ematics, such as low dimensional topology, algebraic geometry, differential geom-
etry and mathematical physics. On the character varieties, there is a tautological
action of the mapping class group – the group of symmetries of the surface, which is expected to be ergodic in certain cases. In this talk, I will review related results
toward proving the ergodicity and introduce two long standing and related conjectures: Goldman’s Conjecture and Bowditch’s Conjecture. It is shown by Marche and Wolff that the two conjectures are equivalent for closed surfaces. For punctured surfaces, we disprove Bowditch’s Conjecture by giving counterexamples, yet prove that Goldman’s Conjecture is still true in this case.
The Sporadic Logic Seminar returns this week at a new place and time
(Fridays, 2:30, K404). This week Mushfeq Khan will speak:
Title: “The Homogeneity Conjecture”
Abstract: It is often said that the theorems and methods of recursion theory
relativize. One might go as far as to say that much of its analytical power
derives from this feature. However, this power is accompanied by definite
drawbacks: There are important examples of theorems and open questions
whose statements are non-relativizing, i.e., they have been shown to be
true relative to some oracles, and false relative to others. It follows
that these questions cannot be settled purely through relativizing methods.
A famous example of such a negative result is Baker, Gill, and Solovay’s
theorem on the P vs. NP question.
The observation that techniques based on diagonalization, effective
numbering, and simulation relativize led some recursion theorists (notably
Hartley Rogers, Jr) to formulate what became known as the “Homogeneity
Conjecture”. It said that for any Turing degree d, the partial order of
degrees that are above d is isomorphic to the entire partial order of the
Turing degrees. In 1979, Richard Shore refuted it in an elegant, one-page
article which will be the subject of this talk.
Title : Removability in Conformal Welding and Koebe’s Uniformization Conjecture
Abstract :
Ever since the seminal work of Ahlfors and Beurling in the middle of the 20th century, the study of removable plane sets with respect to various classes of analytic functions has proven over the years to be of fundamental importance to a wide variety of problems in complex analysis and geometric function theory. Questions revolving around necessary and sufficient geometric conditions for removability have held a prominent role in the development of valuable techniques, leading to deep results in various fields of mathematical analysis.
In recent years, attention has been drawn to the more modern notion of conformal removability, which continues to reveal connections with an ever-growing variety of central problems in complex analysis and related fields. Striking examples include injectivity of conformal welding, as well as the observation by He and Schramm in the 1990′s of the close relationship between conformal removability and Koebe’s uniformization conjecture.
The first part of the talk will consist of a brief introduction to conformal welding. I will discuss how removability appears naturally in the study of the injectivity of the welding correspondence.
In the second part of the talk, I will present new results on the conformal rigidity of circle domains and uniqueness in Koebe’s conjecture, following the work of He and Schramm.
Title: Topology of ends of finite volume, nonpositively curved manifolds
Abstract: The structure of ends of a nonpositively curved, locally symmetric manifold M is very well understood. By Borel-Serre, the thin part of the universal cover of such a manifold is homotopy equivalent to a rational Tits building. This is a simplicial complex built out of the algebra of the locally symmetric space which turns out to have dimension
Speaker: Gitta Kutyniok
Title: Applied Harmonic Analysis meets Sparse Regularization of Inverse Problems
Abstract:
Sparse regularization of inverse problems has already shown its effectiveness both theoretically and practically. The area of applied harmonic analysis offers a variety of systems such as wavelet systems which provide sparse approximations within certain model situations which then allows to apply this general approach provided that the solution belongs to this model class. However, many important problem classes in the multivariate situation are governed by anisotropic structures such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shear layers in solutions of transport dominated equations. Since it was shown that the (isotropic) wavelet systems are not capable of sparsely approximating such anisotropic features, the need arose to introduce appropriate anisotropic representation systems. Among various suggestions, shearlets are the most widely used today. Main reasons for this are their optimal sparse approximation properties within a suitable model situation in combination with their unified treatment of the continuum and digital realm, leading to faithful implementations.
In this talk, we will first provide an introduction to sparse regularization of inverse problems, followed by an introduction to the area of applied harmonic analysis, in particular, discussing the anisotropic representation system of shearlets and presenting the main theoretical results. We will then analyze the effectiveness of using shearlets for sparse regularization of exemplary inverse problems both theoretically and numerically.