Tu, November 26, 2024, 9:00-10:00 am PST via Zoom:
Giovanni Covi (University of Helsinki, Finland).
Title: Nonlocality in inverse problems.
Abstract: We will discuss some general aspects of inverse problems for nonlocal operators. In particular, we will consider the fundamental example of the fractional Calderòn problem, in which an electric potential has to be recovered from nonlocal Dirichlet-to-Neumann data. We will see how the nonlocality of the operator helps in the resolution of the problem, by allowing the use of a surprisingly powerful approximation technique. Finally, we will discuss some interesting applications, results and open problems.
Tu, November 12, 2024, 9:00-10:00 am PST via Zoom:
Lili Yan (University of Minnesota, USA).
Title: Inverse boundary problems for elliptic operators on Riemannian manifolds.
Abstract: In an inverse boundary problem, one seeks to determine the coefficients of a PDE inside a domain, describing internal properties, from the knowledge of boundary values of solutions of the PDE, encoding boundary measurements. Applications of such problems range from medical imaging to non-destructive testing. In this talk, starting with the fundamental Calderon inverse conductivity problem, we shall first discuss a partial data inverse boundary problem for the Magnetic Sch\"odinger operator in the setting of compact Riemannian manifolds with boundary. Next, we discuss first-order perturbations of biharmonic operators in the setting of compact Riemannian manifolds with boundary. Specifically, we shall present a global uniqueness result as well as a reconstruction procedure for the latter inverse boundary problem on conformally transversally anisotropic Riemannian manifolds of dimensions three and higher.
Tu, October 29, 2024, 9:00-10:00 am PDT via Zoom:
Yuzhou (Joey) Zou (Northwestern University, USA).
Title: The X-Ray Transform on Euclidean and Hyperbolic Disks via Projective Equivalence.
Abstract: We discuss recent works studying sharp mapping properties of weighted X-ray transforms on the Euclidean disk and hyperbolic disk. We are particularly interested in the mapping properties of weighted versions of normal operators associated to the X-ray transform and the behavior of such operators up to the boundary; the presence of weights sometimes improves such behavior. We prove a C^\infty isomorphism result (joint with R. Mishra and F. Monard) for certain weighted normal operators on the Euclidean disk by studying the spectrum of a distinguished Keldysh-type degenerate elliptic differential operator. We then discuss how to transfer these results to the hyperbolic disk (joint with N. Eptaminitakis and F. Monard), by using a projective equivalence between the Euclidean and hyperbolic disks via the Beltrami-Klein model, where one can view geodesics in the hyperbolic disk as Euclidean geodesics up to reparametrization.
Tu, October 15, 2024, 9:00-10:00 am PDT via Zoom:
Govanni Granados (The University of North Carolina at Chapel Hill, USA).
Title: Reconstruction of small and extended regions in EIT with a Robin transmission condition.
Abstract: In this talk, we will discuss some applications of the Regularized Factorization Method (RegFM) to a problem coming from Electrical Impedance Tomography (EIT) with a first-order Robin transmission condition. This method falls under the category of qualitative methods for inverse problems. Qualitative Methods are used in non-destructive testing where physical measurements on the surface or exterior of an object are used to infer the interior structure. In general, qualitative methods require little a priori knowledge of the interior structure or physical parameters. We assume that the Dirichlet-to-Neumann (DtN) mapping is given on the exterior boundary from an imposed voltage. Full knowledge of this DtN mapping allows us to reconstruct extended regions. We also discuss the asymptotic analysis of an integral equation involving the DtN mapping and apply a Multiple Signal Classification (MUSIC)-type algorithm to recover regions of small volume. We also consider the problem where we have a second-order Robin condition. For this problem, RegFM will be used to recover extended regions for the separate cases where the boundary parameters are complex-valued and real-valued. Numerical examples will be presented for all cases in two dimensions in the unit circle.
Tu, October 1, 2024, 9:00-10:00 am PDT via Zoom:
Yang Zhang (University of California Irvine, USA).
Title: Inverse Boundary Value Problems Arising in Nonlinear Acoustic Imaging.
Abstract: In nonlinear acoustic imaging, the propagation of ultrasound waves can be modeled using the Westervelt equation, a quasilinear wave equation. In this talk, we will discuss inverse problems related to this equation, particularly focusing on various damping effects. We will talk about determining both the nonlinearity and damping coefficients in two specific contexts: a weakly damped model and a strongly damped one. For the weakly damped Westervelt equation, our approach involves using multi-fold linearization and the nonlinear interactions of distorted plane waves, based on the work by Kurylev, Lassas, and Uhlmann. In the case of the strongly damped Westervelt equation, our strategy involves constructing a complex geometric optics solution and applying Hörmander's fundamental solutions to control the remainder term.
Tu, September 17, 2024, 9:00-10:00 am PDT:
Jan Bohr
(University of Bonn, Germany).
Title: Tomography and holomorphic vector bundles.
Abstract: In non-Abelian X-ray tomography one tries to recover a connection on a vector bundle from measurements of the parallel transport operator. For simple surfaces many aspects of this non-linear problem are now well-understood, including a general injectivity result and a range characterisation. In the talk I will discuss some of these developments from the viewpoint of twistor spaces and their holomorphic vector bundles.