Tu, March 18, 2025, 9:00-10:00 am PDT via Zoom:
Manuel Cañizares (Johann Radon Institute for Computational and Applied Mathematics, Austria).
Title: Indentifying electric potentials via the local near-field scattering pattern at fixed energy.
Abstract: We study the inverse scattering problem with electric potentials. We prove that local measurements of electromagnetic waves at fixed energies can uniquely determine a rough compactly supported potential in dimension n ≥ 3.
By rough, we mean that the potential can be decomposed into a part that
lives in L^{n/2}, a part that is supported in a compact hypersurface, and a part
that corresponds to the sth derivative of an L^∞ function, with s < 1.
We will review how Harmonic Analysis plays into solving the forward problem, but we will center the talk in the solution of the inverse problem. Caro and Garcia proved in 2020 that measuring waves at a fixed energy on a sphere surrounding the potential would give its unique determination. To extend these results to smaller set of measurements, in this case to a small hypersurface in the vicinity of the potential, we prove a Runge approximation result via unique continuation and interior regularity arguments.
Also, solving of a Neumann problem for the Helmholtz equation is key in
the proof of this Runge approximation. We will show how domain perturbation
techniques allow us to find a solution to this boundary value problem.
Tu, March 4, 2025, 9:00-10:00 am PST via Zoom:
Spyridon Filippas (University of Helsinki, Finland).
Title: On unique continuation for Schrödinger operators.
Abstract: We are interested in the following question: a solution of the linear time dependent Schrödinger equation vanishing in a small open set during a small time does it vanishe everywhere? In the case where the operator includes a potential the answer to this question depends on its regularity. We will present a result under the assumption that the potential has a Gevrey 2 regularity with respect to time. This relaxes the analyticity assumption known previously. This is a joint work with Camille Laurent and Matthieu Léautaud.
Tu, February 18, 2025, 9:00-10:00 am PST via Zoom:
Hjørdis Schlüter (University of Helsinki, Finland).
Title: The second step in hybrid inverse problems in limited view.
Abstract: Hybrid inverse problems combine two imaging modalities in order to make the reconstruction procedure more “well-posed”. They typically consist of two steps: One step to obtain internal data and another step to reconstruct the desired material parameter. In this talk I will focus on hybrid inverse problems that combine Electrical Impedance Tomography (EIT) with another imaging modality, ultrasound waves or magnetic resonance imaging, in order to reconstruct the electrical conductivity. Additionally, I consider a limited view setting, where one only has control over a part of the boundary. Relative to classical EIT these hybrid imaging techniques suffice with only two EIT measurements. However, the two boundary functions imposed on a part of the boundary for the EIT procedure should be chosen carefully, so that the corresponding internal data contains enough information for reconstruction of the conductivity. In this talk I will address under what conditions on the boundary functions this is the case, and I will go through a numerical example.
Tu, February 4, 2025, 9:00-10:00 am PST via Zoom:
Antti Kykkänen (Rice University, USA).
Title: Geometry of gas giants and inverse problems.
Abstract: In this talk, I will introduce a Riemannian geometric model for wave propagation in gas giant planets. Terrestrial planets and gas giants have one key difference: in gas, density goes to zero at the surface, and seismic waves come to a full stop. We model the sound speed in a planet by a Riemannian metric. Starting from a polytropic model for the planet, we derive that the vanishing of the density ammounts to a specific conformal-type singularity in the Riemannian metric.
We will highlight the key differences between the arising geometry and its more studied relatives. We finish with an overview of inverse problems results in our new geometry. The talk is based on joint work with Maarten de Hoop (Rice University), Joonas Ilmavirta (University of Jyväskylä), and Rafe Mazzeo (Stanford University).
Tu, January 21, 2025, 9:00-10:00 am PST via Zoom:
Hadrian Quan (University of California, Santa Cruz, USA).
Title: The anisotropic Calderón problem for the fractional Dirac operator.
Abstract: In this talk I will discuss joint work with Gunther Uhlmann regarding the anisotropic fractional Calderon problem for Dirac operators on closed manifolds; these give fractional analogues of Maxwell systems. Namely we show that knowledge of the source-to-solution map of the fractional Dirac operator, for data sources supported in an arbitrary open set in a Riemannian manifold allows one to reconstruct the Riemannian manifold, its Clifford module structure, and the associated connection (up to an isometry fixing the initial set).
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.