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Inverse Problems for Elliptic Operators
Inverse Problems for Elliptic Operators
In this minicourse, we will provide an introduction to the field of inverse problems for elliptic PDEs, covering both classical results and recent advancements. We will start with the renowned Calderon problem, which seeks to determine the electrical conductivity of a medium from voltage and current measurements on its boundary. This problem forms the basis for Electrical Impedance Tomography, an imaging modality with applications in seismic and medical imaging. We will present global uniqueness results for this and related problems, addressing both full and partial data cases, using complex geometric optics solutions, and Carleman estimate techniques.
Next, we will explore inverse boundary problems for elliptic PDEs in transversally anisotropic Riemannian geometries. We will also address inverse problems for nonlinear elliptic PDEs, both in Euclidean domains and on Riemannian manifolds. In particular, we will see that the presence of nonlinearity may actually help, allowing one to solve inverse problems in situations where the corresponding linear counterpart remains open. Finally, we will discuss inverse problems for nonlocal elliptic operators on compact Riemannian manifolds without boundary. Throughout the minicourse, we will also state several open problems in the field.
Introduction to ray transforms
Introduction to ray transforms
This short course gives an introduction to the mathematics of X-ray tomography or the study of the X-ray transform. Instead of diving deep into a single method of inverting the X-ray transform, we will explore a variety of different methods.
The lecture will be based on the lecture notes of Joonas Ilmavirta.
Inverse Problems Arising from Hyperbolic PDEs
Inverse Problems Arising from Hyperbolic PDEs
In Reflection seismology, one uses the seismic energy to probe beneath the surface of the Earth, usually as an aid in searching for economic deposits of oil, gas, or minerals, but also for engineering, archeological, and scientific studies. Typically, this is done by setting up explosive charges or seismic vibrators in some area of the surface and trying to receive the echoes of the reflections of the seismic waves on some measurement area. The geophysical exploration process can be formulated in the language of hyperbolic Partial Differential Equations (PDE), with the goal of finding the unknown coefficients of the PDE from a boundary measurement. Since physical quantities are coordinate invariant, it is convenient to model planet Earth by a compact, connected Riemannian manifold with boundary. Under these assumptions a fundamental hyperbolic inverse boundary value problem is to recover the unknown geometric structure from the hyperbolic Dirichlet-to-Neumann map. This can be accomplished by reducing the PDE-based problem to a geometric problem which carries information about the unknown coefficients.
Research Presentations
Research Presentations
Towards Spectral Convergence of Locally Linear Embedding on Manifolds with Boundary, Andrew Lyon, UNC Chappel Hill
Travel Time Tomography via Microlocal Analysis, Joye Zou, North Western University
Lipschitz Stability of the Travel Time Data, Andrew Shedlock, NC State University
Recovery of Time-dependent Coefficients in Hyperbolic Equations on Riemannian Manifolds from Partial Data, Boya Liu, North Dakota State University
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