Inverse boundary problems for elliptic operators on conformally transversally anisotropic manifolds

Speaker: 

Lili Yan

Speaker Link: 

https://sites.google.com/view/liliyan/

Institution: 

University of Minnesota

Time: 

Friday, May 26, 2023 - 1:00pm to 1:50pm

Location: 

RH 306

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 on CTA manifolds. Next, we discuss first-order perturbations of biharmonic operators in the same geometric. Specifically, we shall present a global uniqueness result as well as a reconstruction procedure for the latter inverse boundary problem. 

Inverse boundary value problems arising in nonlinear acoustic imaging

Speaker: 

Yang Zhang

Institution: 

University of Washington

Time: 

Tuesday, April 18, 2023 - 3:00pm to 3:50pm

Location: 

340P

Inverse problems of recovering the metric and nonlinear terms were originated in the work by Kurylev, Lassas, and Uhlmann for the semi-linear wave equation $\square_g u(x) + a(x)u^2(x) = f(x)$ in a manifold without boundary. The idea is to use the multi-fold linearization and the nonlinear interactions of distorted planes waves to produce point-source-like singularities in an observable set. In this talk, I will discuss joint works with Gunther Uhlmann, which consider the recovery of the nonlinearity for a quasilinear wave equation arising in nonlinear acoustic imaging. The main difficulty that we need to handle here is caused by the presence of the boundary.

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