In this talk, I will discuss the Calderón Problem with magnetic/electromagnetic perturbations in both the Riemannian and Lorentzian settings. For both of them, we consider the Laplace-Beltrami operator with lower order (electro)magnetic terms, and ask what information about the metric can be recovered when a family of Dirichlet-to-Neumann maps are given by perturbing the (electro)magnetic field. The approaches are different for the two settings: for Riemannian, we utilize the rigidity of elliptic equations to uniquely determine the metric without gauge equivalence; for Lorentzian, we rely on microlocal analysis and the propagation of singularity enjoyed by hyperbolic equations to explicitly construct the trajectory of lightlike geodesics.

https://sites.uci.edu/inverse/

https://sites.uci.edu/inverse/

https://sites.uci.edu/inverse/

We consider the following inverse problem: Suppose a (1 + 1)-dimensional wave equation on R+ with zero initial conditions is excited with a Neumann boundary data modelled as a white noise process. Given also the Dirichlet data at the same point, determine the unknown first order coefficient function of the system. The inverse problem is then solved by showing that correlations of the boundary data determine the Neumann-to-Dirichlet operator in the sense of distributions, which is known to uniquely identify the coefficient. The model has potential applications in acoustic measurements of internal cross-sections of fluid pipes such as pressurised water supply pipes and vocal tract shape determination. This talk is based on a joint-work with Emilia Blåsten, Antti Kujanpää and Tapio Helin (LUT), and Lauri Oksanen (Helsinki).