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Inverse Scattering Problems for the Time Dependent Wave Equation

Banff International Research Station for Mathematical Innovation and Discovery

In this presentation we will discuss recent progress in non-iterative methods in the time domain. The use of time dependent data is a remedy for the large spacial aperture that these method need to obtain a reasonable reconstructions. Fist we consider the linear sampling method for solving inverse scattering problem for inhomogeneous media. A fundamental tool for the justification of this method is the solvability of the time domain interior transmission problem that relies on understanding the location on the complex plane of transmission eigenvalues. We present our latest result on the solvability of this problem. As opposed to the frequency domain case, in the time domain there are no known qualitative methods with a complete mathematical justification, such as e.g. the factorization method.This is still a challenging open problem and the second part of the talk addresses this issue. In particular, we discuss the factorization method to obtain explicit characterization of a (possibly non-convex) Dirichlet scattering object from measurements of time-dependent causal scattered waves in the far field regime. In particular, we prove that far fields of solutions to the wave equation due to particularly modified incident waves, characterize the obstacle by a range criterion involving the square root of the time derivative of the corresponding far field operator. Our analysis makes essential use of a coercivity property of the solution of the Dirichlet initial boundary value problem for the wave equation in the Laplace domain that forces us to consider this particular modification of the far field operator. The latter in fact, can be chosen arbitrarily close to the true far field operator given in terms of physical measurements. Finally we discuss some related open questions.

Mathematics; Functional analysis; Calculus of variations and optimal control; optimization; Numerical analysis

video/mp4

Author affiliation: Rutgers University

2019-12-26

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/73001

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/