A recent trend in inverse scattering theory has focused on the development of a qualitative approach, which yield fast reconstructions with very little of a priori information but at the expense of obtaining only limited information of the scatterer such as the support, and estimates on the values of the constitutive parameters. Examples of such an approach are the linear sampling and factorization methods. These two methods are very well developed in the time harmonic regime, more generally for the underlying elliptic PDEs models, however for hyperbolic problems only limited results are available. In inverse scattering, the use of time domain measurements is a remedy for large amount of spatial data typically needed for the application of qualitative approach.
In this presentation we will discuss recent progress in the development of linear sampling and factorization methods in the time domain. 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 some latest results in this regard. The second problem addresses the lack of mathematical rigorousness of the linear sampling method. In this context 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.