# POEMS seminar on Inverse Problems

Classical uncertainty principles in signal processing limit the amount of simultaneous concentration of a signal with respect to time and frequency. In the inverse source problem, the far field radiated by a source f is its restricted (to the unit sphere) Fourier transform, and the operator that maps the restricted Fourier transform of f(x) to the restricted Fourier transform of its translate f(x + c) is called the far field translation operator. In this talk we discuss an uncertainty principle, where the role of the Fourier transform is replaced by the far field translation operator. Combining this principle with a regularized Picard criterion, which characterizes the non-evanescent far fields radiated by a compactly supported limited power source provides extensions of several results about splitting a far field radiated by well-separated sources into the far fields radiated by each source component. We also combine the regularized Picard criterion with a more conventional uncertainty principle for the map from a far field to its Fourier coefficients. This leads to a data completion algorithm which tells us that we can deduce missing data if we know a priori that the source has small support. All of these results can be combined so that we can simultaneously complete the data and split the far fields into the components radiated by well-separated sources. We discuss both l2 (least squares) and l1 (basis pursuit) algorithms to accomplish this. Perhaps the most significant point is that all of these algorithms come with explicit bounds on their condition numbers which are sharp in their dependence on geometry and wavenumber.

This is joint work with John Sylvester (University of Washington).

####15:30 - Lorenzo Audibert (EDF, France) ####Transmission eigenvalues with artificial background and their determination from scattering data

We are interested in the problem of retrieving information on the refractive index, n, of a penetrable inclusion embedded in a reference medium from farfield data associated with incident plane waves. Our approach relies on the use of transmission eigenvalues (TEs) that carry information on n and that can be determined from the knowledge of the farfield operator F. We explain how to modify F into a farfield operator F^{art}=F-\tilde{F}, where \tilde{F} is computed numerically, corresponding to well chosen artificial background and for which the associated TEs provide more accessible information on n. We will emphasis that the artificial background simplified the analysis of interior transmission eigenvalue problem while maintaining our abilities to detect transmission eigenvalues efficiently using sampling method.