The first edition of the book "Inverse Acoustic and Electromagnetic Scattering Theory" by D. Colton and R. Kress appeared in 1992. It was a comprehensive exposition of fundamental mathematical background as well as exciting developments happening at the time in inverse scattering theory, from uniqueness results to reconstruction algorithms. The book became a classic in the field. The fourth edition of this book was published in 2019, about 30 years later, in a much extended version. The added 200 pages represent a part of the myriad directions that the research in inverse scattering has taken. This includes development of novel non-iterative reconstruction approaches, such as factorization, generalized linear sampling and other direct imaging methods, the design and analysis of more advanced and efficient optimization algorithms, the investigation of special sets of frequencies, namely transmission eigenvalues, non-scattering wave numbers and scattering poles, and their applications in solving inverse scattering problems. We shall review some of the key moments, places and anecdotes that contributed to this achievement.

We report on the determination of the shape and location of scattering obstacles by passive imaging techniques. More precisely, we assume that the available data are correlations of randomly excited waves with zero mean. Passive imaging techniques are employed in seismology, ocean acoustics, experimental aeroacoustics, ultrasonics, and local helioseismology. They have also been thoroughly investigated mathematically, typically as a qualitative imaging modality, but the study of inverse obstacle problems seems to be new in this context.

We assume that wave propagation is described by the Helmholtz equation in two or three space dimensions. Furthermore, the random source is assumed to be uncorrelated and either compactly supported or at infinite distance. The source strength is considered as an additional unknown of the inverse problem.

As a main theoretical result, we show that both the shape of a smooth obstacle without holes and the source strength are uniquely determined by correlation data, both in the near-field and in the far-field. We also show numerical simulations supporting our theoretical results.

In 1994, just within the 30 years mentioned in the title of this minisymposium, Colton and Kirsch proposed a set of target signatures for imperfectly conducting obstacles at fixed frequency [1]. These are characterized by using the far field equation. Today there are many families of target signatures including transmission eigenvalues, Steklov eigenvalues and modified transmission eigenvalues. All of these relate to scattering by a target of non-zero volume, and they can all be determined from scattering data using appropriate modifications of the far field equation [2].

In this presentation I will continue by describing recently developments target signatures for screens. Screen are open surfaces, and hence have no volume. A typical example is a resistive screen modeled using transmission conditions across the screen. The goal is to design target signatures that are computable from scattering data in order to detect changes in the material properties of the screen. This target signature is characterized by a mixed Steklov eigenvalue problem for a domain whose boundary contains the screen.

Following [3], I shall show that the corresponding eigenvalues can be determined from an appropriately modified far field equation. Numerical experiments using the classical linear sampling method are presented to support our theoretical results.

[1] D.L. Colton, A. Kirsch. Target signatures for imperfectly conducting obstacles at fixed frequency. Quart. J. Mech. Appl. Math. 47:1--15, 1994.

[2] D.L. Colton, F. Cakoni, H. Haddar. Inverse Scattering Theory and Transmission Eigenvalues, 2nd edition, CBMS-NSF, Regional Conference Series in Applied Mathematics, SIAM Publications, 98, 2022.

[3] F. Cakoni, P. Monk, Y. Zhang. Target signatures for thin surfaces. Inverse Probl. 38, 025011, 28 pp, 2021. doi: 10.1088/1361-6420/ac4154

Dynamical models are the basis for forecasting in important application regimes such as weather or climate forecasting. Numerical models are based on a combination of PDEs from fluid flow, simulation of electromagnetic radiation and microphysics. For synchronization of such systems with reality data assimilation methods are used. These methods combine observations with short range forecasts into so-called analysis of components of the earth system, e.g. the atmosphere, land or the ocean. This is repeated for global atmospheric models every three hours, for high-resolution atmospheric models every hour, for ocean forecasting once per day. In climate science monthly means are assimilated for seasonal or decadal forecasting. The cycled run of short range forecasts and assimilation steps is known as data assimilation cycle. Observations include radiative transfer codes for microwave or infrared measurements, leading to integral-equation type observation operators as the basis of high-resolution global or regional data assimilation.

Here, we will address the task to learn dynamical models or model components iteratively which running such a data assimilation cycle. To this end we will employ either iterated Tikhonov regularization or its more elaborate version, the Kalman filter. We will demonstrate that model learning can be carried out very efficiently based on a particular representation of the model based on a sufficiently large variety of observations to be exploited in each step of the assimilation cycle. Examples from popular academic models such as the Lorenz 63 or 96 systems and more real-word systems will be demonstrated.