Variational regularization is a well-established technique to tackle instability in inverse problems, and it requires solving a minimization problem in which a mismatch functional is endowed with a suitable regularization term. The choice of such a functional is a crucial task, and it usually relies on theoretical suggestions as well as a priori information on the desired solution. A promising approach to this task is provided by data-driven strategies, based on statistical learning: supposing that the exact solution and the measurements are distributed according to a joint probability distribution, which is partially known thanks to a training sample, we can take advantage of this statistical model to design regularization operators. In this talk, I will present a hybrid approach, which first assumes that the desired regularizer belongs to a class of operators (suitably described by a set of parameters) and then learns the optimal one within the class. In the context of linear inverse problems, I will first briefly recap the main results obtained for the family of generalized Tikhonov regularizers: a characterization of the optimal regularizer, and two learning-based techniques to approximate it, with guaranteed error estimates. Then, I will focus on a class of sparsity-promotion regularizers, which essentially leads to the task of learning a sparsifying transform for the considered data. Also in this case, it is possible to deduce theoretical error bounds between the optimal regularizer and its supervised-learning approximation as the size of the training sample grows.

Diffusion models have recently received significant interest in the imaging sciences. After achieving impressive results in image generation, focus has shifted towards finding ways to exploit the encoded prior knowledge in classical inverse problems. In this talk, we highlight another intriguing viewpoint: Instead of focusing on image reconstruction, we propose to use tractable diffusion models which also allow to estimate the noise in an image. In particular, we utilize a fields-of-experts-type model with Gaussian mixture experts that admits an analytic expression for a normalized density under diffusion, and can be trained with empirical Bayes. The normalized model can be used for noise estimation of a given image by maximizing it w.r.t. diffusion time, and simultaneously gives a Bayesian least-squares estimator for the clean image. We show results on denoising problems and propose possible applications to more involved inverse problems.

In this talk, we address the image denoising problem in presence of speckle degradation typically arising in ultra-sound images. Variational methods and Convolutional Neural Networks (CNNs) are considered well-established methods for specific noise types, such as Gaussian and Poisson noise. Nonetheless, the advances achieved by these two classes of strategies are limited when tackling the de-speckle problem. In fact, variational methods for speckle removal typically amounts to solve a non-convex functional with the related issues from the convergence viewpoint; on the other hand, the lack of large datasets of noise-free ultra-sound images has not allowed the extension of the state-of-the-art CNN denoiser methods to the case of speckle degradation. Here, we aim at combining the classical variational methods with the predictive properties of CNNs by considering a weighted total variation regularized model; the local weights are obtained as the output of a statistically inspired neural network that is trained on a small and composite dataset of natural and synthetic images. The resulting non-convex variational model, which is minimized by means of the Alternating Direction Method of Multipliers (ADMM) is proven to converge to a stationary point. Numerical tests show the effectiveness of our approach for the denoising of natural and satellite images.

Dictionary learning methods have been used in recent years to solve inverse problems without using the forward model in the traditional optimization algorithms. When the dictionaries are large, and a sparse representation of the data in terms of the atoms is desired, computationally efficient sparse optimization algorithms are needed. Furthermore, reduced dictionaries can represent the data only up to a model reduction error, and Bayesian methods for estimating modeling errors turn out to be useful in this context. In this talk, the ideas of using Bayesian hierarchical models and modeling error methods are discussed.

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.