Two types of new results of the presenter will be discussed:

1. Holder and Lipschitz stability estimates for coefficient inverse problem and inverse source problem with the final overdetermination [1]. The solution of the parabolic equation is known at $t=0$ and $t=T$. Both Dirichlet and Neumann boundary conditions are known either on part of the boundary or on the entire boundary. A new Carleman estimate for the parabolic operator is the key here. Unlike standard Carleman estimates in this one, the Carleman Weight Function is independent on $t$. The Holder stability estimate is in the case of incomplete boundary data and the Lipshitz stability is in the case of complete boundary data. Both results and the methodology are significantly different from previous ones.

2. Stability estimates and uniquness theorems for some inverse problems for the Mean Field Games system [2]. These results are also new. The Mean Field Games system is a system of two parabolic equations, which was originally proposed by J.-M. Lasry and P.-L. Lions in 2006-2007 and became quite popular nowadays due to a number of very exciting applications. The main challenge here is that the time t is running in two opposite directions in these equations. Therefore, the Volterra-like property of conventional systems of parabolic PDEs is not kept here.

[1] M. V. Klibanov, Stability estimates for some parabolic inverse problems with the final overdetermination via a new Carleman estimate, arxiv: 2301.09735, 2023.

[2] M. V. Klibanov, Yu. V. Aveboukh, Stability and uniqueness of two inverse problems for the Mean Field Games system, in preparation.

Hyperbolic conservation laws are central in the theory of PDEs. One of their typical features is the development of shock waves. This poses many challenges to the mathematical theory of both forward and inverse problems. It is well-known that two different initial data may involve into the same solution. In this talk, we will present a number of ways to overcome this difficulty, with emphasis on the Bayesian approach, and survey some recent results.

Holographic coherent x-ray imaging enables nanoscale imaging of biological cells and tissue, rendering both phase and absorption contrast, i.e. real and imaginary part of the refractive index. A main challenge of this imaging technique is radiation damage. We present a different modality of this imaging technique using a partially incoherent incident beam and time-resolved intensity measurements based on new measurement technologies. This enables the acquisition of intensity correlations in addition to the commonly used expectations of intensities. In this talk we explore information content of holographic intensity correlation data, analytically showing that in the linearized model both phase and absorption contrast can uniquely be determined by the intensity correlation data. The uniqueness theorem is derived using multi-dimensional Kramers-Kronig relations. We also deduce a uniqueness theorem for ghost holography imaging as an unconventional X-ray imaging scheme.

For regularized reconstruction it is important to take into account the statistical distribution of the correlation data. The measured intensity data are described by a so-called Cox-processes, roughly speaking a Poisson process with random intensity. For medium-size data sets, we use adaptations of the iteratively regularized Gauss-Newton method and the FISTA method as reconstruction methods. Our numerical results even in the full nonlinear model confirm that both phase and absorption contrast can jointly be reconstructed from only intensity correlations without the use of average intensities. Although these results are encouraging concerning the information content of the new intensity correlation data, the increased dimensionality of these data causes severe computational challenges.