Motivated from a biological application, we study mesoscopic velocity jump (run-and-tumble) models for particle motion in the phase space. The motion is characterized by a sudden change of direction which is governed by the turning rate. The inverse problem is to determining this mesoscopic turning rate from macroscopic, i.e. directionally averaged data. The lack of directional information in the measurements poses problems in the reconstruction of the mesoscopic quantity. These problems can be leveraged by the use of time dependent interior domain data, as theoretical results on the reconstructability suggest. We then investigate the macroscopic limit behaviour for the inverse problem on the macroscopic regime and present first results on the numerical inversion.

This is joint work with Christian Klingenberg (Würzburg, Germany), Qin Li (Madison, Wisc., USA) and Min Tang (Shanghai, China).

In this talk, I will present a framework to study inverse problems involving forward-peaked diffusive transport. More specifically, we will study Fokker-Planck approximations to highly forward-peaked scattering and the accuracy of its respective approximations via Fermi pencil-beams, in a metric based on the 1-Wasserstein distance. We argue that metrics of this kind are suitable for analyzing the stability of related inverse problems, for instance, in inverse transport, microscopy, and off-axis laser detection. This is joint work with Guillaume Bal.

In this talk, we will discuss the determination of coefficients in time-dependent nonlinear transport equations. We consider both cases of time-independent and time-dependent coefficients. The talk is based on joint work with Ru-Yu Lai and Gunther Uhlmann.

I will discuss recent results on the range characterization of the X-ray transform on the hyperbolic disk, along with functional relations with distinguished differential operators, and mapping properties in adapted scales of Sobolev spaces.

This is based on joint work with Nikolaos Eptaminitakis (Leibniz University Hannover) and Yuzhou Zou (Northwestern).

In this talk we show how the singular decomposition method can be applied to recover the doping profile in the Vlasov-Poisson equation.

In many practical applications of inverse problems, the data measured contain unknown normalization constants due to the unknown strength of the illumination sources. In such cases, it is usually impossible to quantitatively reconstruct the coefficients of interest. We show, mainly computationally, that one can actually have quantitative reconstructions for some inverse transport problems where redundancy in data helps us to eliminate the unknown normalization constant encoded in the illumination sources.

In this talk, we briefly talk about the inverse boundary/medium problems with the semilinear transport model which has a natural application in two-photon absorption optics. For the inverse boundary problem, we adopted the usual linearization technique and prove the uniqueness of reconstruction. For the inverse medium problems, we consider the problem from photoacoustic imaging specifically in static and time-dependent settings and prove the uniqueness of the reconstruction for the absorption coefficients.

Based on work by Cohen, Damen and Devore [1] and Bourrier et al. [2], we propose a framework that highlights the importance of knowing the measurement model $F$ and model class $\mathcal{M}_1$, when solving ill-posed (non-)linear inverse problems, by introducing the concept of kernel size. Previous work has assumed that the problem is injective on the model class $\mathcal{M}_1$ and we obviate the need for this assumption. Thus, it is applicable in Deep Larning (DL) based settings where $\mathcal{M}_1$ can be an arbitrary data set.

$\textbf{Setting and initial main result}$ Let $\mathcal{X}$, $\mathcal{Y}$ and $\mathcal{Z}$ be non-empt sets, $\mathcal{M}_1 \subset \mathcal{X}$ be the $\textit{model class}$, $\mathcal{E}\subset \mathcal{Z}$ be the $\textit{noise class}$" and $F \colon \mathcal{M}_1\times \mathcal{E} \to \mathcal{Y}$ be the $\textit{forward map}$. An inverse problem has the form: $$ \text{Given noisy measurements } y = F(x,e) \text{ of } x \in \mathcal{M}_1\text{ and } e \in \mathcal{E}, \text{ recover } x.\quad (1) $$ Here $e$ represent the model noise and $x$ the signal (function or vector) we wish to recover or approximate. This also includes linear cases with additive noise, where $\mathcal{Y}=\mathcal{Z}$, $A \colon \mathcal{X} \to \mathcal{Y}$ is a linear map between vector spaces and the forward map is $$ F(x,e) = Ax+e. \quad (2) $$

To measure accuracy we equip $\mathcal{X}$, $\mathcal{Y}$ and $\mathcal{Z}$ with metrics $d_{\mathcal{X}}$, $d_{\mathcal{Y}}$ and $d_{\mathcal{Z}}$, such that the induced topology is second countable (it admits a countable base). We assume that the metrics $d_\mathcal{X}$ and $d_\mathcal{Z}$ satisfy the Heine-Borel property, i.e., all closed and bounded sets are compact, and that for every $y \in \mathcal{M}_2^{\mathcal{E}}$ the $\textit{feasible set}$ $$ \pi_1(F^{-1}(y)) = \{x \in \mathcal{M}_1: \exists e\in\mathcal{E} \text{ s.t. } F(x,e)=y\} $$ is compact. Under these assumptions we define the optimality constant as the smallest error any reconstruction map can achieve, and study both the case of worst-case noise (critical when there is risk of adversarial attacks) and average random noise (more typical in practical applications). We prove that the optimality constant is bounded from below and from above (sharply) by the $\textit{kernel size}$, which is a quantity intrinsic to the inverse problem. For simplicity, we restrict to the best worst-case reconstruction error here. As we consider set-valued reconstruction maps $\varphi\colon \mathcal{M}_2^{\mathcal{E}} \rightrightarrows \mathcal{X}$, we use the Hausdorff distance $d_{\mathcal{X}}^H(\cdot, \cdot)$.

$\textbf{Definition: Optimality constant under worst-case noise}$ The $\textit{optimality constant}$ of $(1)$ is $$ c_{\mathrm{opt}}(F,\mathcal{M}_1, \mathcal{E}) = \inf_{\varphi\colon \mathcal{M}_2^{\mathcal{E}} \rightrightarrows \mathcal{X}} \sup_{x\in \mathcal{M}_1} \sup_{e \in \mathcal{E}} \ d_{\mathcal{X}}^H(x, \varphi(F(x,e))) $$ A mapping $\varphi\colon \mathcal{M}_2^{\mathcal{E}} \rightrightarrows \mathcal{X}$ that attains such an infimum is called an $\textit{optimal map}$.

$\textbf{Definition: Kernel size with worst-case noise}$ The kernel size of the problem $(1)$ is $$ \operatorname{kersize}(F,\mathcal{M}_1, \mathcal{E}) = \sup_{\substack{ (x,e),(x',e')\in\mathcal{M}_1\times\mathcal{E} \text{ s.t. }\\ F(x,e) = F(x',e')} } d_{\mathcal{X}}(x,x'). $$

$\textbf{Theorem: Worst case optimality bounds}$ Under the stated assumptions, the following holds.

$(i)$ We have that $$ \operatorname{kersize}(F,\mathcal{M}_1, \mathcal{E})/2 \leq c_{\mathrm{opt}}(F,\mathcal{M}_1, \mathcal{E}) \leq \operatorname{kersize}(F,\mathcal{M}_1, \mathcal{E}). \quad (3) $$

$(ii)$ Moreover, the map

$$ \Psi(y) = \operatorname*{argmin}_{z \in \mathcal{X}}\sup_{(x,e) \in F^{-1}(y)} d_{\mathcal{X}}(x,z) = \operatorname*{argmin}_{z \in \mathcal{X}} d_{\mathcal{X}}^H(z, \pi_1(F^{-1}(y))), $$ has non-empty, compact values and it is an optimal map.

This illustrates a fundamental limit for the inverse problem $(1)$. Indeed, one would hope to find a solution for $(1)$ whose error is as small as possible. The lower bound in $(3)$ shows that there is a constant intrinsic to the problem -- the kernel size -- such that no reconstruction error can be made smaller than this constant for all possible choices of $x \in \mathcal{M}_1$. Note that the above theorem is extended to the average reconstruction error in our work.

$\textbf{Background and related work}$

Linear inverse problems $(2)$ arise in image reconstruction for scientific, industrial and medical applications [3-7]. Traditional image reconstruction methods are model-based, and they have also been studied in a Bayesian setting [8]. Less studied are non-linear inverse problems, which appear in geophysics [9,10] and in inverse scattering problems [11,12]. Accuracy and error bounds have been studied in [13]. In many cases, today, DL-based methods obtain higher accuracy than traditional methods, and an overview is given in [14-17]. The key point of DL based methods for solving inverse problems in imaging is that given enough data a neural network can be trained to approximate a decoder to solve $(2)$.

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[2] A. Bourrier, M. E. Davies, T. Peleg, P. Perez, R. Gribonval. Fundamental performance limits for ideal decoders in high- dimensional linear inverse problems. IEEE Transactions on Information Theory, 60(12):7928–7946, 2014.

[3] C. A. Bouman. Foundations of Computational Imaging: A Model-Based Approach. SIAM, Philadelphia, PA, 2022.

[4] H. H. Barrett, K. J. Myers. Foundations of image science. John Wiley & Sons, 2013.

[5] C. L. Epstein. Introduction to the mathematics of medical imaging. SIAM, 2007.

[6] P. Kuchment. The Radon transform and medical imaging. SIAM, 2013.

[7] F. Natterer, F. Wubbeling. Mathematical methods in image reconstruction. SIAM, 2001.

[8] A. M. Stuart. Inverse problems: A Bayesian perspective. Acta numerica, 19:451–559, 2010.

[9] C. G. Farquharson, D. W. Oldenburg. Non-linear inversion using general measures of data misfit and model structure. Geophysical Journal International, 134(1):213–227, 1998.

[10] C. G. Farquharson, D. W. Oldenburg. A comparison of automatic techniques for estimating the regularization parameter in non-linear inverse problems. Geophysical Journal International, 156(3):411–425, 2004.

[11] J. L. Mueller, S. Siltanen. Linear and nonlinear inverse problems with practical applications. SIAM, 2012.

[12] M. T. Bevacqua, L. Crocco, L. Di Donato, T. Isernia. An algebraic solution method for nonlinear inverse scattering. IEEE Transactions on Antennas and Propagation, 63(2):601–610, 2014.

[13] N. Keriven, R. Gribonval. Instance optimal decoding and the restricted isometry property. In Journal of Physics: Conference Series, IOP Publishing:012002. 2018.

[14] M. T. McCann, M. Unser. Biomedical image reconstruction: From the foundations to deep neural networks. Foundations and Trends in Signal Processing, 13(3):283–359, 2019.

[15] S. Arridge, P. Maass, O. Oktem, C.-B. Sch onlieb. Solving inverse problems using data-driven models. ¨ Acta Numer., 28:1–174, 2019.

[16] G. Wang, J. C. Ye, K. Mueller, J. A. Fessler. Image reconstruction is a new frontier of machine learning. IEEE Trans. Med. Imaging, 37(6):1289–1296, 2018.

[17] H. Ben Yedder, B. Cardoen, G. Hamarneh. Deep learning for biomedical image reconstruction: A survey. Artificial intelligence review, 54(1):215–251, 2021.

Mathematical modeling of infectious propagation is strongly connected with social and economic processes. The different types of mathematical models (time-series, differential, agent-based and mean-field games ones) are investigated to formulate adequate models. The parameters (contagiousness, probability of progression of severe disease, mortality, asymptomatic carriers, probability of testing, and others) of models are, as a rule, unknown and should be estimated by solving inverse problems. Inverse problems are ill-posed, i.e. its solutions are unstable and/or non-unique due to incomplete and noisy input data (epidemiological statistics, socio-economic characteristics, etc.). Therefore we use regularization ideas and special technique to achieve the appropriate estimation of the model parameters.

The first step in construction of the mathematical model of epidemiology and social processes propagation consists in data processing using machine learning methods. It helps to fix key characteristics of both processes. For detailed inverse problem data time-series, differential (ordinary or partial equations to account for migration), agent-based models are combined to describe epidemiology situation in a considered region with influence of social processes [1, 2]. Otherwise, mean field approach is used for optimal control of epidemiology and social processes that consists in combination of Kolmogorov-Fokker-Plank (KFP) equation for propagation of the density of representative agent for considered epidemiological status and Hamilton-Jacobi-Bellman (HJB) equation for the optimal control strategy [3]. In [3] it was showed that an incorrect assessment of current social processes in the population leads to significant errors in predicting morbidity. The search for a correct description of the socio-epidemiological situation in mathematical terms leads to the formulation of the inverse mean field problem, where, according to epidemiological statistics and the rate of increase in the number of infected, factors affecting the development of the incidence (for example, antiviral restrictions) can be estimated [5].

The second step based on sensitivity-based identifiability analysis with using Bayesian approach, Monte-Carlo method, and singular value decomposition [4]. It provides the sequence of sensitivity parameters from the more sensitive to the less ones and reduces the boundaries of variation of the unknown parameters for further development of the regularization algorithm.

The inverse problems consist in (1) identification of less sensitive epidemic parameters for models based on differential equations and after that (2) identification of more sensitive epidemic parameters for agent-based models using known approximation of other parameters and processes data. Inverse problems are formulated as a least-squares minimization problem that solved by combination of global (Tree-Structured Parzen Estimator, tensor optimization, differential evolution) and gradient type optimization methods with a priory information about inverse problem solution.

Deep neural and generative adversarial networks for data processing and forecasting of time-series data as well as alternative approach for the direct and inverse problem solutions are applied and investigated. The key features of epidemiological and social processes and Wasserstein metrics are used for construction neural networks.

The numerical results are demonstrated the effectiveness of combination models and approaches to the COVID-19 propagation in different regions using socio-economic processes and optimal control programs of epidemic propagation in different economics.

The work is supported by the Mathematical Center in Akademgorodok under the agreement No. 075-15-2022-281 with the Ministry of Science and Higher Education of the Russian Federation.

[1] O. Krivorotko, M. Sosnovskaia, I. Vashchenko, C. Kerr, D. Lesnic. Agent-based modeling of COVID-19 outbreaks for New York state and UK: parameter identification algorithm. Infectious Disease Modelling. 7: 30-44, 2022.

[2] O.I. Krivorotko, N.Y. Zyatkov. Data-driven regularization of inverse problem for SEIR-HCD model of COVID-19 propagation in Novosibirsk region. Eurasian Journal of Mathematical and Computer Applications. 10(1): 51-68, 2022.

[3] V. Petrakova, O. Krivorotko. Mean field game for modeling of COVID-19 spread. Journal of Mathematical Analysis and Application. 514: 126271, 2022.

[4] O.I. Krivorotko, S.I. Kabanikhin, M.I. Sosnovskaya, D.V. Andornaya. Sensitivity and identifiability analysis of COVID-19 pandemic models. Vavilovskii Zhurnal Genetiki i Selektsiithis. 25(1): 82-91, 2021.

[5] L. Ding, W. W., S. Osher et al. A Mean Field Game Inverse Problem. Journal of Scientific Computing. 92:7, 2021.

Solutions of the evolution equation generally lie in certain Bochner-Sobolev spaces, in which the solutions may have regularity and integrability properties for the time variable that can be different for the space variables. Therefore, in our paper, we developed a framework that shows that deep neural networks can approximate Sobolev-regular functions with respect to Bochner-Sobolev spaces. In this talk we will present the power of using the so-called Rectified Cubic Unit (ReCU) as an activation function in the networks. This activation function allows us to deduce approximation results of the neural networks. While avoiding issues caused by the non regularity of the most commonly used Rectified Linear Unit (ReLU) activation function. This is a joint work with Philipp Grohs.

[1] A. Abdeljawad, P. Grohs. Approximations with deep neural networks in Sobolev time-space. Analysis and Applications 20.03 (2022): 499-541.