Talks

В данной работе исследуются градиентные подходы к обратным и некорректным задачам, которые могут интерпретироваться как задачи оптимального управления. В работе исследуются обратные задачи для уравнения гиперболической теплопроводности и параболического уравнения с точечными источниками. Цель данной работы показать другим гибкость и широту применения градиентных методов в задачах, в которых, казалось бы, у них нет очевидного применения, однако оно является достаточно эффективным. Также исследуются вопросы выбора оптимальных процедур вычисления разностных схем, и то, какие схемы лучше выбирать. Были предложены компактные неявные схемы для такого рода задач. Данные схемы вычисляются при помощи метода Ньютона, где за начальное значение выбирается приближение по явной схеме Эйлера. Данный подход показывает большую устойчивость по сравнению с явными и другии неявными схемами.

We investigate the inverse problem of identifying a polyhedral inclusion embedded in a homogeneous isotropic background medium using boundary measurements. Both the cases of conductive and elastic media are considered. By relying on local Dirichlet-to-Neumann maps, we establish a global Lipschitz stability estimate for the reconstruction of the inclusion, ensuring robustness with respect to perturbations in the data.

The talk addresses a nonlocal balance equation $$\partial_t\mu_t+\operatorname{div}(b(x,\mu_t)\mu_t)=g(x,\mu_t)\mu_t,\ \ \mu_0=\mu_*,$$ considered in the class of flows of nonnegative measures, meaning that for each time instant $t$, $\mu_t$ is a nonnegative measure. This equation describes the dynamics of a system of interacting particles with possible particle creation/annihilation. The following aspects are planned for discussion: 1. existence and uniqueness of solutions to the balance equation; 2. representation of the balance equation solution via a nonlinear Markov process; 3. approximation of solutions to the nonlocal balance equation using systems of ordinary differential equations.

In $R^n$, we will consider a set-valued integral over the interval [0,t] of a matrix of a certain smoothness onto a convex compact subset containing zero. We will consider the ball of the largest radius $r(t)$ with the center at zero, contained in the integral. We will discuss properties of the function $r$ in the context of 1) continuity of the Bellmann function and 2) sufficiency of the maximum principle both for a linear time-optimal control problem.

\begin{abstract} The talk provides a coherent presentation of an operator scheme, which is used in an approach to inverse problems of mathematical physics (the boundary control method). The scheme is based on the triangular factorization of operators. It not only solves inverse problems but provides the functional models of a class of symmetric semi-bounded operators. These models are constructed via an evolutionary dynamical system associated with the operator. \end{abstract}

We study robotization in economic growth models with heterogeneous agents. The asymptotic behavior of equilibrium paths is fully described both in the case where robotization is impossible and in the case of robotization. It is shown that the transition from an economy without robots to an economy with robots will not necessarily lead to an increase in welfare for all dynasties, but it will almost certainly lead to an increase in wealth and income inequality.

Population dynamics of many biological species defined mutations and appearance new genotypes. Mathematical descriptions of this process occur difficulties for the reason of simultaneous combination discrete process of appearance new genotypes in a random moment of time and continuous dynamics of growth of population. Combination discrete process mutation and growth in the famous Eigen's and Crow-Kimura models of population realized with help matrices of mutations each element of that define the probability appearance new genotypes. However, these models don't consider death rates and inter competitions of species as well during all time total number of all species is fixed. We present two mathematical models based of new principal. First one based on sequential solutions system of nonlinear ODE on the time period inter mutations where dimension of the system increased on unit after each new mutation. We suppose that time of appearance each mutation coincides with time of doubling species and has Gauss distribution. Initial data for new genotypes defined probability of its surviving. Thus, the total value of population presented a mathematical expectation of all surviving genotypes. Second model description of dynamics population basis on the flow equation. In this case the problem reduced to solution of quasilinear partial parabolic equation with nonlocal component. We also present numerical results of applying these models to case populations of cancer cells and viruses.

We study the nonlinear inverse source problem of detecting, localizing and identifying unknown accidental disturbances on a driven and damped network. A first result is that strategic observation sets are enough to guarantee the detection of disturbances. To localize and identify them, we additionally need the observation set to be absorbent. If this set is dominantly absorbent, then detection, localization and identification can be done in "quasi real-time". We illustrate these results with numerical experiments. This is joint work with Adel Hamdi.

This work considers methods for solving the eikonal equation in problems of medical tomography, aimed at improving the diagnosis of malignant inclusions in human soft tissues at early stages using ultrasound probing. The eikonal equation, which describes the propagation of wavefronts in media with variable velocity, plays an important role in image reconstruction in ultrasound tomography. An accurate solution of this equation makes it possible to improve the quality of pathology visualization, which is critically important for the early detection of oncological diseases. The study presents various approaches to solving the eikonal equation, including classical schemes, fast marching methods (Fast Marching Method) \cite{Dudar:Article}, as well as modern approaches using neural networks. Each method is analyzed in terms of computational efficiency, accuracy, and applicability to medical imaging problems. Classical finite-difference schemes provide high accuracy in simple cases but encounter limitations when processing complex heterogeneous media. Fast marching methods demonstrate robustness and speed, but require careful parameter tuning. Neural networks, in turn, open up prospects for adaptive modeling of complex media, but their training requires significant computational resources and high-quality data, and they have poor adaptability to new data. The paper discusses the advantages and disadvantages of each approach, as well as their potential for integration into medical diagnostic systems. Particular attention is paid to the possibility of applying the methods in real clinical conditions, including limitations related to data noise and tissue heterogeneity. The results of the study may serve as a basis for the development of new algorithms and software solutions intended for engineers and researchers in the field of medical tomography, and may also contribute to improving the accuracy and accessibility of diagnostic procedures.

The model depicts an economy where agents optimally split their time between producing final goods and expanding production capacity as technology evolves. The emerging mean-field game system consists of two Burgers-type equations capturing Schumpeterian dynamics with heterogeneous imitation, coupled with two Hamilton-Jacobi-Bellman (HJB) equations arising from the maximization of discounted total output.

When describing the group behavior of high-frequency traders, there arises a boundary value problem based on the concept of mean field games. The system consists of two coupled partial differential equations: the Hamilton–Jacobi–Bellman equation which describes the evolution of the average payoff function in backward time and the Kolmogorov–Fokker–Planck equation which describes the evolution of the distribution density of traders in forward time. The system is inherently ill-conditioned due to the turnpike effect. Under certain assumptions, it is possible to perform reduction to a system of Riccati equations; however, the question of well-posedness of the reduced problem remains open. This work investigates this question, namely, the conditions for the existence and uniqueness of the solution to the boundary value problem depending on the model parameters.

ACCELERATED ALGORITHM FOR SOLVING THE SCATTERING PROBLEM FOR THE MANAKOV MODEL L. L. Frumin*, A. E. Chernyavsky** Institute of Automation and Electrometry SD RAS Novosibirsk, Academician Koptyug Avenue 1, Russia, *e-mail: lfrumin@yandex.ru **e-mail: alexander.cher.99@gmail.com An algorithm for accelerated ("super-fast") solution of direct scattering problems for the continuous spectrum of the Manakov system associated with the vector nonlinear Schrödinger equation of the Manakov model is proposed, requiring for a discrete grid of size asymptotically of the order of ($N \log^2⁡ N$) arithmetic operations. For smooth localized (decreasing at infinity) solutions of the direct scattering problem, the algorithm has the second order of accuracy. The algorithm finds a polynomial approximation of the dependence of the scattering coefficients on the spectral parameter, which determine the spectral reflection and transmission coefficients. To speed up the calculations of the direct scattering problem for the Manakov system, the duplication strategy, the convolution theorem, and the fast Fourier transform are used. The resulting polynomial approximation of the spectral dependence of the scattering coefficients can be used in the future to search for the discrete spectrum of the Manakov system.

This work addresses the Cauchy problem for the metaharmonic equation arising in thermal modeling with blood perfusion. The problem is considered in a cylindrical domain of rectangular cross–section with Cauchy data prescribed on a surface of general type and homogeneous conditions imposed on the lateral boundaries. Such a formulation is ill–posed in the sense of Hadamard, which necessitates the use of special regularization techniques. An approach to constructing the Carleman function is proposed, which provides a stable integral representation of the solution. Its main properties are established: representation via the fundamental solution and a metaharmonic part, fulfillment of the boundary conditions, and vanishing of integral contributions from the upper plane as $ \alpha \to 0$. It is shown that the use of the Carleman function in combination with Tikhonov regularization ensures a stable approximate solution of the inverse thermography problem.

The system of PDE, describes the motion of two immiscible fluids (oil–water) in a poroelastic medium, includes the following equations: $$ \frac{\partial s_{i}\phi\rho_i^0}{\partial{t}} + \nabla \cdot (s_{i}\phi \vec{u}_i\rho_i^0) = 0, \quad i = 1,2, $$ $$ s_{i} \phi (\vec{u}_i - \vec{u}_3) = -K_0\frac{k_{0i}}{\mu_i}( \nabla{ p_i} - \vec{g}\rho_i), \quad i = 1,2, $$ $$ p_2-p_1 = p_c(x,s_1), $$ $$ \frac{\partial(1 - \phi)\rho_3^0}{\partial{t}} + \nabla \cdot {((1 - \phi)\vec{u}_3\rho_3^0)} = 0, $$ $$ \nabla \cdot \vec{u}_3 = -a_1(\phi)p_e - a_2(\phi)(\frac{\partial{p_e}}{\partial{t}}-\vec{u}_3 \cdot \nabla p_e), $$ $$ \nabla p_{tot} - div \left(\eta(1-\phi) \left( \frac{\partial \vec{u_3}}{\partial \vec{x}} + \frac{\partial \vec{u_3}}{\partial \vec{x}}^*\right) \right) = \rho_{tot} \vec{g}. $$ Here $\rho_i^0, \vec{u}_i s_i,$ and $p_i$ are, respectively, the true density, velocity, saturation, and pressure of the $i$-th phase ($i = 1$ is the wetting phase, $i = 2$ is the nonwetting phase, $s_1 + s_2 = 1$, and $i = 3$ is the solid deformable skeleton), $K_0(\phi)$ is the permeability tensor, $k_{0i}$ is relative phase permeability, $\mu_i$ is the dynamic viscosity of the $i$-th fluid, $\vec{g}$ is the mass force density; $\phi$ is the porosity, $p_c$ capillary pressure, $\eta$, $a_1(\phi)$ and $a_2(\phi)$ are the coefficients of shear viscosity, bulk viscosity, and bulk compressibility of the medium, $p_e = p_{tot} - (s_1 p_1 + s_2 p_2)$ is the effective pressure, $p_{tot} = (1-\phi)p_s + \phi(s_1p_1 + s_2p_2)$ is the total pressure, $\rho_{tot} = (1-\phi)\rho_{3}^0 + \phi(s_1 \rho_{1}^0 + s_2\rho_2^0)$ is the total density. $$$$ The system is a generalization of the Muskett-Leverett model[1][2]. This paper presents a theoretical[1][3] and computational[4] research of initial-boundary problem for the system. $$$$ This work was completed as part of a state assignment from the Ministry of Science and Higher Education of the Russian Federation on the topic "Modern Hydrodynamic Models for Nature Management, Industrial Systems, and Polar Mechanics" (topic number: FZMV-2024-0003). $$$$ [1] P. V. Gilev, A. A. Papin. Filtration of Two Immiscible Incompressible Fluids in a Thin Poroelastic Layer // Journal of Applied and Industrial Mathematics, 18:2 (2024), p. 234–245; [2] S. N. Antontsev, A. V. Kazhikhov, and V. N. Monakhov. Boundary Value Problems in the Mechanics of Inhomogeneous Fluids, Nauka Publishing House, Novosibirsk, 1983. (In Russian); [3] P.V. Gilev. Uniqueness of the solution to the problem of two-phase filtration in a poroelastic thin layer // Mathematical Notes, 2025, Volume 118, Issue 4, pp. 630–634. (In Russian); [4] V.B. Pogosyan, M.A. Tokareva, A.A. Papin. Calculation of the physical characteristics of a poroelastic medium during gas filtration // Applied Mechanics and Theoretical Physics, Novosibirsk, 2024. DOI: 10.15372/PMTF202415539. (In Russian);

Currently, the task of providing seismic protection for extended objects, such as industrial buildings and airfield runways, is relevant. During operation, they can be exposed to various seismic waves, including Rayleigh and Rayleigh-Lamb surface waves, evanescent and head SP waves. One of the options for their protection is the installation of special seismic barriers [1] made of metamaterials with special mechanical properties and designed to dissipate the wave energy of surface seismic waves. The physical feature of these granular materials is their different resistance to tension and compression [2]. In this paper, we consider a full-wave solution to the dynamic problem of loading a heterogeneous geological massif. To perform computer modeling of seismic wave propagation in a heterogeneous medium and their interaction with protective barriers, we use the grid-characteristic method on structured computational grids [3]. A distinctive feature of this method is the high-precision reproduction of all types of volume and surface waves, which is critically important for modeling the problems and processes under consideration. This work was carried out with the financial support of the Russian Science Foundation, project No. 25-19-00404, https://rscf.ru/project/25-19-00404/. References 1. Morozov, N.F., Bratov, V.A., Kuznetsov, S.V. Seismic barriers for protection against surface and head waves. Multiple scatters and metamaterials // Mech. Solids. – M., 2021. – V.56. – P. 911 – 921. 2. Sadovskii, V.M., Sadovskaya, O.V., Petrakov, I.E. On the theory of constitutive equations for composites with different resistance in compression and tension // Composite Structures. – M., 2021. – V.268. – paper 113921. 3. Petrov, I.B., Golubev, V.I., Shevchenko, A.V. et al. Three-Dimensional Grid-Characteristic Schemes of High Order of Approximation // Dokl. Math. 110, 457–463 (2024).

In the talk it will be presented the theory, numerical methods and neural networks in inverse and ill-posed problems in geosciences.

An optimal control problem with mixed constraints is addressed. The case is considered when the conventional regularity hypothesis with respect to the given constraints is not valid. It is also assumed that the data is not continuous with respect to the time variable. Pontryagin's maximum principle is derived in this general setting. One example of optimal control problem is also considered which demonstrates how the obtained conditions can be applied.

The work M. V. Klibanov and Y. Averboukh, Lipschitz stability estimate and uniqueness in the retrospective analysis for the mean field games system via two Carleman estimates, SIAM J. Mathematical Analysis, 56, 616--636, 2024 is the first one, where the tool of Carleman estimates was introduced in the field of Mean Field Games. This tool has allowed to prove first stability estimates and uniqueness theorems for many forward and inverse problems of Mean Field Games. Furthermore, Carleman estimates allow to construct globally convergent numerical methods for both forward and inverse problems. In particular, numerical results for a practically important problem of predicting of public opinions via Mean Field Games will be presented. Probably even for experimental data (if available). All numerical results are supported by the rigorous convergence analysis.

We study the direct and inverse scattering problems for the general Zakharov-Shabat system. Representations for the Jost solutions are obtained in the form of the power series in terms of a transformed spectral parameter. In terms of that parameter, the Jost solutions are convergent power series in the unit disk. The coefficients of the series are computed following a simple recurrent integration procedure. This essentially reduces the solution of the direct scattering problem to the computation of the coefficients and location of zeros of an analytic function inside of the unit disk. The solution of the inverse scattering problem reduces to the solution of a system of linear algebraic equations for the power series coefficients, while the potential is recovered from the first coefficient. The overall approach leads to a simple and efficient method for the numerical solution of both direct and inverse scattering problems, which is illustrated by numerical examples.

Multi-population mean-field control (MFC) provides a mathematically tractable framework for optimizing intervention strategies across heterogeneous populations in epidemiology [1-2]. However, solving high-dimensional stochastic MFC problems remains challenging due to complex cross-population interactions, curse of dimensionality, and stochastic epidemic dynamics. We propose a coupled adversarial network approach for numerical solutions to multi-population MFCs in epidemic management. First, we establish a compartmental-structured MFC model capturing heterogeneous transmission dynamics, vaccination policies, and mobility-driven interactions between subpopulations [2-3]. Leveraging the variational structure of MFCs, we reformulate the problem as a saddle-point optimization. Our architecture employs multiple generators to parameterize both population-specific controls (e.g., vaccination rates) and infection state distributions, while cooperative discriminators enforce consistency with epidemiological constraints and mean-field couplings. Adversarial training minimizes residuals of the coupled Hamilton-Jacobi-Bellman (HJB) and Kolmogorov forward equations. \textit{This work is supported by the Russian Science Foundation (project No. 23-71-10068).} [1] Bensoussan A., Tao H., Laurière M. Mean field control and mean field game models with several populations. Minimax Theory Its Applications. 2018. V. 3. P. 173–209. [2] Wang G., Fang J., Jiang L., Yao W., Li N. Coupled alternating neural networks for solving multi-population high-dimensional mean-field games. Mathematics. 2024. V. 12. Article 3803. [3] Petrakova V., Krivorotko O. Comparison of two mean field approaches to modeling an epidemic spread. Journal of Optimization Theory and Applications. 2025. V. 204. Article 39.

The report will present two methods for constructing exact solutions of nonlinear partial differential equations: the contact linearization method and the finite-dimensional dynamics method. Both methods are constructive. Their application will be illustrated on problems of continuum mechanics.

Inverse problems of reconstructing several key parameters of the one-phase Stefan problem with fractional time derivative, such as the order of the fractional derivative and latent heat, are solved numerically. The enthalpy formulation of the problem is used, and the phase transition boundary in the discrete problem is constructed based on the values of temperature and enthalpy. A distinctive feature of the problem being solved is the choice of one or several points of the phase transition boundary as an observation. The correctness of the inverse problem statements is due to the monotonic dependence of the phase transition boundary velocity on the above parameters. The methods for solving a one-dimensional problem with observation of the position of one or several points of the phase transition boundary for reconstructing one or a set of parameters are studied in detail theoretically and numerically. The developed methods are adapted to solving the grid approximation of two-dimensional one-phase Stefan problem.

The gradient descent (GD) method – is a fundamental and likely the most popular optimization algorithm in machine learning (ML), with a history traced back to a paper in 1847 (Cauchy, 1847). In this paper, we provide an improved convergence analysis of gradient descent and its variants, assuming generalized smoothness. In particular, we show that GD has the following behavior of convergence in the convex setup: At first, the algorithm has linear convergence, and approaching the solution, has standard sublinear rate. Moreover, we show that this behavior of convergence is also common for its variants using different types of oracle: Normalized Gradient Descent as well as Clipped Gradient Descent (the case when the oracle has access to the full gradient); Random Coordinate Descent (when the oracle has access only to the gradient component); Random Coordinate Descent with Order Oracle (when the oracle has access only to the comparison value of the objective function). In addition, we also analyze the behavior of convergence rate of GD algorithm in a strongly convex setup.

We apply the method of geometrization of random vectors to turbulent media, which we understand as random vector fields on base manifolds. This gives rise to various geometric structures on the tangent as well as cotangent bundles. Among these, the most important is the Mahalanobis metric on the tangent bundle, which allows us to obtain all the necessary ingredients for implementing the scheme to the description of flows in turbulent media. As an illustration, we consider the applications to flows of real gasses based on Maxwell-Boltzmann statistics are given.

\begin{document} \begin{center} \title{The Inverse Spectral Problem for Dirac Type Systems on the Half Line.} \vspace{0.5cm} \author{M.\,M. Malamud} {e-mail: malamud3m@gmail.com }\\ \end{center} On the half line $\Bbb R_+$ we will discuss first order differential equation of the form % \begin{equation}\label{eq:Dir1} \frac{1}{i}B\frac{dy}{dx} + Q(x)y = \lambda y, \qquad y=\binom{y_1}{y_2}, \quad y_j\in\Bbb C^{n} \end{equation} % subject to the boundary condition \begin{equation}\label{eq:Dir2} H_2 y_2(0)=H_1 y_1(0), \qquad \text{where} \qquad H_1,H_2\in\Bbb C^{n\times n}. \end{equation} % Here $y_1(0),y_2(0)\in\Bbb C^n$ denote the first $n$, resp. last $n$, components of the vector $y(0)$. Besides, % \begin{equation}\label{eq:Dir3} B= \text{diag}(B_1,-B_2) = \begin{pmatrix}B_1& \Bbb O_n\\ \Bbb O_n &-B_2 \end{pmatrix} =B^* \in\Bbb C^{2n\times 2n}, \end{equation} % and matrices $B_j(\in\Bbb C^{n\times n})$, $j\in \{1,2\}$, are positive definite. We also assume that a potential matrix $Q(\cdot)$ is selfadjoint block off diagonal matrix function, $Q(\cdot) = Q(\cdot)^*$. It happens that the minimal operator $L_{H_1,H_2}$ generated in $L^2(\Bbb R_+; \Bbb C^{2n})$ by the problem \eqref{eq:Dir1} -- \eqref{eq:Dir3} is self-adjoint if and ony if the matrices $H_1$ and $H_2$ are invertible and $B_1 = H^*B_2H$, where $H=H_2^{-1}H_1$. We present necessary and sufficient conditions for a $n\times n$ matrix function $\sigma(\cdot)$ to be the spectral (non-orthogonal) measure of the problem \eqref{eq:Dir1} -- \eqref{eq:Dir3} (of the operator $L_{H_1,H_2}$). Using the exitance of triangular transformation operator for appropriate matrix solutions of the problem \eqref{eq:Dir1} -- \eqref{eq:Dir3} we discuss a version of the Gelfand-Levitan type equation. Moreover, we present an explicit procedure for solvability of this equation that allows to determination the potential matrix $Q(\cdot) = Q(\cdot)^*$ from $\sigma(\cdot)$. In $2\times 2$-Dirac operators these results complete and improve the results due to Gasymov and Levitan (see [1]). We will also discuss the problem of describing possible spectral types of Dirac type systems by indicating admissible spectral multiplicities $\le n$. In particular, we will discuss the existence of $2n\times 2n$ Dirac type systems on the half line with purely absolute continuous, purely singular continuous, and purely discrete spectrum with prescribed multiplicity function $N_L(t)=p(t)\le n$. The talk is partially based on results of the works [2]-[3] and some new results. \vspace{0.5cm} {\bf Список литературы.} %\begin{thebibliography}{9} \begin{enumerate} \item B.M. Levitan and I.S. Sargsjan, Sturm-Liouville and Dirac operators. Kluwer, Dordrecht, 1991. \item M. Lesch M. Malamud, The Inverse Spectral Problem for First Order Systems on the Half Line. Operator Theory: Advances and Appl., Vol. 117 (2000), p. 199-238. Birkhauser Verlag Basel/Switzerland \item Malamud M.M., {Unique Determination of a System by a Part of the Monodromy Matrix.} Funct. Anal. and Its Applications, Vol. 49, No. 4 (2015), p. 264--278. \end{enumerate} \end{document}

One of the main examples of a quantum graph of the "tree" type is an electrical network consisting of wires (edges) connected in transformer substations (vertices). Differential equations describe electrical oscillations in wires of length $l_k$ with distributed capacitance $C_k$ and inductance $L_k$. The vertices of degree one are numbered from $1$ to $N$. Suppose that one of the ends $e_k$ of the wire is grounded through a concentrated induction $\~{L}_k$ and a capacitance $\~{C}_k$ connected in series, and the current flows through them in the opposite direction to the direction of increase $U_k$. Two gluing conditions are set for each of the $M = P + 1 – N$ inner vertices. The condition of potential continuity is that if the same vertex v is incident to several edges $e_j \in E_v$, then the values of the components of the function $U_j$ on these edges at the ends corresponding to the vertex $v$ coincide. The Kirchhoff condition or current balance is that the sum of the normal derivatives of the components of the function $U_k$ at the inner vertex $v$ is zero. The second example is to consider a system of homogeneous rods in the form of an arbitrary graph of the "tree" type. On each of the $𝑃$ edges, the equation of thermal conductivity in a rod of length $l_j$ is set in the absence of external heat sources with a thermal conductivity coefficient $k_j$, specific heat capacity $c_j$, and material density $p_j$. Concentrated heat capacities $\~{c}_j$ are placed at the free ends of the rods $x_j=l_j, j=(1,N)$, and heat exchange occurs with the corresponding coefficient $h_j$ with an external medium of zero temperature. The conditions of temperature continuity and thermal balance are set in each of the $𝑀 = 𝑃 + 1 − 𝑁$ inner vertices. For both examples, boundary value problems for eigenvalues are obtained, which coincide and are equivalent to the operator equation $K_1 (p_{11}, ..., p_{N1} )=\lambda K_2 (p_{12}, …, p_{N2} )$. Prove the monotonic dependence of the eigenvalues $\lambda$ of the boundary value problem on the parameters of the boundary conditions $\vec{p}=(p_{11}, ..., p_{N1}, p_{12}, ..., p_{N2} )$. The inverse spectral problem for the boundary value problem consists in finding the values of the vector $\vec{p}$ of the coefficients of the boundary conditions under which the pre-set numbers $\lambda_1, ..., \lambda_2N$ are the eigenvalues of the boundary value problem. This problem is reduced to a multiparametric inverse spectral problem for a finite-dimensional operator defined in a finite-dimensional real Euclidean space $E^{2N}$, for which an algorithm for numerical construction of solutions based on the monotonic dependence of eigenvalues on the parameters of boundary conditions is proposed. The obtained results make it possible to restore the parameters of boundary conditions, for example, distributed inductance and capacitance connected in series for electrical networks in areas difficult to access for visual inspection, to select the parameters of boundary conditions to ensure the desired frequency spectrum of alternating current or voltage fluctuations in the network, as well as the values of concentrated heat capacities and heat transfer coefficients at the ends of the rods.

The Hamiltonian formulation of the equations of ideal hydrodynamics in Cartesian coordinates is considered. It is shown that in arbitrary curvilinear coordinates the system can be transformed to a form with the same Poisson bracket as for Cartesian coordinates. Reductions of the system with respect to curvilinear coordinates and the connection with Casimir functionals are considered.

Relations are established between the dynamic inverse problem for discrete-time dynamical systems associated with complex Jacobi matrices and the complex moment problem. For the trancated problem, we give sufficient conditions for a sequence of complex numbers to be a sequence of moments of some Borel measure on $\mathbb{C}$.

A dynamic scattering problem for the Schrödinger operator on the half-line is considered. Relationships between inverse data for quantum and acoustic scattering problems are established, and an answer is given to the question of characterizing inverse dynamic data.

The report will consider an approach to the problem of integrating hyperbolic systems of first-order differential equations based on the procedure of contacting symplectic spaces. This class includes both quasilinear equations and equations containing Jacobian-type nonlinearities. According to [1], such systems can be considered as almost product structures on a four-dimensional space. Our approach is based on the transformation to a five-dimensional contact space, which allows us to use contact geometry methods [2]. This approach significantly expands the class of admissible transformations of systems and, in some cases important for applications, allows us to obtain their exact solutions. The method will be illustrated using nonlinear filtration equations. 1. Lychagin V. V. Differential equations on two-dimensional manifolds // Izvestiya vuzov. Matematika. - 1992. - No. 5. - P. 43-57 [in Russian]. 2. Kushner A. G., Lychagin V. V., Rubtsov V. N. Contact geometry and nonlinear differential equations. Encyclopedia of Mathematics and Its Applications. - Cambridge: Cambridge University Press, 2007.

We consider a radiation solution $\psi$ for the Helmholtz equation in an exterior region in $\mathbb{R}^2$. We show that $\psi$ in the exterior region is uniquely determined by its imaginary part $Im(\psi)$ on an interval of a line $L$ lying in the exterior region. This result has a holographic prototype in the recent work Nair, Novikov (2025, J. Geom. Anal. 35, 123). Some other curves for measurements, instead of the lines $L$, are also considered. Applications to the Gelfand-Krein-Levitan inverse problem (from boundary values of the spectral measure in $\mathbb{R}^2$) and passive imaging are also indicated.

We consider a radiation solution $\psi$ for the Helmholtz equation in an exterior region in $R^3$. We show that the restriction of $\psi$ to any ray $L$ in the exterior region is uniquely determined by its imaginary part Im $\psi$ on an interval of this ray. As a corollary, the restriction of $\psi$ to any plane $X$ in the exterior region is uniquely determined by Im $\psi$ on an open domain in this plane. These results have holographic prototypes in the recent work Novikov (2024, Proc. Steklov Inst. Math. 325, 218-223). In particular, these and known results imply a holographic type global uniqueness in passive imaging and for the Gelfand-Krein-Levitan inverse problem (from boundary values of the spectral measure in the whole space) in the monochromatic case. Some other surfaces for measurements instead of the planes $X$ are also considered.

Polarimetric imaging has emerged as a promising technique for enhancing tissue contrast and structural information in medical applications, particularly in neurosurgery. Our prior studies have demonstrated the ability of wide-field imaging Mueller polarimetry to highlight microstructural differences between healthy and pathological brain tissue. However, its adoption in brain tumor surgery has been limited by the computational complexity of Mueller matrix (MM) analysis and the time-consuming nature of conventional data processing pipelines. In this talk, we present a novel framework for fast post-processing of polarimetric data, enabling real-time visualization of brain tumor border during neurosurgery. Our approach integrates optimized algorithms and machine learning models for checking the physical realizability of the MM and reconstructing diagnostic polarimetric maps, leveraging GPU acceleration to achieve significant speed-up without compromising accuracy. This advancement represents a crucial step toward the clinical translation of polarimetric imaging systems that will provide surgeons with immediate feedback during critical decision-making stages. We will discuss the technical innovations, validation results, and current limitations, as well as outline future directions for integrating this technology into standard neurosurgical workflows.

This work presents recent advances in the formulation and solution of inverse problems for a novel class of nonlinear input-output models incorporating inputs substitution and resource constraints in network economies. Unlike the traditional linear Leontief interindustry balance framework, our generalized nonlinear approach enables equilibrium analysis of production networks while accounting for interdependencies between financial flows and goods and services prices. We propose a duality-based formulation, where the nonlinear input-output model is represented as a pair of Young dual variational inequalities. A key advantage of this framework is its ability to decouple the computation of equilibrium prices and financial flows under given scenario conditions. Additionally, we address challenges in scenario analysis and forecasting of primary inputs (non-produced resources such as labour and imported inputs) within the proposed resource-constrained framework.

Closing moment chains in the transition from kinetic to hydrodynamic models of continuous media is a classical, very complex problem. We consider it for the electron plasma model in the simplest one-dimensional case for which analytical results exist. In the kinetic formulation, the model was written out by Landau. The Cauchy problem for the kinetic model was investigated by Iordanskii, and it was found, in particular, that the solution preserves global smoothness for smooth initial data. However, the hydrodynamic analogue of this problem has completely different properties, namely, it is known that there is a wide class of smooth initial conditions for which smoothness is lost in a finite time. The hydrodynamic model of cold plasma is the result of closing moment chains at the first step. In this paper, an attempt is made to explain the change in the properties of the solution depending on the methods of closing the moment chain. In contrast to the traditional closure method, where the hyperbolicity of the system is lost already at the third step, we propose a method that leads to a non-strictly hyperbolic system at each step.

We consider a model of the evolution of the Lorenz curve describing the distribution of income between economic agents. In this model, the Pigou-Dalton transfer system generates a steady-state income distribution chosen by the welfare state. The model is described by a mixed problem whose main equation is a partial integro-differential equation. We prove that the steady-state solution of this equation is asymptotically stable. The proof involves studying the spectrum of the integral operator obtained by linearization of the integro-differential operator. It appears that this study of the spectrum can be reduced to finding zeros of an integral as a function of a complex parameter. Sufficient conditions for the absence of zeros of the integral in the right half-plane are given.

We consider the inverse problem: recovering a 2-nd order homogeneous and even w.r.t. momenta Hamiltonian by the scattering map and travel times.

We consider the particular case of a Radon transform (RT) with an axially symmetrical attenuation. We assume the geometry of the parallel beam tomography. A circular harmonic expansion leads to the generalized Cormack equation. The properties and results of the injectivity for this transform were investigated [1]. Application of Novikov's algorithm to a medium without refraction and with axially symmetric attenuation [2] allows us to construct a similar solution for a medium with axially symmetric refraction. The previously obtained solution [3] to the external problem is unstable. The compatibility conditions presented in the reprints [4] allow us to improve the stability of the previously obtained solution. Examples of laws of refraction that allow the inversion to be carried out in analytical form include the generalized Cormack type family of curves [5], [6]. References [1] de Hoop M., Ilmavirta J. 2017 Abel transforms with low regularity with applications to x-ray tomography on spherically symmetric manifolds. Inverse Problems 33, 124003. [2] Puro A., Garin A. 2013 Cormack-type inversion of attenuated Radon transform. Inverse problems. 29 (6) 065004 [3] Puro A. E. Tomography in Optically Axisymmetric Media. Optics and Spectroscopy. 2018, 124 (2), 278–284 [4] Puro A. Properties of Cormac-type inversion Radon transform in axially symmetric refraction media. August 2024 DOI: 10.13140/RG.2.2.11967.93604 [5] M. K. Nguyen and T. T. Truong, Inversion of a new circular-arc Radon transform for Compton tomography, Inverse Problems 26 (2010), no. 065005. [6] Rigaud. G. On the inversion of the Radon transform on a generalized Cormack-type class of curves. Inverse Problems 29( 2013) 115010.

We consider discrete Schr\"odinger operators $H=\Delta+Q$ with $\Gamma$-periodic potentials $Q$ on $\Gamma$-periodic graphs $\mathcal{G}$, where $\Delta$ is the discrete Laplacian and $\Gamma$ is a lattice of rank $d$. The spectrum of the operator $H$ is a union of spectra of Floquet operators (matrices) $H(k)$ acting on the finite quotient graph $\mathcal{G}/\Gamma$ and depending on the parameter $k\in[-\pi,\pi]^d$ called quasimomentum. The family of spectra of all Floquet operators $H(k)$ is referred to as the Floquet spectrum of $H$. It is known that the Floquet spectrum of the Schr\"odinger operator $H$ on the lattice $\mathbb{Z}^d$, $d\geq2$, generically (with respect to perturbations of $Q$) determines the potential $Q$ uniquely up to the obvious symmetries (translation and reflection) in $\mathbb{Z}^d$. We provide examples of other periodic graphs $\mathcal{G}$ for which the periodic potential $Q$ is uniquely (up to symmetries in the periodic graph) determined by the Floquet spectrum of the Schr\"odinger operator $H=\Delta+Q$ on $\mathcal{G}$. The proof is based on spectral invariants for the Schr\"odinger operator on periodic graphs. This work is supported by the Russian Science Foundation (project No. 25-21-00157).

1. For a compact Riemannian manifold $(M,g)$ with boundary $\partial M$, the Dirichlet-to-Neumann operator $\Lambda_g:C^\infty(\partial M)\longrightarrow C^\infty(\partial M)$ is defined by $\Lambda_gf=\left.\frac{\partial u}{\partial\nu}\right|_{\partial M}$, where $\nu$ is the unit outer normal vector to the boundary and $u$ is the solution to the Dirichlet problem $\Delta_gu=0,\ u|_{\partial M}=f$. Let $g_\partial$ be the Riemannian metric on $\partial M$ induced by $g$. The Calder\'{o}n problem is posed as follows: To what extent is $(M,g)$ determined by the data $(\partial M,g_\partial,\Lambda_g)$? In the two-dimensional case a surface $(M,g)$ is determined by the data uniquely up to conformal equivalence. We suggest two approaches to numerical solution of the two-dimensional Calder\'{o}n problem for simply connected sufaces. Some numerical results are presented. It is the joint work with Alexander Samokhin. 2. The Doppler transform $I$ measures the job of a vector field over lines. The operator $I$ has a nontrivial kernel, only the solenoidal part of a vector field $f$ can be recovered from $If$. In the two-dimensional case, we derive an analogous of the Cormack inversion formula which recovers a solenoidal vector field from integrals measured over lines that do not intersect a certain disk. Then we study the exterior problem for the two-dimensional Doppler transform in two cases: (1) for vector fields defined in a bounded annulus and (2) for vector fields in an unbounded annulus. The theorem on decomposition of a vector field into solenoidal and potential parts is proved in both cases. These two theorems are very different; in particular, the decomposition is not unique in the case of an unbounded annulus. The algorithm of recovering the solenoidal part of a vector field is presented in both cases. Finally a numerical example of reconstructing a solenoidal vector field is presented. It is a joint work with Nikita Vaitsel.

Bifurcation theory attempts to explain the effects of various kinds of nonlinearities in the description of natural phenomena. Currently, there is a growing interest in utilizing the nature and consequences of bifurcation, particularly in some engineering applications where such analysis can be used to tune the system parameters for beneficial purposes. From a phenomenological viewpoint, the observed states of a physical system often correspond to solutions of certain nonlinear equation. A state in such phenomenon can be observed if the system is stable, in some intuitive sense. Usually a slight change of a parameter in the system does not alter the solution to a great extent, and rather the solutions change continuously in a suitable manner. However, as a specific parameter crosses certain threshold, the system may organize itself into a new state that differs considerably from one observed before. The buckling of an Euler rod is a prototypical example. In general, the literature in three-dimensional nonlinear elasticity, and nonlinear mechanics, contains many examples in which a necessary condition for bifurcation is worked out by linearizing the nonlinear problem about a trivial line of (known) solutions; a critical value or `buckling load' at which the linearization fails to be injective is then determined as the threshold. In this talk, I will discuss specific examples as some recent applications of such analysis, namely, a local bifurcation analysis of circular von-Kármán plate with Kirchhoff rod boundary (in collaboration with Deepankar Das) and, if time permits, analysis of instabilities in colloidal crystals on fluid membranes (in collaboration with Sanjay Dharmavaram). The detailed calculations in case of the former are rather involved as they utilise a technique of symmetry based reduction for local post-buckling analysis, while the latter one addresses a simplified case of an original differential geometric problem but takes only first step towards understanding the bifurcations. Useful references: https://arxiv.org/abs/2501.18166, https://arxiv.org/abs/2501.10442, https://arxiv.org/abs/2506.17098

Tomography has been developing intensively over the past two decades. The development of ultrasound methods for early diagnosis of soft tissue malignancies is one of the key problems in medicine, and the absence of exposure to ionized radiation has become an urgent trend in medicine, which makes ultrasound tomography very popular. Ultrasound tomographs are currently being developed by various research groups around the world. One of the serious problems of ultrasound tomography is the development of effective and stable methods for solving nonlinear coefficient inverse problems. The most adequate model is a three-dimensional inverse problem in which the speed of wave propagation, the density of the medium, and other acoustic parameters must be reconstructed from data recorded by detectors located at the boundary of the region. The solution of the inverse problem is reduced to the minimizing the cost functional using the accelerated gradient methods. Formulas for the gradient of the functional are obtained using solutions to the corresponding direct and adjoint problems. The results of numerical calculations and a comparative analysis of numerical algorithms, including neural network approaches, are presented.

In this talk, we consider three-dimensional tomographic scheme for reconstructing parameters of inhomogeneous geophysical media. Input data for solving the inverse problem are propagation times of Rayleigh-type surface waves, estimated from microseismic noise correlations [1]. Such estimations are possible due to the well-known possibility of estimating the Green’s function from noise cross-correlations [2]. By using the linearized formulation of inverse problem, perturbations of surface wave propagation times are inverted into perturbations of medium parameters. As an example, reconstruction of shear-wave speed inhomogeneities is considered. A distinctive feature of our approach is the absence of an intermediate step for recovering two-dimensional distributions of surface wave velocities. A stripe basis [3] is used to describe three-dimensional shear-wave speed anomalies . This allows us to take into account the smoothness conditions of reconstructed functions in horizontal plane and in depth, thereby improving the conditionality of the inverse problem. Results of a numerical study of sensitivity, resolution ability and noise immunity of the regarded tomographic scheme are presented. Based on experimental data of international PLUME experiment [4], we have also obtained and present in this talk results of tomographic reconstruction of shear-wave speed inhomogeneities in the lithosphere and upper asthenosphere near the Hawaiian archipelago. These reconstruction results correspond to the expected parameters of the Hawaiian hot spot, thereby validating the robustness of the developed scheme. The reported study was funded by the Russian Science Foundation, project number 25-61-00027. 1. Tikhotskii S.A., Presnov D.A., Sobisevich A.L., Shurup A.S. The Use of Low-Frequency Noise in Passive Seismoacoustic Tomography of the Ocean Floor // Acoustical Physics. 2021. V. 67. N. 1. 2. Agaltsov A.D., Hohage T., Novikov R. G. Monochromatic Identities for the Green Function and Uniqueness Results for Passive Imaging // SIAM Journal on Applied Mathematics. 2018. V. 78. N. 5. 3. Burov V.A., Sergeev S.N., Shurup A.S. A three dimensional tomography model for reconstruction of oceanic inhomogeneities under unknown antenna positioning // Acoustical Physics. 2011. V. 57. N 3. 4. Laske G., Collins J.A., Wolfe C.J., Solomon S.C., Detrick R.S., Orcutt J.A., Bercovici D., Hauri E.H. Probing the Hawaiian hot spot with new ocean bottom instruments // EOS Trans. AGU. 2009. N. 90. http://www.fdsn.org/networks/detail/YS_2004/

The talk is dedicated to Dirichlet problem for the Sturm-Liouville equation perturbed by an integral operator with a convolution kernel. Sharp asymptotic formulas for the eigenvalues are found. The asymptotic contains information about the Fourier coefficients of the potential and the kernel, and estimates of the remainder which take into account both the rate of decrease as the eigenvalues tend to infinity and the rate of decrease as the norms of the potential and kernel tend to zero. The formulas are new even in the case of the Sturm-Liouville operator, when the convolution kernel is zero.

The determination of the dielectric constant of an unknown object Q is carried out using a two-step process [1-3]. In the first step, we solve a linear integro-differential equation to find the polarization current J(x) inside the object, using the known values of the incident E0(x) and total E(x) fields in a certain area D outside the object. By analyzing the field distribution outside the object, we can derive the integro-differential equation of thr first kind for J(x) in D [3]. Then, in the second step, the dielectric constant of the object is calculated directly from the found polarization current using the volume integro-differential equation in Q [3]. The results of solving the problem using the numerical method are presented in [4]. The research was carried out with the support of the Ministry of Science and Higher Education of the Russian Federation within the framework of the state assignment for research, registration number 124020200015-7. References 1. M.Yu. Medvedik, Yu.G. Smirnov, A.A. Tsupak. The two-step method for determining a piecewisecontinuous refractive index of a 2D scatterer by near field measurements. Inverse Problems in Science and Engineering, 28(3), 427–447 (2020) 2. M.Yu. Medvedik, Yu.G. Smirnov, A.A. Tsupak. Non-iterative two-step method for solving scalar inverse 3D diffraction problem. Inverse Problems in Science and Engineering, 28(1), 1474–1492 (2020) 3. M.Y. Medvedik, Y.G. Smirnov, A.A. Tsupak. Inverse vector problem of diffraction by inhomogeneous body with a piecewise smooth permittivity. J. Inverse Ill-Posed Probl. 32(3), 453–465 (2024) 4. Smolkin E., Smirnov Y., Snegur M. Solution of the Vector Three-Dimensional Inverse Problem on an Inhomogeneous Dielectric Hemisphere Using a Two-Step Method. Computation, 12 (11) 213 (2024)

The report considers the results of the study of interconnected 2D models of sediment transport and 3D models of suspended matter transport consisting of several fractions that may differ in average particle sizes, density and granulometric size, which determines the rate of particle sedimentation, as well as other characteristics. On the time grid, the linearization of the initial-boundary value problem of sediment transport is performed by setting nonlinear coefficients on the previous time layer. For the original problem of suspended matter transport, a "delay" transformation is performed on the constructed time grid, in which for the functions - concentrations of suspended matter included in the right-hand sides of the equations of the problem and not related to the fraction for which the initial-boundary value problem for the diffusion-convection equation is formulated, the values ​​of these concentrations were determined on the previous time layer. The use of these transformations allows us to solve them at each time step as systems of independent linear grid equations of parabolic type with lower derivatives for each of the fractions of suspended matter. The conditions for correctness of the stated initial-boundary value problems of sediment and suspension transport are investigated. Sufficient conditions are obtained for the convergence of the solution of the linearized sediment transport problem to the solution of the original nonlinear problem in the norm of the Banach space L1 as the step τ of the time grid tends to zero at a rate of O(τ). Conditions are also obtained for the convergence of the solution of the transformed "with delay" problem of suspension transport to the solution of the original problem in the norm of the Hilbert space L2 at a rate of O(τ) as the step of the time grid tends to zero.

We discuss PT-symmetric finite gap one-dimensional Schrodinger operators, the inverse problem for them and the perturbations of the spectra of these operators.

Спрос на первичном рынке жилья в России почти полностью удовлетворяется за счет выдачи ипотечных кредитов. Условия на рынке ипотечного кредитования постоянно изменяются: вводятся новые программы льготного кредитования, меняются ставки по кредитам, растет стоимость типового жилья. Результаты введения ипотечных программ в 2020 году не являются однозначными, в частности в результате их введения в среднем произошло снижение покупательной способности населения на рынке жилья. В связи с этим, представляет интерес моделирование спроса на ипотеку при различных вариантах введения льготных программ, ставок кредитования.

A system of quantum mechanical particles is characterized by its density matrix. In particular, this matrix allows to compute the probabilities of the outcomes of measurements of the physical system. Natural inverse problems (sometimes referred to as quantum tomography) concern the reconstruction of the density matrix of a quantum mechanical system given a sufficiently rich set of measurements. Density matrices are positive semidefinite operator with trace equal to 1. We demonstrate that these properties may have a strongly regularizing effect, and may even turn an ill-posed operator equation into a well-posed one. Moreover, we will discuss regularization methods enforcing these properties, e.g. based on the von-Neumann entropy, prove their regularizing effect and discuss their implementations.

A new inverse model of acoustic imaging is presented. It was constructed by transforming an original inverse problem of an acoustic wave equation to the Lavrent’ev linear integral equation of the first kind, and then to a system of two linear elliptic problems as was first done in [ M.V.Klibanov and A.Timonov, SIAM J.Imaging Sciences, 18(2), 2025, 1002-1027, https://doi.org/10.1137/24M1671402 ]. This model is utilized for determining an approximate coefficient in the acoustic wave equation by the regularized variational quasi-reversibility method that exploits the total variation regularization. Some numerical results of comparative studies of three techniques, including also a regularized version of the Boundary Control Method, are demonstrated.

For the first time, the problem of determining the gradient, not the Fr\'{e}chet derivative, of a functional $J(u)$ for numerical optimization problems with non stationary quasi-linear systems under control $u(x)$ is discussed. It is shown that control should be considered as a function of both space $x$ and time $t$. The controllability of such a task is investigated, taking into account the mapping of the space-time gradient $\nabla J(u;x;t)\to\nabla J(u;x)$ by traditional time integration and projection onto the line $x$ at the right moment $t$. Examples for the quasi-linear hyperbolic systems are considered: identification of the roughness of an open channel; optimal design of the nozzle shape of a hydraulic gun. It is revealed that optimization with a new gradient form on the line implements the best approximation to the optimum. When optimizing the nozzle shape, new optimal shapes were found.

We present a mathematical description of the economic behavior of a representative borrower in an imperfect market of consumer loans. The model is formalized as a hybrid optimal control problem: a borrower chooses the strategy of the annuity payments in order to maximize his discounted consumptions, and, after paying off the debt, chooses the strategy of his consumptions based on a modified Ramsey-type model under current parameters of the economic conjuncture. A classification of the borrowers' behavior of different social layers depending on market condition parameters is studied.

The impact of the large number of high-frequency traders (HFTs) can be modelled via mean field term. A Mean Field Game is a coupled system of PDEs: a Kolmogorov–Fokker–Planck equation, evolving forward in time and describing evolution of the HFTs probability density function spread by the amount of asset shares; and a Hamilton–Jacobi–Bellman equation, evolving backwards in time and defining the strategy of the HFTs. These equations form a boundary value problem. We consider the model of the HFTs’ behavior in the stock market with reference to Fatone L., Mariani F., Recchioni M.C., Zirilli F. "A trading execution model based on mean field games and optimal controlAppl. Math. 5, 3091-3116 (2014). We apply the Mean Field Games approach to describe the Chinese stock market crash in 2015. The HFTs wanted to make profit through the transactions. The behavior of the professional traders can be described as the behavior of one rational representative trader. We solve an inverse problem of reproducing the asset price and explain the strategies of the HFTs and the professional traders in the stock market during the Chinese stock market crisis in 2015.

The Doppler transform $I$ measures the job of a vector field over lines. The operator $I$ has a nontrivial kernel, only the solenoidal part of a vector field $f$ can be recovered from $If$. In the two-dimensional case, we derive an analogous of the Cormack inversion formula which recovers a solenoidal vector field from integrals measured over lines that do not intersect a certain disk. Then we study the exterior problem for the two-dimensional Doppler transform in two cases: (1) for vector fields defined in a bounded annulus and (2) for vector fields in an unbounded annulus. The theorem on decomposition of a vector field into solenoidal and potential parts is proved in both cases. These two theorems are very different; in particular, the decomposition is not unique in the case of an unbounded annulus. The algorithm of recovering the solenoidal part of a vector field is presented in both cases. Finally a numerical example of reconstructing a solenoidal vector field is presented.

We discuss image-processing techniques for personalization of hemodynamic models demanded in two clinical applications. The first application is related to diagnosis of coronary heart disease. The second application is related to hemodynamic correction of congenital heart defects. {Danilov A., Ivanov Yu., Pryamonosov R., Vassilevski Yu.} {Methods of graph network reconstruction in personalized medicine // Int.J.Numer.Meth.Biomed.Engng., 2016, V.~32, No.8, e02754.} {Dobroserdova T., Yurpolskaya L., Vassilevski Yu., Svobodov A.} {Patient-specific input data for predictive modelling of the Fontan procedure // Math. Model. Nat. Phenom., 2024, V.~19, P.~16.}

The report considers the problem of modeling medical ultrasound imaging in relation to the study of brain structures through the skull wall. The bone tissue of the skull wall distorts the wave fronts, creating artifacts and aberrations in the image. The work describes mathematical models and numerical methods for solving a direct problem - calculating the formation of an image (B-scan) for heterogeneous media. The work also considers the inverse problem - restoring the real configuration of the medium and using the determined configuration to correct aberrations.

The presentation describes the expreience in the study, both analytical and numerical, of one specific type of linear optimal control problems with a state constraint of order k. A specific feature of this problem is that the extremal trajectories may have accumulated contact points with the boundary of the state constraint. The number of such contact points, or so-called tangency points, is determined entirely by the boundary conditions. In this work, we apply a special type of the Pontryagin maximum principle, which was derived to solve the problems of the specified class. The results of the theoretical analysis of the optimality conditions are applied to a computational scheme, which is then used to solve the problem numerically for different numbers of tangency points and different boundary conditions.

Scattering data recorded during tomography of an object under study require preliminary correction. This is due to the fact that characteristics of an antenna array, in general, deviate from the ideal characteristics assumed at the stage of recorded data processing by one or another wave algorithm for solving the inverse scattering problem. For example, ultrasound medical tomographs currently being developed [1–5] are intended for diagnosing soft biological tissues, primarily the mammary gland. The ultimate goal of the acoustic tomography process is to obtain tomograms of the object under study in the form of estimates of spatial distributions of sound speed and absorption coefficient, which characterize an internal structure of the object. It is necessary to ensure both high resolution of the tomograms and the correctness of quantitative values of the reconstructed characteristics for purposes of early diagnostics of neoplasms. This is ensured, firstly, by using strict wave algorithms for processing the scattering data and, secondly, by data correction, which allows compensating for antenna imperfections. Otherwise, the resolution of the tomograms will deteriorate sharply, and small details with a size commensurate with a wavelength and less will not be reproduced. At the same time, such small details are the most informative for the purposes of early diagnostics of pathologies in medical applications. Displacements of geometric positions of receiving and transmitting transducers of the antenna array from their ideal positions are one of the types of errors introduced into the recorded fields in practical conditions. The magnitude of such displacements can be comparable to the wavelength, which is about 1 mm in the megahertz frequency range. Thus, if such displacements are not taken into account at the data processing stage, this will lead to the aforementioned sharp decrease in the information capacity of the reconstructed tomograms. In connection with the designated problem, a method for correcting the recorded fields has been developed and numerically tested [6]. The displacements of the transducers from their ideal positions are determined in advance. The correction method involves finding the expansion coefficients over angular harmonics for the fields measured by displaced transducers. Then the found expansion coefficients allow us to recalculate the fields for ideal positions of the transducers and then to use the already corrected fields in the processing algorithm. In addition, the direct relationship has been obtained between the expansion coefficients of the fields transmitted and received by the quasi-point transducers and the scattering amplitude. The established relationship allows us to quite simply recalculate the complete scattering data of one type into data of another type. This is convenient, since different algorithms for solving the inverse problem must be fed with data of a certain fixed type to the input. The developed correction algorithm can be used for both the two-dimensional and three-dimensional tomography schemes. The efficiency and noise resistance of the correction algorithm are confirmed by numerical modeling, which has been performed by the high-precision two-dimensional functional algorithm [7]. This functional algorithm, using the ideas of solving the quantum-mechanical problems [8], allows us to take into account the processes of multiple waves scattering almost strictly. It has shown high efficiency in numerical implementation for solving the acoustic inverse problems of the tomographic type [9, 10]. At the same time, numerical modeling has shown that the tomograms can be destroyed in the absence of the correction of data measured in real conditions. This indicates the need to perform the correction procedure. The developed and tested algorithm for finding the coefficients of the field expansion over the angular harmonics can be used not only in the medical tomography problems, but also in the ocean tomography problems, after the geometric displacements of the transmitting and receiving transducers from their ideal positions have been preliminarily estimated – see, for example, [4 (section 8.2)]. The research was funded by the Russian Science Foundation (project No. № 25-61-00027, https://rscf.ru/project/25-61-00027/ REFERENCES 1. Duric N., Sak M., Fan S., Pfeiffer R. M., Littrup P. J., Simon M. S., Gorski D. H., Ali H., Purrington K. S., Brem R. F., Sherman M. E., Gierach G. L. Using whole breast ultrasound tomography to improve breast cancer risk assessment: a novel risk factor based on the quantitative tissue property of sound speed // Journal of Clinical Medicine. 2020. V. 9. N 2. Art. N 367. 2. Littrup P. J., Mehrmohammadi M., Duric N. Breast tomographic ultrasound: the spectrum from current dense breast cancer screenings to future theranostic treatments // Tomography. 2024. V. 10. N 4. P. 554–573. 3. Wiskin J. W., Borup D. T., Iuanow E., Klock J., Lenox M. W. 3-D nonlinear acoustic inverse scattering: Algorithm and quantitative results // IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control. 2017. V. 64. N 8. P. 1161–1174. 4. Буров В. А., Румянцева О. Д. Обратные волновые задачи акустической томографии. Ч. II: Обратные задачи акустического рассеяния. М.: ЛЕНАНД, 2020, 2021. 768 с. Burov V. A., Rumyantseva O. D. Inverse wave problems of acoustic tomography. Pt. 2: Inverse acoustic scattering problems. M.: URSS, 2020, 2021. 768 p. [in Russian]. 5. Dmitriev K. V., Zotov D. I., Rumyantseva O. D. Principles of obtaining and processing of acoustic signals in linear and nonlinear tomographs // Bulletin of the Russian Academy of Sciences: Physics. 2017. V. 81. N 8. P. 915–919. 6. Zotov D.I., Rumyantseva O.D. Compensating the effect of the displacement of array transducers on tomography data // Acoustical Physics. 2025. V. 71. N 2. P. 282–299. 7. Novikov R. G. Approximate inverse quantum scattering at fixed energy in dimension 2 // Proc. V. A. Steklov Inst. Math. 1999. V. 225. P. 285–302. 8. Faddeev L. D. Inverse problem of quantum scattering theory II // J. of Soviet Math. 1976. V. 5. P. 334–396. 9. Burov V. A., Rumyantseva O. D. Inverse wave problems of acoustic tomography. Pt. 4: Functional-analytical methods for solving the multidimensional acoustic inverse scattering problem. M.: URSS, 2024. 504 p. [in Russian]. 10. Zotov D. I., Rumyantseva O. D., Cherniaev A. S. Reconstructing the spatial distribution of acoustic characteristics using the angular harmonic technique // Bulletin of the Russian Academy of Sciences: Physics. 2025. V. 89. N 1. P. 109–114.

The process of information diffusion in online social networks characterizes by nonlinear diffusion logistic equation. Adding control parameter that provides a Nash equilibrium in the system of interacting agents and minimizes the cost functional transforms to the mean-field control. We consider a large number of users who can take states from 0 (means that the user is involved in the process of information dissemination) to 1 (means the opposite). Then the density of users obeys the Kolmogorov-Fokker-Planck equation. Using the Lagrange multiplier method a system similar to the Hamilton-Jacobi-Bellman equation and the optimality conditions are constructed. The numerical calculations were analyzed and discussed. The purpose of this work is to identify the cost function based on real data of influenced users in online social networks. Firstly, we propose the control function from epidemiology in population to the information dissemination problem. Then we develop the control function used real social data. The work was performed according to the Government research assignment for Sobolev Institute of Mathematics SB RAS, project FWNF-2024-0002.