We study ill-posed continuation problems for partial differential equations, with an emphasis on the mechanisms of ill-posedness and their mitigation via conditional stability and regularization. Three canonical examples of elliptic, parabolic, and hyperbolic type are used to illustrate the underlying ill-posedness. For a second-order elliptic continuation problem, we summarize well-posedness results for the associated direct and adjoint problems, establish conditional stability estimates, and develop an adjoint-based iterative reconstruction method with convergence-rate guarantees. For a parabolic continuation problem, we present corresponding well-posedness results and an adjoint-based iterative scheme. For a hyperbolic continuation problem, we derive a conditional stability result. We further analyze the singular numbers of the continuation operator for a complex-valued Helmholtz equation, thereby characterizing the frequency dependence of the ill-posedness. Finally, we compare Tikhonov regularization with linear neural networks for ill-posed Helmholtz inverse problems, highlighting their complementary strengths.

In this paper, we develop the explicit finite difference method (FDM) to solve an ill-posed Cauchy problem for the 3D acoustic wave equation in a time domain with the data on a part of the boundary given (continuation problem) in a cube. FDM is one of the numerical methods used to compute the solutions of hyperbolic partial differential equations (PDEs) by discretizing the given domain into a finite number of regions and a consequent reduction in given PDEs into a system of linear algebraic equations (SLAE). We present a theory, and through Matlab Version: 9.14.0.2286388 (R2023a), we find an efficient solution of a dense system of equations by implementing the numerical solution of this approach using several iterative techniques. We extend the formulation of the Jacobi, Gauss–Seidel, and successive over-relaxation (SOR) iterative methods in solving the linear system for computational efficiency and for the properties of the convergence of the proposed method. Numerical experiments are conducted, and we compare the analytical solution and numerical solution for different time phenomena.

In this paper, we revisit Linear Neural Networks (LNNs) with single-output neurons performing linear operations. The study focuses on constructing an optimal regularized weight matrix Q from training pairs { G , H } {\{G,H\}} , reformulating the LNNs framework as matrix equations, and addressing it as a linear inverse problem. The ill-posedness of linear machine learning problems is analyzed through the lens of inverse problems. Furthermore, classical and modern regularization techniques from both the machine learning and inverse problems communities are reviewed. The effectiveness of LNNs is demonstrated through a real-world application in blood test classification, highlighting their practical value in solving real-life problems.

The paper considers an inverse problem and a direct problem for the Burgers equation in a domain with movable boundaries. With the help of an additional condition, a formula is obtained for determining the desired function from a direct problem for the loaded Burgers equation for the solvability of which we require a condition on the functions according to which the boundaries of the domain change. The solvability of direct problems is proved using a priori estimates and the methods of Faedo-Galerkin and functional analysis.

. In this paper, we commence by providing a succinct overview of the mechanics behind Physics-Informed Neural Networks (PINNs) tailored for resolving partial differential equations (PDEs), along with an introduction to certain nonlinear Schrodinger equations pertinent to applications in semiconductor optical amplifier fiber lasers. Subsequently, we introduce an innovative architecture for PINNs, which incorporates an adaptive activation function and Latin Hypercube Sampling, aimed at the efficient approximation of soliton solutions to nonlinear Schrodinger equations. A series of numerical examples are presented to illustrate the straightforwardness and effectiveness of our proposed approach.

Overdetermined boundary value problems and the minimal operators generated by them are extremely important in the description of regular boundary value problems for differential equations, and are also widely used in the study of local properties of solutions. In addition, for inverse problems of mathematical physics arising from applications, when determining unknown data, it is necessary to study problems with overdetermined boundary conditions, which is reflected in the study of problems, including for hyperbolic equations and systems, arising in physics, geophysics, seismic tomography, geoelectrics, electrodynamics, medicine, ecology, economics and many other practical areas. Thus, the study of overdetermined boundary value problems is of both theoretical and applied interest. In this paper, a criterion for the regular solvability of the overdetermined Cauchy problem for the Gellerstedt equation and the minimal differential operator generated by it in a hyperbolic domain is established, as which both the case of a characteristic triangle and the case of a more general domain with fairly general assumptions about the boundary of the domain are considered. Due to overdetermined boundary conditions, the problem under consideration will be ill-posed in the general case, therefore, for its regular solvability, additional conditions must be imposed on the initial data. In other words, we have considered the inverse problem: to determine what requirements the initial data of the problem, in particular the right part of the Gellerstedt equation, should meet, in question, so that the overdetermined Cauchy problem is regularly solvable. The proof is based on the Gellerstedt potential, the properties of solutions of the Goursat problem in the characteristic triangle, and the properties of special functions.

This paper proposes and analyzes a mathematical model of tuberculosis and HIV co-infection that specifies for Russian Federation regions, based on mass balance law and described by eight ordinary differential equations. A sensitivity-based identifiability analysis of this mathematical model was performed, which revealed the sensitivity of the averaged parameters of the models to statistical real data of infectious individuals based on the Sobol method. The problem of identifying the sensitive epidemiological parameters (contagiousness, the rate of tuberculosis activation, additional mortality rate, etc.) for the model was reduced to the problem of minimization of the quadratic misfit function. The numerical results of the modeling of the number of people expected to be infected with tuberculosis and HIV were shown and discussed for the Sverdlovsk and Moscow regions of the Russian Federation. It has been shown that increasing the capacity of the medical system by 10% will make it possible to reduce the number of diagnosed cases of active tuberculosis by 2 times over the next 3 years in some regions of Russian Federation.

In this paper, we consider the simplest version of a linear neural network (LNN). Assuming that for training (constructing an optimal weight matrix Q) we have a set of training pairs, i.e. we know the input data G={g^(1),g^(2),⋯,g^(K)}, as well as the correct answers to these input data H={h^(1),h^(2),⋯,h^(K)}. We will study the possibilities of constructing a weight matrix Q of a neural network that will give correct answers to arbitrary input data based on the connection of the specified problem with a system of linear algebraic equations (SLAE). Consider a class of neural networks in which each neuron has only one output signal and performs linear operations. We will show how such LNEs are reduced to SLAEs. Since the questions G and the correct answers H are known to us, the desired weight matrix Q must satisfy the equations Qg^(k)=h^(k), k=1,2,⋯,K. It is required to restore Q. In the general case, the matrix Q is rectangular Q=Q_MN={q_mn}, m is the row number, and g^(k)∈ℝ^N, h^(k)∈ℝ^M. Let G_NK be a matrix composed of columns g^(1), g^(2),⋯,g^(k), and H_MK be a matrix composed of columns h^(1),h^(2),⋯,h^(k). Then, with respect to Q_MN, we obtain a matrix SLAE. Q_MNG_NK=H_MK. This paper will present methods for regularizing the constructed system.

В статье рассматривается алгоритм получения изображения самоподобного объекта, который является результатом вычисления относительной погрешности различных конечно-разностных схем решения задачи Коши второго порядка с помощью итерационного процесса. Построенный графический алгоритм позволил моделировать изображение множества для изучения, например, для выявления областей устойчивости решения задачи. С помощью программы можно наблюдать при каких условиях и на каких точках значение погрешности может стремиться к бесконечности или оставаться в области определенных значений. Полученная модель позволяет определить характер изменений множества в зависимости от исходных параметров, таких как шаг дискретизации, точность оценки, области на комплексной плоскости. Приводится компьютерный графический анализ указанных явлений. Компьютер можно превратить в своеобразный микроскоп и наблюдать с его помощью за поведением границ области. Мақалада итерациялық процессті пайдалана отырып, екінші ретті Коши есебін шешуге арналған әр түрлі шекті-айырымдық сұлбаларының салыстырмалы қателігін есептеу арқылы қалыптасатын өзіне-өзіұқсас объектінің кескінін алу алгоритмі қарастырылады. Құрылған графикалық алгоритм зерттеуге арналған жиынтық бейнесін имитациялауға, мысалы, есепті шешу үшін тұрақтылық аймақтарын анықтауға мүмкіндік берді. Бағдарламаның көмегімен қандай жағдайларда және қандай нүктелерде қателікмәні шексіздікке ұмтылатынын немесе белгілі бір мәндер аймағында қалатынын байқауға болады. Алынған модель дискретизация қадамы, бағалау дәлдігі, күрделі жазықтықтың аумақтары сияқты бастапқы параметрлерге байланысты жиынтықтағы өзгерістердің сипатын анықтауға мүмкіндік береді. Бұл құбылыстардың компьютерлік графикалық талдауы берілген. Компьютерді микроскоптың бір түріне айналдырып, оны аймақтың шекараларының мінез-құлқын бақылау үшін пайдалануға болады. The article considers the algorithm of obtaining self-similar object, which forms by calculating the relative error of various finite-difference schemes for solving the second-order Cauchy problem using iterative process. The constructed graphical algorithm made it possible to simulate the image of the set for study, for example, to identify areas of stability for solving the problem. Using the program, it is possible to observe under what conditions and at what points the relative error value tends to infinity or remains in the area of certain values. The resulting model allows you to determine the nature of the changes in the set depending on the initial parameters, such as discretization step, estimation accuracy, areas of the complex plane. A computer graphical analysis of these phenomena is given. The computer can be turned into a kind of microscope and use it to observe the behavior of the boundaries of the region.

This paper presents classification and analysis of the mathematical models of the spread of COVID-19 in different groups of population such as family, school, office (3-100 people), town (100-5000 people), city, region (0.5-15 million people), country, continent, and the world. The classification covers major types of models (time-series, differential, imitation ones, neural networks models and their combinations). The time-series models are based on analysis of time series using filtration, regression and network methods. The differential models are those derived from systems of ordinary and stochastic differential equations as well as partial differential equations. The imitation models include cellular automata and agent-based models. The fourth group in the classification consists of combinations of nonlinear Markov chains and optimal control theory, derived by methods of the mean-field game theory. COVID-19 is a novel and complicated disease, and the parameters of most models are, as a rule, unknown and estimated by solving inverse problems. The paper contains an analysis of major algorithms of solving inverse problems: stochastic optimization, nature-inspired algorithms (genetic, differential evolution, particle swarm, etc.), assimilation methods, big-data analysis, and machine learning.

The work is devoted to some generalization of the regularization method for ill-conditioned and degenerate systems of linear algebraic equations with symmetric non-negative definite matrices. For regularization, a shift of matrix is used with the unit operator multiplied by a small positive parameter. In addition to the regularization with one small parameter, two variants of linear combinations of regularized solutions with different parameters are considered. In the first of them, for a consistent system of equations, the choice of weights for a linear combination increases its achievable order of accuracy in comparison with the one-parameter regularization. In the second variant, a special choice of weights allows one to find an approximate normal pseudo-solution of an inconsistent system with the usual order of accuracy without orthogonalization of the right-hand side to the matrix kernel. A computational experiment illustrates the theoretical result.

Abstract The horizontally diagonalize and fit (HDF) method is proposed to solve the ill-posed Cauchy problem for the three-dimensional Poisson equation with data given on the part of the boundary (a continuation problem). The HDF method consists in discretization over horizontal variables and transformation of the system of differential equations to a diagonal form. This allows to uncouple the original three-dimensional continuation problem into a moderate number of one-dimensional problems in the vertical dimension. The problem size reduction can be carried taking into account the noise level, so that the number k of one-dimensional problems appears to be a regularization parameter. Our experiments show that HDF is applicable to large-scale problems and for n ≤ 2500 {n\leq 2500} is significantly more efficient than Landweber iteration.

Earlier, a method for constructing an initial approximation for solving the inverse problem of acoustics by a gradient method based on a convolutional neural network trained to predict the distribution of velocities in a medium from wave response was proposed [9]. It was shown that the neural network trained on responses from simple layered media can be successfully used for solving the inverse problem for a significantly more complex model. In this paper, we present algorithms for processing data about epidemics and an example of applying a neural network for modeling the propagation of COVID-19 in Novosibirsk region (Russia) based only on data. A neural network NN-COVID-19 that uses data about the epidemics is constructed. It is shown that this neural network predicts the propagation of COVID-19 for five days by an order of magnitude better than SEIR-HCD. When a new variant (Omicron) appeared, this neural network was able to predict (after retraining) the propagation of the epidemics more accurately. Note that the proposed neural network uses not only epidemiological data but also social ones (such as holidays, restrictive measures, etc.). The proposed approach makes it possible to refine mathematical models. A comparison of the curves constructed by SEIR-HCD model and by the neural network shows that the plots of solutions of the direct problem almost coincide with the plots constructed by the neural network. This helps refine coefficients of the differential model.

The paper is devoted to the short review and application of sensitivity-based identifiability approaches for analyzing mathematical models of epidemiology and related processes described by systems of differential equations and agent-based models. It is shown that for structural identifiability of basic SIR models (describe the dynamic of Susceptible, Infected and Removed groups based on nonlinear ordinary differential equations) of epidemic spread and linear compartmental models it is possible to use a priori information about the process. It is demonstrated that a model can be structurally identifiable but be practically non-identifiable due to incomplete data. The paper uses methods for analyzing the sensitivity of parameters to data variation, as well as analyzing the sensitivity of model states to parameter variation, based on linear and differential algebra, Bayesian, and Monte Carlo approaches. It was shown that in the SEIR-HCD model of COVID-19 propagation, described by a system of seven ordinary differential equations and based on the mass balance law, the parameter of humoral immunity acquisition is the least sensitive to changes in the number of diagnosed, critical and mortality cases of COVID-19. The spatial SEIR-HCD model of COVID-19 propagation demonstrated an increase the sensitivity of the partial immunity duration parameter over time, as well as a decrease in the limits of change in the infectivity and infection parameters. In the case of the SEIR-HCD mean-field model of COVID-19 propagation, the sensitivity of the system to the self-isolation index and the lack of sensitivity of the stochastic parameters of the system are shown. In the case of the agent-based COVID-19 propagation model, the change in the infectivity parameter was reduced by more than a factor of 2 compared to the statistics. A differential model of co-infection HIV and tuberculosis spread with multiple drug resistance was developed and its local identifiability was shown.

We propose an algorithm for modeling scenarios for newly diagnosed cases of COVID-19 in the Republic of Kazakhstan. The algorithm is based on treating incomplete epidemiological data and solving the inverse problem of reconstructing the parameters of the agent-based model (ABM) using the set of available epidemiological data. The main tool for constructing the ABM is the Covasim open library. In the event of a drastic change in the situation (appearance of a new strain, removal or introduction of restrictive measures, etc.), the model parameters are updated taking into account additional information for the previous month (online data assimilation). The inverse problem is solved by stochastic global optimization (of tree-structured Parzen estimators). As an example, we give two scenarios of COVID-19 propagation calculated on December 12, 2021 for the period up to January 20, 2022. The scenario that took into account the New Year holidays (published on December 12, 2021 on http://covid19-modeling.ru ) almost coincided with what happened in reality (the error was 0.2

An algorithm has been developed for numerically solving the source determination problem in the model of information dissemination in synthetic online social networks, described by reaction–diffusion-type equations, using additional information about the process at fixed time points. The degree of ill-posedness of the source determination problem for a parabolic equation is studied based on the analysis of singular values of the linearized operator of the inverse problem. The algorithm developed is based on a combination of the tensor optimization method and gradient descent supplemented with the A.N. Tikhonov regularization. Numerical calculations demonstrate the smallest relative error in the reconstructed source obtained by the developed algorithm in comparison with classical approaches.

In this paper, we consider the Gelfand–Levitan–Marchenko–Krein approach. It is used for solving a variety of inverse problems, like inverse scattering or inverse problems for wave-type equations in both spectral and dynamic formulations. The approach is based on a reduction of the problem to the set of integral equations. While it is used in a wide range of applications, one of the most famous parts of the approach is given via the inverse scattering method, which utilizes solving the inverse problem for integrating the nonlinear Schrodinger equation. In this work, we present a short historical review that reflects the development of the approach, provide the variations of the method for 1D and 2D problems and consider some aspects of numerical solutions of the corresponding integral equations.

Abstract The problem of identification of unknown epidemiological parameters (contagiosity, the initial number of infected individuals, probability of being tested) of an agent-based model of COVID-19 spread in Novosibirsk region is solved and analyzed. The first stage of modeling involves data analysis based on the machine learning approach that allows one to determine correlated datasets of performed PCR tests and number of daily diagnoses and detect some features (seasonality, stationarity, data correlation) to be used for COVID-19 spread modeling. At the second stage, the unknown model parameters that depend on the date of introducing of containment measures are calibrated with the usage of additional measurements such as the number of daily diagnosed and tested people using PCR, their daily mortality rate and other statistical information about the disease. The calibration is based on minimization of the misfit function for daily diagnosed data. The OPTUNA optimization framework with tree-structured Parzen estimator and covariance matrix adaptation evolution strategy is used to minimize the misfit function. Due to ill-posedness of identification problem, the identifiability analysis is carried out to construct the regularization algorithm. At the third stage, the identified parameters of COVID-19 for Novosibirsk region and different scenarios of COVID-19 spread are analyzed in relation to introduced quarantine measures. This kind of modeling can be used to select effective anti-pandemic programs.

In this paper, numerical methods for solving multidimensional equations of hyperbolic type by the Gelfand-Levitan method are proposed and implemented. The Gelfand-Levitan method is one of the most widely used in the theory of inverse problems and consists in reducing a nonlinear inverse problem to a one-parameter family of linear Fredholm integral equations of the first and second kind. In the class of generalized functions, the initial-boundary value problem for a multidimensional hyperbolic equation is reduced to the Goursat problem. Discretization and numerical implementation of the direct Goursat problem are obtained to obtain additional information for solving a multidimensional inverse problem of hyperbolic type. For the numerical solution, a sequence of Goursat problems is used for each giveny. A comparative analysis of numerical experiments of the two-dimensional Gelfand-Levitan equation is performed. Numerical experiments are presented in the form of tables and figures for various continuous functions q(x, y).

In the theory of partial differential equations, an example constructed by J. Hadamard, which shows the instability of the solution of the Cauchy problem for the Laplace equation with respect to small changes in the initial data, is of great importance. Hadamard's example served as the beginning of a systematic study of ill-posed problems in mathematical physics. On the other hand, the study of the Cauchy problem for the Laplace equation arises from problems of geophysics. At the same time, the question arises whether the Cauchy problem is correct for other elliptic equations including degenerate elliptic equations. We have constructed analogs of Hadamard's example and established the incorrectness of the solution of the Cauchy problem for the Gellerstedt equation in two-dimensional and multidimensional cases. The condition of strong solvability of the mixed Cauchy problem for the multidimensional Gellerstedt equation in a cylindrical domain is found. The proof is based on the spectral properties of the Laplace operator and the properties of special functions.