arXiv subjects

Michael V. Klibanov

Publications and source records attributed to Michael V. Klibanov.

At least 19 recordsLinked to original sources

arXiv2026

Simultaneous Reconstructions of (x,t) -Dependent Coefficients and Initial Conditions of Parabolic Equations

This is the first publication, in which an inverse problem of the simultaneous reconstruction of an unknown (x,t) - dependent coefficient and an unknown initial condition in a general linear parabolic equation of the second order is considered. The input data depend on (n+1) variables and are, therefore, formally determined ones. Stability estimates and uniqueness theorem are obtained for this inverse problem. As a by-product, logarithmic stability estimates for initial conditions of parabolic equations and inequalities with time reversed data are obtained for the first time. All known publications about inverse problems for parabolic equations with formally determined input data and unknown coefficients assume that those coefficients depend either only on x or only on t. In addition, the initial condition is assumed to be known, except of two publications cited in the text. The inverse problem of this paper has two potential applications. The first one is in forecasting of public opinions in the framework of the Mean Field Games theory. The second one is in tracking spatiotemporal outbreaks of epidemics.

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arXiv2026

On an $n-$Dimensional Travel Time Tomography Problem

In their seminal works Herglotz (1905) and Wiechert and Zoeppritz (1907) have solved the so-called Travel Time Tomography Problem (TTTP) in the 1-D case. However, the question about stability estimates and uniqueness theorems for an n-D n>= 2 TTTP with formally determined incomplete input data still mostly stands open after more than one hundred years period. \textquotedblleft Formally determined input data" means that the number p of free variables in the input data equals the number $n$ of free variables in the unknown right hand side of the governing nonliniear eikonal PDE, p=n. Some previous publications demonstrate that it is possible to develop well performed numerical methods for the TTTP with formally determined input data, which indicates the importance of such data for practical applications. This is the first publication in which the above question is addressed. More precisely, we consider a semi-discrete case, in which a PDE generated by the eikonal equation is written in finite differences with respect to n-1 variables. In addition, it is assumed that the solution of that semi-discrete PDE is represented via a truncated Fourier-like series with respect to a special orthonormal basis of functions, which depend only on the position of the point source. Under these conditions, Lipschitz stability estimate is proven, and this estimate implies uniqueness. An important tool of this paper is a new Carleman estimate. Carleman Weighted Spaces are introduced. Carleman estimates were not applied previously to address questions about stability estimates and uniqueness theorems for the TTTP.

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arXiv2026

Toward Practical Forecasts of Public Sentiments via Convexification for Mean Field Games: Evidence from Real World COVID-19 Discussion Data

We apply a convexification-based numerical method to forecast public sentiment dynamics using Mean Field Games (MFGs). The theoretical foundation for the convexification approach, established in our prior work, guarantees global convergence to the unique solution to the MFG system. The present work demonstrates the practical potential of this framework using real-world sentiment data extracted from social media public discussion during the COVID-19 pandemic. The results show that the MFG model with appropriate parameters and convexification yields sentiment density predictions that align closely with observed data and satisfy the governing equations. While current parameter selection relies on manual calibration, our findings establish the first proof-of-concept evidence that MFG models can capture complex temporal patterns in public sentiment, laying the groundwork for future work on systematic parameter identification methods, i.e. solutions of coefficient inverse problems for the MFG system.

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arXiv2026

Global Convergence and Uniqueness for an Inverse Problem Posed by Gelfand

The first globally convergent numerical method is developed for a coefficient inverse problem (CIP) for the $n-$d, $n\geq 2$ wave equation with the unknown potential in the most challenging case when the $δ-$ function is present in the initial condition with a single location of the point source. In fact, an approximate mathematical model for that CIP is derived. That globally convergent numerical method is developed for this model. This is a new version of the so-called convexification numerical method. Uniqueness theorem is proven as well within the framework of that approximate mathematical model. The question about uniqueness of this CIP was first posed by a famous mathematician I. M. Gelfand in 1954 as an $n-$d ($n=2,3$) extension of the fundamental theorem of V.A. Marchenko in the 1-d case (1950). Based on a Carleman estimate, convergence analysis is carried out. This analysis ensures the global convergence of the proposed numerical method, i.e. it is not necessary to have a good first guess for the solution. Exhaustive computational experiments with noisy data demonstrate a high reconstruction accuracy of complicated structures. In particular, this accuracy points towards a high adequacy of that approximate mathematical model.

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arXiv2026

A Carleman Semi-Discrete Convexification Method Combined With Deep Learning for Electrical Impedance Tomography

In this paper, a new semi-discrete version of the Carleman estimate-based convexification globally convergent numerical method is developed. It is used for the delivery of the starting point for the training procedure of deep learning. An important feature of the continuous version of the convexification method is that its convergence to the true solution is independent on the availability of a good first guess about this solution. A new concept of the h-strong convexity is introduced, where h is the grid step size in the semi-discrete version of the convexification method. The h -strong convexity allows to obtain an a priori accuracy estimate of the starting point for the training step of the deep learning procedure. This approach is demonstrated for a highly nonlinear problem of Electrical Impedance Tomography. Results of numerical experiments for complicated media structures demonstrate the computational feasibility of this procedure.

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arXiv2025

Convexification Numerical Method for Imaging of Moving Targets

The problem of imaging of a moving target is formulated as a Coefficient Inverse Problem for a hyperbolic equation with its coefficient depending on all three spatial variables and time. As the initial condition, the point source running along a straight line is used. Lateral Cauchy data are known for each position of the point source. A truncated Fourier series with respect to a special orthonormal basis is used. First, Lipschitz stability estimate is obtained. Next, a globally convergent numerical method, the so-called convexification method, is developed and its convergence analysis is carried out. The convexification method is based on a Carleman estimate. Results of numerical experiments are presented.

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arXiv2025

Forecasting Public Sentiments via Mean Field Games

Motivated by the goal of forecasting public sentiments, we consider a forecasting problem in the context of the Mean Field Games theory. We develop a numerical method, which is a version of the so-called convexification method. We provide theoretical convergence analysis that establishes global convergence of the method with a convergence rate. We also conduct numerical experiments that demonstrate the accurate performance of the convexification technique and highlight some promising features of this approach.

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arXiv2025

Corruption via Mean Field Games

A new mathematical model governing the development of a corrupted hierarchy is derived. This model is based on the Mean Field Games theory. A retrospective problem for that model is considered. From the applied standpoint, this problem amounts to figuring out the past activity of the corrupted hierarchy using the present data for this community. Three new Carleman estimates are derived. These estimates lead to Hölder stability estimates and uniqueness results for both that retrospective problem and its generalized version. Hölder stability estimates characterize the dependence of the error in the solution of the retrospective problem from the error in the input data.

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arXiv2025

The Carleman Contraction Mapping Method for a Coefficient Inverse Problem of the Epidemiology

It is proposed to monitor spatial and temporal spreads of epidemics via solution of a Coefficient Inverse Problem for a system of three coupled nonlinear parabolic equations. To solve this problem numerically, a version of the so-called Carleman contraction mapping method is developed for this problem. On each iteration, a linear problem with the incomplete lateral Cauchy data is solved by the weighted Quasi-Reversibility Method, where the weight is the Carleman Weight Function. This is the function, which is involved as the weight in the Carleman estimate for the corresponding parabolic operator. Convergence analysis ensures the global convergence of this procedure. Numerical results demonstrate an accurate performance of this technique for noisy data.

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arXiv2025

Convexification With the Viscocity Term for Electrical Impedance Tomography

A version of the globally convergent convexification numerical method is constructed for the problem of Electrical Impedance Tomography in the 2D case. An important element of this version is the presence of the viscosity term. Global convergence analysis is carried out. Results of numerical experiments are presented.

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arXiv2025

Convexification With Viscosity Term for an Inverse Problem of Tikhonov

In 1965 A.N. Tikhonov, the founder of the theory of Ill-Posed and Inverse Problems, has posed an coefficient inverse problem of the recovery of the unknown electric conductivity coefficient from measurements of the back reflected electrical signal. In the geophysical application targeted by Tikhonov, this coefficient depends only on the depth and characterizes the electrical conductivity of the ground. The goal of this paper is to construct for this problem a version of the globally convergent convexification numerical method for this problem. In this version, the viscosity term is introduced. A Carleman estimate allows to prove global convergence of this method.

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arXiv2024

Convexification for the 3D Problem of Travel Time Tomography

The travel time tomography problem is a coefficient inverse problem for the eikonal equation. This problem has well known applications in seismic. The eikonal equation is considered here in the circular cylinder, where point sources run along its axis and measurements of travel times are conductes on the whole surface of this cylinder. A new version of the globally convergent convexification numerical method for this problem is developed. Results of numerical studies are presented.

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arXiv2024

A New Type of Ill-Posed and Inverse Problems for Parabolic Equations

The time dependent experimental data are always collected at discrete grids with respect to the time t. The step size h of such a grid is always separated from zero by a certain positive number. The same is true for all computations, which are always done on discrete grids with their grid step sizes being not too small. These applied considerations prompt us to introduce a new type of Ill-Posed Problems and Coefficient Inverse Problems (CIP)for parabolic equations. In these problems the t-derivatives of corresponding parabolic operators are written in finite differences with the grid step size being separated from zero. We call this the "t-finite difference framework" (TFD). We address three long standing open questions within the TFD framework. Finally, a numerical method is developed for the CIP of monitoring epidemics. The global convergence of this method is proven.

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arXiv2024

Convexification for a Coefficient Inverse Problem for a System of Two Coupled Nonlinear Parabolic Equations

A system of two coupled nonlinear parabolic partial differential equations with two opposite directions of time is considered. In fact, this is the so-called "Mean Field Games System" (MFGS), which is derived in the mean field games (MFG) theory. This theory has numerous applications in social sciences. The topic of Coefficient Inverse Problems (CIPs) in the MFG theory is in its infant age, both in theory and computations. A numerical method for this CIP is developed. Convergence analysis ensures the global convergence of this method. Numerical experiments are presented.

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arXiv2024

An Inverse Problem With the Final Overdetermination for the Mean Field Games System

The mean field games (MFG) theory has broad application in mathematical modeling of social phenomena. The Mean Field Games System (MFGS) is the key to the MFG theory. This is a system of two nonlinear parabolic partial differential equations with two opposite directions of time 0<t<T. The topic of Coefficient Inverse Problem (CIPs) for the MFGS is a newly emerging one. A CIP for the MFGS is studied. The input data are Dirichlet and Neumann boundary conditions either on a part of the lateral boundary (incomplete data) or on the whole lateral boundary (complete data). In addition to the initial conditions at t=0, terminal conditions at t=T are given. The terminal conditions mean the final overdetermination. The necessity of assigning all these input data is explained. Holder and Lipschitz stability estimates are obtained for the cases of incomplete and complete data respectively. These estimates imply uniqueness of the CIP.

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arXiv2024

Stability Estimates for Some Parabolic Inverse Problems With the Final Overdetermination via a New Carleman Estimate

This paper is about Holder and Lipschitz stability estimates and uniqueness theorems for some coefficient inverse problems and associated inverse source problems for a general linear parabolic equation of the second order with variable coefficients. The data for the inverse problem are given at the final moment of time {t=T}. In addition, both Dirichlet and Neumann boundary conditions are given either on a part or on the entire lateral boundary. Thus, if these boundary conditions are given only at a part of the boundary, then even if the target coefficient is known, still the forward problem is not a classical initial boundary value problem.

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arXiv2024

Spatiotemporal Monitoring of Epidemics via Solution of a Coefficient Inverse Problem

Let S,I and R be susceptible, infected and recovered populations in a city affected by an epidemic. The SIR model of Lee, Liu, Tembine, Li and Osher, \emph{SIAM J. Appl. Math.},~81, 190--207, 2021 of the spatiotemoral spread of epidemics is considered. This model consists of a system of three nonlinear coupled parabolic Partial Differential Equations with respect to the space and time dependent functions S,I and R. For the first time, a Coefficient Inverse Problem (CIP) for this system is posed. The so-called \textquotedblleft convexification" numerical method for this inverse problem is constructed. The presence of the Carleman Weight Function (CWF) in the resulting regularization functional ensures the global convergence of the gradient descent method of the minimization of this functional to the true solution of the CIP, as long as the noise level tends to zero. The CWF is the function, which is used as the weight in the Carleman estimate for the corresponding Partial Differential Operator. Numerical studies demonstrate an accurate reconstruction of unknown coefficients as well as S,I,R functions inside of that city. As a by-product, uniqueness theorem for this CIP is proven. Since the minimal measured input data are required, then the proposed methodology has a potential of a significant decrease of the cost of monitoring of epidemics.

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arXiv2023

On The Mean Field Games System With the Lateral Cauchy Data via Carleman Estimates

The second order Mean Field Games system (MFGS) in a bounded domain with the lateral Cauchy data is considered. This means that both Dirichlet and Neumann boundary data for the solution the MFGS are given. Two Hölder stability estimates for two slightly diffeent cases are derived. These estimates indicate how stable the solution of the MFGS is with respect to the possible noise in the lateral Cauchy data. Our stability estimates imply uniqueness. The key mathematical apparatus is the apparatus of two new Carleman estimates.

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