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In this paper, we study the well-posedness of the solution of an optimal control problem related to a multi-parameters identification problem in degenerate parabolic equations. Problems of this type have important applications in several fields of applied science. Unlike other inverse coefficient problems for classical parabolic equations, the mathematical model discussed in the paper is degenerate on both lateral boundaries of the domain. Moreover, the status of the two unknown coefficients are different, namely that the reconstruction of the source term is mildly ill-posed, while the inverse initial value problem is severely ill-posed. On the basis of optimal control framework, the problem is transformed into an optimization problem. The existence of the minimizer is proved and the necessary conditions which must be satisfied by the minimizer are also established. Due to the difference between ill-posedness degrees of the two unknown coefficients, the extensively used conjugate theory for parabolic equations cannot be directly applied for our problem. By carefully analyzing the necessary conditions and the direct problem, the uniqueness, stability and convergence of the minimizer are obtained. The results obtained in the paper are interesting and useful, and can be extended to more general parabolic equations with degenerate coefficients. [ABSTRACT FROM AUTHOR]

In the present paper, the unique solvability of the direct and inverse source problems for the pseudo-parabolic equation involving the bi-ordinal Hilfer fractional derivative and 2D Landau Hamiltonian operator is considered. Applying the Fourier analysis for the operator Landau Hamiltonian, the theorems of uniqueness and existence of solutions to direct and inverse source problems are proved. In the investigation of the inverse source problem, we have used the value of unknown function at the final time in order to find the right-hand side of the equation. It is also presented the stability result of the inverse problem. [ABSTRACT FROM AUTHOR]

In this paper, we apply a Landweber iterative regularization method to determine a space-dependent source for a time-space fractional diffusion-wave equation from the final measurement. In general, this problem is ill-posed, and a Landweber iterative regularization method is used to obtain the regularization solution. Under the a priori parameter choice rule and the a posteriori parameter choice rule, we give the error estimates between the regularization solution and the exact solution, respectively. Some numerical results in the one-dimensional and two-dimensional cases show the utility of the used regularization method. [ABSTRACT FROM AUTHOR]

In this paper, using Sinc-Galerkin method and TSVD regularization, an approximation of the quasi-solution to an inverse source problem is obtained. To do so, the solution of direct problem is obtained by the Sinc-Galerkin method, and this solution is applied in a least squares cost functional. Then, to obtain an approximation of the quasi-solution, we minimize the cost functional by TSVD regularization. Error analysis and convergence of the proposed method are investigated. In addition, at the end, four numerical examples are given in details to show the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

This paper deals with an inverse problem on simultaneously determining a time-dependent potential term and a time source function from two-point measured data in a multi-term time-fractional diffusion equation. First we study the existence, uniqueness and some regularities of the solution for the direct problem by using the fixed point theorem. Then a nice conditional stability estimate of inversion coefficients problem is obtained based on the regularity of the solution to the direct problem and a fine property of the Caputo fractional derivative. In addition, the ill-posedness of the inverse problem is illustrated and we transfer the inverse problem into a variational problem. Moreover, the existence and convergence of the minimizer for the variational problem are given. Finally, we use a modified Levenberg–Marquardt method to reconstruct numerically the approximate functions of two unknown time-dependent coefficients effectively. Numerical experiments for three examples in one- and two-dimensional cases are provided to show the validity and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse scattering of acoustic waves by an unbounded periodic elastic medium in the three-dimensional case. A novel uniqueness theorem is proved for the inverse problem of recovering a bi-periodic interface between acoustic and elastic waves using the near-field data measured only from the acoustic side of the interface, corresponding to a countably infinite number of quasi-periodic incident acoustic waves. The proposed method depends only on a fundamental a priori estimate established for the acoustic and elastic wave fields and a new mixed-reciprocity relation established in this paper for the solutions of the fluid-solid interaction scattering problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider Carleman estimates and inverse problems for the coupled quantitative thermoacoustic equations. In part I, we establish Carleman estimates for the coupled quantitative thermoacoustic equations by assuming that the coefficients satisfy suitable conditions and taking the usual weight function φ ⁢ (x , t) = e λ ⁢ ψ ⁢ (x , t) , ψ ⁢ (x , t) = | x - x 0 | 2 - β ⁢ | t - t 0 | 2 + β ⁢ t 0 2 for x in a bounded domain in ℝ n with C 3 -boundary and t ∈ (0 , T) , where t 0 = T 2 . We will discuss applications of the Carleman estimates to some inverse problems for the coupled quantitative thermoacoustic equations in the succeeding part II paper [M. Cristofol, S. Li and Y. Shang, Carleman estimates and inverse problems for the coupled quantitative thermoacoustic equations. Part II: Inverse problems, preprint 2020, https://hal.archives-ouvertes.fr/hal-02863385]. [ABSTRACT FROM AUTHOR]

Identification of physical properties of materials is very important because they are in general unknown. Furthermore, their direct experimental measurement could be costly and inaccurate. In such a situation, a cheap and efficient alternative is to mathematically formulate an inverse, but difficult, problem that can be solved, in general, numerically; the challenge being that the problem is, in general, nonlinear and ill-posed. In this paper, the reconstruction of a lower-order unknown time-dependent coefficient in a Cahn–Hilliard-type fourth-order equation from an additional integral observation, which has application to characterizing the nonlinear saturation of the collisional trapped-ion mode in a tokamak, is investigated. The local existence and uniqueness of the solution to such inverse problem is established by utilizing the Rothe method. Moreover, the continuous dependence of the unknown coefficient upon the measured data is derived. Next, the Tikhonov regularization method is applied to recover the unknown coefficient from noisy measurements. The stability estimate of the minimizer is derived by investigating an auxiliary linear fourth-order inverse source problem. Henceforth, the variational source condition can be verified. Then the convergence rate is obtained under such source condition. [ABSTRACT FROM AUTHOR]

It is demonstrated in this paper that the propagation of the electric wave field in a heterogeneous medium in 3D can sometimes be governed well by a single scalar second order PDE, which is derived from Maxwell's equations. The corresponding component of the electric field dominates two other components. This justifies some past results of the second author with coauthors about numerical solutions of coefficient inverse problems with experimental electromagnetic data. In addition, since it is simpler to work in applications with a single scalar second order PDE rather than with the complete Maxwell system, the result of this paper might be useful to researchers working on applied issues of the propagation of electromagnetic waves in inhomogeneous media. [ABSTRACT FROM AUTHOR]

The theory of inverse problems is an actively studied area of modern differential equation theory. This paper studies the solvability of the inverse problem for a linearized system of Navier–Stokes equations in a cylindrical domain with a final overdetermination condition. Our approach is to reduce the inverse problem to a direct problem for a loaded equation. In contrast to the well-known works in this field, our approach is to find an equation for a loaded term whose solvability condition provides the solvability of the original inverse problem. At the same time, the classical theory of spectral decomposition of unbounded self-adjoint operators is actively used. Concrete examples demonstrate that the assertions of our theorems naturally develop and complement the known results on inverse problems. Various cases are considered when the known coefficient on the right-hand side of the equation depends only on time or both on time and a spatial variable. Theorems establishing new sufficient conditions for the unique solvability of the inverse problem under consideration are proved. [ABSTRACT FROM AUTHOR]

The inverse problem for simultaneously identifying the space-dependent source term and the initial value in a time-fractional diffusion equation is studied in this paper. The simultaneous inversion is formulated into a system of two operator equations based on the Fourier method to the time-fractional diffusion equation. Under some suitable assumptions, the conditional stability of simultaneous inversion solutions is established, and the exponential Tikhonov regularization method is proposed to obtain the good approximations of simultaneous inversion solutions. Then the convergence estimations of inversion solutions are presented for a priori and a posteriori selections of regularization parameters. Finally, numerical experiments are conducted to illustrate effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The (interior) transmission eigenvalue problems are a type of non-elliptic, non-selfadjoint and nonlinear spectral problems that arise in the theory of wave scattering. They connect to the direct and inverse scattering problems in many aspects in a delicate way. The properties of the transmission eigenvalues have been extensively and intensively studied over the years, whereas the intrinsic properties of the transmission eigenfunctions are much less studied. Recently, in a series of papers, several intriguing local and global geometric structures of the transmission eigenfunctions are discovered. Moreover, those longly unveiled geometric properties produce some interesting applications of both theoretical and practical importance to direct and inverse scattering problems. This paper reviews those developments in the literature by summarizing the results obtained so far and discussing the rationales behind them. There are some side results of this paper including the general formulations of several types of transmission eigenvalue problems, some interesting observations on the connection between the transmission eigenvalue problems and several challenging inverse scattering problems, and several conjectures on the spectral properties of transmission eigenvalues and eigenfunctions, with most of them are new to the literature. [ABSTRACT FROM AUTHOR]

A fast numerical algorithm for solving the Cauchy problem for elliptic equations with variable coefficients in standard calculation domains (rectangles, circles, or rings) is proposed. The algorithm is designed to calculate the heat flux at the inaccessible boundary. It is based on the separation of variables method. This approach employs a finite difference approximation and allows obtaining a solution to a discrete problem in arithmetic operations of the order of N ⁢ ln ⁡ N , where 푁 is the number of grid points. As a rule, iterative procedures are needed to solve the Cauchy problem for elliptic equations. The currently available direct algorithms for solving the Cauchy problem have been developed only for (Laplace, Helmholtz) operators with constant coefficients and for use of analytical solutions for problems with such operators. A novel feature of the results of the present paper is that the direct algorithm can be used for an elliptic operator with variable coefficients (of a special form). It is important that in this case no analytical solution to the problem can be obtained. The algorithm significantly increases the range of problems that can be solved. It can be used to create devices for determining in real time heat fluxes on the parts of inhomogeneous constructions that cannot be measured. For example, to determine the heat flux on the inner radius of a pipe made of different materials. [ABSTRACT FROM AUTHOR]

The paper considers the scattering problem for the first-order system of hyperbolic equations on the half-axis with a nonhomogeneous boundary condition. This problem models the phnomennon of wave propagation in a nonstationary medium where an incoming wave unaffected by a potential field. The scattering operator on the half-axis with a nonzero boundary condition is defined and the uniqueness of the inverse scattering problem (the problem of finding the potential with respect to scattering operator) is studied. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse scattering problem of identifying a two-layered cavity by internal acoustic measurements under the condition that the interior interface has a mixed transmission boundary condition. We focus on the mathematical analysis of recovering the shape of the interior interface by using the linear sampling method, including reconstructing the surface conductivity by the same measurements. [ABSTRACT FROM AUTHOR]

In this paper, we address stability results in determining the time-dependent scalar and vector potentials appearing in the convection-diffusion equation from the knowledge of the Cauchy data set. We prove Hölder-type stability estimates. The key tool used in this work is the geometric optics solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse problem of vibro-acoustography, a technique for enhancing ultrasound imaging by making use of nonlinear effects. It amounts to determining two spatially variable coefficients in a system of PDEs describing propagation of two directed sound beams and the wave resulting from their nonlinear interaction. To justify the use of Newton's method for solving this inverse problem, on one hand, we verify well-definedness and differentiability of the forward operator corresponding to two versions of the PDE model; on the other hand, we consider an all-at-once formulation of the inverse problem and prove convergence of Newton's method for its solution. [ABSTRACT FROM AUTHOR]

The time domain enclosure method is one of the analytical methods for inverse obstacle problems governed by partial differential equations in the time domain. This paper considers the case when the governing equation is given by the Maxwell system and consists of two parts. The first part establishes the base of the time domain enclosure method for the Maxwell system using a single set of the solutions over a finite time interval for a general (isotropic) inhomogeneous medium in the whole space. It is a system of asymptotic inequalities for the indicator function which may enable us to apply the time domain enclosure method to the problem of finding unknown penetrable obstacles embedded in various background media. As a first step of its expected applications, the case when the background medium is homogeneous and isotropic is considered and the time domain enclosure method is realized. This is the second part. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability of the inverse source problem for Maxwell's equations in an inhomogeneous background medium. We show that the stability estimate consists of the Lipschitz-type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. The analysis employs scattering theory to obtain the holomorphic domain and an upper bound for the resolvent of the elliptic operator. [ABSTRACT FROM AUTHOR]

In this paper, we study the inverse problems of determining the unknown transverse shear force g ⁢ (t) in a system governed by the damped Euler–Bernoulli equation ρ ⁢ (x) ⁢ u t ⁢ t + μ ⁢ (x) ⁢ u t + (r ⁢ (x) ⁢ u x ⁢ x ) x ⁢ x + (κ ⁢ (x) ⁢ u x ⁢ x ⁢ t ) x ⁢ x = 0 , (x , t) ∈ (0 , ℓ) × (0 , T ] , subject to the boundary conditions u ⁢ (0 , t) = 0 , u x ⁢ (0 , t) = 0 , [ r ⁢ (x) ⁢ u x ⁢ x + κ ⁢ (x) ⁢ u x ⁢ x ⁢ t ] x = ℓ = 0 , - [ (r ⁢ (x) ⁢ u x ⁢ x + κ ⁢ (x) ⁢ u x ⁢ x ⁢ t ) x ] x = ℓ = g ⁢ (t) , for t ∈ [ 0 , T ] , from the measured deflection ν ⁢ (t) := u ⁢ (ℓ , t) , t ∈ [ 0 , T ] , and from the bending moment ω ⁢ (t) := - (r ⁢ (0) ⁢ u x ⁢ x ⁢ (0 , t) + κ ⁢ (0) ⁢ u x ⁢ x ⁢ t ⁢ (0 , t) ) , t ∈ [ 0 , T ] , where the terms (κ ⁢ (x) ⁢ u x ⁢ x ⁢ t ) x ⁢ x and μ ⁢ (x) ⁢ u t account for the Kelvin–Voigt damping and external damping, respectively. The main purpose of this study is to analyze the Kelvin–Voigt damping effect on determining the unknown transverse shear force (boundary input) through the given boundary measurements. The inverse problems are transformed into minimization problems for Tikhonov functionals, and it is shown that the regularized functionals admit unique solutions for the inverse problems. By suitable regularity on the admissible class of shear force g ⁢ (t) , we prove that these functionals are Fréchet differentiable, and the derivatives are expressed through the solutions of corresponding adjoint problems posed with measured data as boundary data associated with the direct problem. The solvability of these adjoint problems is obtained under the minimal regularity of the boundary data g ⁢ (t) , which turns out to be the regularizing effect of the Kelvin–Voigt damping in the direct problem. Furthermore, using the Fréchet derivative of the more regularized Tikhonov functionals, we obtain remarkable Lipschitz stability estimates for the transverse shear force in terms of the given measurement by a feasible condition only on the Kelvin–Voigt damping coefficient. [ABSTRACT FROM AUTHOR]

In this paper, we intend to identify two significant parameters – expected return and absolute risk aversion – in the Merton portfolio optimization problem under an exponential utility function where volatility is driven by a slow mean-reverting diffusion process. First, we find the approximate solution of the fully nonlinear Hamilton–Jacobi–Bellman equation for the Merton model by the stochastic asymptotic approximation method. Second, we estimate parameters – expected return and absolute risk aversion – through the approximate solution and prove the uniqueness and stability of the parameter identification problem. Finally, we provide an illustrative example to demonstrate the capacity and efficiency of our method. [ABSTRACT FROM AUTHOR]

This paper is devoted to recovering simultaneously the fractional order and the space-dependent source term from partial Cauchy's boundary data in a multidimensional time-fractional diffusion equation. The uniqueness of the inverse problem is obtained by employing analytic continuation and the Laplace transform. Then a modified non-stationary iterative Tikhonov regularization method with a regularization parameter chosen by a sigmoid-type function is used to find a stable approximate solution for the source term and the fractional order. Numerical examples in one-dimensional and two-dimensional cases are provided to illustrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The means to obtain the adsorption isotherms is a fundamental open problem in competitive chromatography. A modern technique of estimating adsorption isotherms is to solve a nonlinear inverse problem in a partial differential equation so that the simulated batch separation coincides with actual experimental results. However, this identification process is usually ill-posed in the sense that the uniqueness of adsorption isotherms cannot be guaranteed, and moreover, the small noise in the measured response can lead to a large fluctuation in the traditional estimation of adsorption isotherms. The conventional mathematical method of solving this problem is the variational regularization, which is formulated as a non-convex minimization problem with a regularized objective functional. However, in this method, the choice of regularization parameter and the design of a convergent solution algorithm are quite difficult in practice. Moreover, due to the restricted number of injection profiles in experiments, the types of measured data are extremely limited, which may lead to a biased estimation. In order to overcome these difficulties, in this paper, we develop a new inversion method – the virtual injection promoting double feed-forward neural network (VIP-DFNN). In this approach, the training data contain various types of artificial injections and synthetic noisy measurement at outlet, generated by a conventional physics model – a time-dependent convection-diffusion system. Numerical experiments with both artificial and real data from laboratory experiments show that the proposed VIP-DFNN is an efficient and robust algorithm. [ABSTRACT FROM AUTHOR]

This work presents an application of tensor field tomography for non-destructive reconstructions of axially symmetric residual stresses in a graded-index YAG single crystal for the case of beam deflection. The axis of the cylinder coincides with the crystallographic axis [001] of the single crystal and it has an axially symmetric refractive index distribution. The transformation of the polarization of light is measured in a plane orthogonal to the axis of the cylinder. Stresses are determined within the framework of the Maxwell piezo-optic law (linear dependence of the permittivity tensor on stresses) and small rotation of quasi principal stress axes. This paper generalizes the method of integrated photoelasticity for the case of ray deflection. [ABSTRACT FROM AUTHOR]

The modified transmission eigenvalue (MTE) problem was introduced in [S. Cogar, D. Colton, S. Meng and P. Monk, Modified transmission eigenvalues in inverse scattering theory, Inverse Problems 33 2017, 12, Article ID 125002] and used as a target signature for nondestructive testing. In this paper, we study the inverse spectral problem to reconstruct the modified transmission eigenvalues using Cauchy data. We propose a reciprocity gap functional method and show that the MTEs can be determined by solving some linear ill-posed integral equations. Numerical examples for both absorbing and non-absorbing media are presented to validate the effectiveness and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is devoted to the inverse problem of determining the spatially dependent source in a time fractional diffusion-wave equation, with the aid of extra measurement data at a subboundary. A uniqueness result is obtained by using the analyticity and the newly established unique continuation principle provided that the coefficients are all temporally independent. We also derive a Lipschitz stability of our inverse source problem under a suitable topology whose norm is given via the adjoint system of the fractional diffusion-wave equation. [ABSTRACT FROM AUTHOR]

In this paper, the existence and uniqueness of a strong solution of the inverse problem of determining a coefficient of right-hand side of the integro-differential Kelvin–Voigt equation are investigated. The unknown coefficient that we search defends on space variables. Additional information on a solution of the inverse problem is given here as an integral overdetermination condition. The original inverse problem is reduced to study an equivalent inverse problem with homogeneous initial condition. Then the equivalences of the last inverse problem to an operator equation of second kind is proved. We establish the sufficient conditions for the unique solvability of the operator equation of second kind. [ABSTRACT FROM AUTHOR]

This paper considers a class of nonlinear time-harmonic Maxwell systems at fixed frequency, with nonlinear terms taking the form X ⁢ (x , | E → ⁢ (x) | 2) ⁢ E → ⁢ (x) , Y ⁢ (x , | H → ⁢ (x) | 2) ⁢ H → ⁢ (x) such that X ⁢ (x , s) , Y ⁢ (x , s) are both real analytic in 푠. Such nonlinear terms appear in nonlinear optics theoretical models. Under certain regularity conditions for 풳 and 풴, it can be shown that boundary measurements of tangent components of the electric and magnetic fields determine the electric permittivity and magnetic permeability functions as well as the form of the nonlinear terms. [ABSTRACT FROM AUTHOR]

Consider the scattering of electromagnetic waves by a penetrable homogeneous cylinder at oblique incident. The Maxwell equations are then reduced to a system of a pair of the two-dimensional Helmholtz equations for 푧-components of the electric and magnetic field through coupled oblique boundary conditions. This paper studies an inverse problem of recovering the penetrable obstacle from the far-field pattern of the electric field. The well-known linear sampling method is used to solve this problem. Compared with the usual inverse scattering problem, the coupled system and the oblique derivative boundary condition bring difficulties in theoretical analysis. Some numerical examples are presented to illustrate the validity and feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

Research presented in this paper was initiated by the publication [A. D. Agaltsov and R. G. Novikov, Examples of solving the inverse scattering problem and the equations of the Veselov–Novikov hierarchy from the scattering data of point potentials, Russian Math. Surveys74 (2019), 3, 373–386] and is based on its results. Two sets of the complex monopole scattering coefficients are distinguished among the possible values of these coefficients for nonabsorbing inhomogeneities. These sets differ in phases of the scattering coefficients. In order to analyze the features and possibilities of reconstructing the inhomogeneities of both sets, on the one hand, the inverse problem is solved for each given value of the monopole scattering coefficient using the Novikov functional algorithm. On the other hand, the scatterer is selected in the form of a homogeneous cylinder with the monopole scattering coefficient that coincides with the given one. The results obtained for the monopole inhomogeneity and for the corresponding cylindrical scatterer are compared in the coordinate and spatial-spectral spaces. The physical reasons for the similarities and differences in these results are discussed when the amplitude of the scattering coefficient changes, as well as when passing from one set to another. [ABSTRACT FROM AUTHOR]

In this paper, we provide a rigorous derivation of an asymptotic formula for the perturbation of eigenvalues associated to the Stokes eigenvalue problem with Dirichlet conditions and in the presence of small deformable inclusions. Taking advantage of the small sizes of the inclusions immersed in an incompressible Newtonian fluid having kinematic viscosity different from the background one, we show that our asymptotic formula can be expressed in terms of the eigenvalue in the absence of the inclusions and in terms of the viscous moment tensor (VMT). [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse scattering problem for the three-dimensional Maxwell equations in bi-anisotropic periodic structures. The inverse scattering problem aims to determine the shape of bi-anisotropic periodic scatterers from electromagnetic near-field data at a fixed frequency. The factorization method is studied as an analytical and numerical tool for solving the inverse problem. We provide a rigorous justification of the factorization method which results in the unique determination and a fast imaging algorithm for the periodic scatterer. Numerical examples for imaging three-dimensional periodic structures are presented to examine the efficiency of the method. [ABSTRACT FROM AUTHOR]

This paper considers the inverse problem of identifying an unknown space- and time-dependent source function F ⁢ (x , t) F(x,t) in the variable coefficient advection-diffusion equation u t = (D ⁢ (x) ⁢ u x) x - (V ⁢ (x) ⁢ u) x + F ⁢ (x , t) u_{t}=(D(x)u_{x})_{x}-(V(x)u)_{x}+F(x,t) from the Dirichlet ν ⁢ (t) := u ⁢ (ℓ , t) \nu(t):=u(\ell,t) and Neumann f ⁢ (t) := - D ⁢ (0) ⁢ u x ⁢ (0 , t) f(t):=-D(0)u_{x}(0,t) , t ∈ (0 , T ] t\in(0,T] , boundary measured outputs. This problem was motivated by several important real-world applications in the field of contaminant hydrogeology, and the novel analysis presented here is highly relevant to problems of practical interest. The input-output operators corresponding to the Dirichlet and Neumann measured boundary data are introduced. The inverse problem is then formulated as a system of operator equations consisting of these operators and the measured outputs. The compactness and Lipschitz continuity of the input-output operators are proved in the relevant classes of admissible source functions ℱ and F r \mathcal{F}_{r}. These results together with the derived trace estimates allow us to show the existence of a quasi-solution of the inverse source problem as a minimum of the Tikhonov functional, under minimal regularity assumptions with respect to the source function and other inputs. An explicit gradient formula for the Fréchet gradient of the Tikhonov functional is also derived by means of an appropriate adjoint problem. [ABSTRACT FROM AUTHOR]

This paper concerns the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumptions which we call basic assumptions, but also some technical assumptions which we call further assumptions. It is shown as usual by first applying the Holmgren transform to this equation/inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate is given via a partition of unity and the Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point. A little more details about this are as follows. Factorize this operator with constant coefficients into two first order differential operators. Conjugate each factor by a Carleman weight, and derive an estimate which is uniform with respect to the point at which we froze the coefficients for each conjugated factor by constructing a parametrix for its adjoint operator. [ABSTRACT FROM AUTHOR]

In this paper, we consider a seventh-order generalized Korteweg–de Vries (GKdV) equation and study the boundary stability results concerning the inverse problem of recovering a space-dependent source term. We establish a new boundary Carleman estimate for the seventh-order linear operator with the Dirichlet–Neumann type boundary conditions. Using this crucial estimate along with regularity result of the nonlinear GKdV equation, we establish a Lipschitz stability estimate of GKdV equation. [ABSTRACT FROM AUTHOR]

In this paper, we study an inverse source problem for a degenerate and singular parabolic system where the boundary conditions are of Neumann type. We consider a problem with degenerate diffusion coefficients and singular lower-order terms, both vanishing at an interior point of the space domain. In particular, we address the question of well-posedness of the problem, and then we prove a stability estimate of Lipschitz type in determining the source term by data of only one component. Our method is based on Carleman estimates, cut-off procedures and a reflection technique. [ABSTRACT FROM AUTHOR]

In this paper, we discuss how to partially determine the Fourier transform F ⁢ (z) = ∫ - 1 1 f ⁢ (t) ⁢ e i ⁢ z ⁢ t ⁢ 𝑑 t , z ∈ ℂ , F(z)=\int_{-1}^{1}f(t)e^{izt}\,dt,\quad z\in\mathbb{C}, given the data | F ⁢ (z) | {\lvert F(z)\rvert} or arg ⁡ F ⁢ (z) {\arg F(z)} for z ∈ ℝ {z\in\mathbb{R}}. Initially, we assume [ - 1 , 1 ] {[-1,1]} to be the convex hull of the support of the signal f. We start with reviewing the computation of the indicator function and indicator diagram of a finite-typed complex-valued entire function, and then connect to the spectral invariant of F ⁢ (z) {F(z)}. Then we focus to derive the unimodular part of the entire function up to certain non-uniqueness. We elaborate on the translation of the signal including the non-uniqueness associates of the Fourier transform. We show that the phase retrieval and magnitude retrieval are conjugate problems in the scattering theory of waves. [ABSTRACT FROM AUTHOR]

This is a review article on the development of the probe and enclosure methods from past to present, focused on their central ideas together with various applications. [ABSTRACT FROM AUTHOR]

This article concerns the uniqueness of an inverse coefficient problem of identifying a spatially varying potential in a one-dimensional time-fractional diffusion equation. The input sources are given by a complete system in L 2 ⁢ (0 , 1) , and measurements are observed at the end point of the spatial interval. Firstly, we provide the positive lower bound of the Green function for the differential operator with different boundary conditions. Then, based on the positive lower bound estimation of the Green function, the relationship between the Green function, the solution of the forward problem, and the potential, such measurements uniquely determine the potential on the entire interval under different boundary conditions. [ABSTRACT FROM AUTHOR]

An overdetermination is introduced in an initial condition for the second order mean field games system (MFGS). This makes the resulting problem close to the classical ill-posed Cauchy problems for PDEs. Indeed, in such a problem an overdetermination in boundary conditions usually takes place. A Lipschitz stability estimate is obtained. This estimate implies uniqueness. A new Carleman estimate is derived. This latter estimate is called "quasi-Carleman estimate", since it contains two test functions rather than a single one in conventional Carleman estimates. These two estimates play the key role. Carleman estimates were not applied to the MFGS prior to the recent work of Klibanov and Averboukh in [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, preprint 2023, https://arxiv.org/abs/2302.10709]. [ABSTRACT FROM AUTHOR]

We establish the stabilization of the strong solution (u ⁢ (t , x) , k ⁢ (t)) to the inverse problem for a pseudoparabolic equation (u + η ⁢ M ⁢ u) t + M ⁢ u + k ⁢ (t) ⁢ u = f with an unknown coefficient k ⁢ (t) to the solution (u ∞ , k ∞) of the appropriate stationary inverse problem. The operator M = - div ⁡ (ℳ ⁢ (x) ⁢ ∇) + m ⁢ (x) ⁢ I is supposed to be elliptic and self-adjoint. The asymptotic behavior of the solution (u ⁢ (t , x) , k ⁢ (t)) is investigated as t → + ∞ . [ABSTRACT FROM AUTHOR]

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 is significantly more efficient than Landweber iteration. [ABSTRACT FROM AUTHOR]

A 3D coefficient inverse problem for a hyperbolic equation with non-overdetermined data is considered. The forward problem is the Cauchy problem with the initial condition being the delta function concentrated at a single plane (i.e. the plane wave). A certain associated operator is written in finite differences with respect to two out of three spatial variables, i.e. "partial finite differences". The grid step size is bounded from below by a fixed number. A Carleman estimate is applied to obtain, for the first time, a Hölder stability estimate for this problem. Another new result is an estimate from below the amplitude of the first term of the expansion of the solution of the forward problem near the characteristic wedge. [ABSTRACT FROM AUTHOR]

Suppose that (M , g) is a compact Riemann surface with metric g and boundary ∂ ⁡ M , and Λ is its DN-map. Let M ′ be diffeomorphic to M, let ∂ ⁡ M = ∂ ⁡ M ′ and let Λ ′ be the DN-map of (M ′ , g ′) . Put (M ′ , g ′) ∈ 필 t if ∥ Λ ′ - Λ ∥ H 1 ⁢ (∂ ⁡ M) → L 2 ⁢ (∂ ⁡ M) ⩽ t holds. We show that, for any holomorphic immersion ℰ : M → ℂ n ( n ⩾ 1 ), the relation sup M ′ ∈ 필 t ⁡ inf ℰ ′ ⁡ d H ⁢ (ℰ ′ ⁢ (M ′) , ℰ ⁢ (M)) → t → 0 0 holds, where d H is the Hausdorff distance in ℂ n and the infimum is taken over all holomorphic immersions ℰ ′ : M ′ ↦ ℂ n . As it is known, Λ determines not the surface (M , g) but its conformal class { (M , ρ ⁢ g) ⁢ ∣ ρ > ⁢ 0 , ρ | ∂ ⁡ M = 1 } , while holomorphic immersions are determined by this class. In the mean time, (M , g) is conformally equivalent to ℰ ⁢ (M) , and (M ′ , g ′) is conformally equivalent to ℰ ′ ⁢ (M ′) . Thus, the closeness of the surfaces ℰ ′ ⁢ (M ′) and ℰ ⁢ (M) in ℂ n reflects the closeness of the corresponding conformal classes for close DN-maps. [ABSTRACT FROM AUTHOR]

A new method for solving inverse spectral problems on quantum star graphs is proposed. The method is based on Neumann series of Bessel function representations for solutions of Sturm–Liouville equations. The representations admit estimates for the series remainders which are independent of the real part of the square root of the spectral parameter. This feature makes them especially useful for solving direct and inverse spectral problems requiring calculation of solutions on large intervals in the spectral parameter. Moreover, the first coefficient of the representation is sufficient for the recovery of the potential. The method for solving the inverse spectral problem on the graph consists in reducing the problem to a two-spectra inverse Sturm–Liouville problem on each edge. Then a system of linear algebraic equations is derived for computing the first coefficient of the series representation for the solution on each edge and hence for recovering the potential. The proposed method leads to an efficient numerical algorithm that is illustrated by a number of numerical tests. [ABSTRACT FROM AUTHOR]

In this article, we investigate the inverse problem of identification of a discontinuous parameter and a discontinuous boundary datum to an elliptic inclusion problem involving a double phase differential operator, a multivalued convection term (a multivalued reaction term depending on the gradient), a multivalued boundary condition and an obstacle constraint. First, we apply a surjectivity theorem for multivalued mappings, which is formulated by the sum of a maximal monotone multivalued operator and a multivalued pseudomonotone mapping to examine the existence of a nontrivial solution to the double phase obstacle problem, which exactly relies on the first eigenvalue of the Steklov eigenvalue problem for the p -Laplacian. Then, a nonlinear inverse problem driven by the double phase obstacle equation is considered. Finally, by introducing the parameter-to-solution-map, we establish a continuous result of Kuratowski type and prove the solvability of the inverse problem. [ABSTRACT FROM AUTHOR]

We study the uniqueness of some inverse source problems arising in thermoelastic models of type-III. We suppose that the source terms can be decomposed as a product of a time dependent and a space dependent function, i.e. g ⁢ (t) ⁢ 퐟 ⁢ (퐱) for the load source and g ⁢ (t) ⁢ f ⁢ (퐱) for the heat source. In the first inverse source problem, the source 퐟 ⁢ (퐱) has to be determined from the final in time measurement of the displacement 퐮 ⁢ (퐱 , T) , or from the time-average measurement ∫ 0 T 퐮 ⁢ (퐱 , t) ⁢ d t . In the second inverse source problem, the source f ⁢ (퐱) has to be determined from the time-average measurement of the temperature ∫ 0 T θ ⁢ (퐱 , t) ⁢ d t . We show the uniqueness of a solution to these problems under suitable assumptions on the function g ⁢ (t) . Moreover, we provide some examples showing the necessity of these assumptions. Finally, we conclude the article by studying two combined problems of determining both sources. [ABSTRACT FROM AUTHOR]

The inverse problem in gravimetry is to find a domain 퐷 inside the reference domain Ω from measurements of gravitational force outside Ω. We consider the problem in three dimensions where we found that a few parameters of the unknown 퐷 can be stably determined given data noise in practical situations. An ellipsoid is the best approximation of 퐷. We prove uniqueness of recovering an ellipsoid in a particular case for the inverse problem from minimal amount of data which are the approximated gravitational force at nine boundary points. In the proofs, we derive and use simple systems of linear and nonlinear algebraic equations for the parameters of an ellipsoid. Similarly, a rectangular parallelepiped 퐷 is considered. To support our theory, we use numerical examples with different location of measurements points on ∂ ⁡ Ω . [ABSTRACT FROM AUTHOR]

In this article, we study the inverse problem of recovering a space-dependent coefficient of the Moore–Gibson–Thompson (MGT) equation from knowledge of the trace of the solution on some open subset of the boundary. We obtain the Lipschitz stability for this inverse problem, and we design a convergent algorithm for the reconstruction of the unknown coefficient. The techniques used are based on Carleman inequalities for wave equations and properties of the MGT equation. [ABSTRACT FROM AUTHOR]

We consider a fixed angle inverse scattering problem in the presence of a known Riemannian metric. First, assuming a no caustics condition, we study the direct problem by utilizing the progressing wave expansion. Under a symmetry assumption on the metric, we obtain uniqueness and stability results in the inverse scattering problem for a potential with data generated by two incident waves from opposite directions. Further, similar results are given using one measurement provided the potential also satisfies a symmetry assumption. This work extends the results of [Rakesh and M. Salo, Fixed angle inverse scattering for almost symmetric or controlled perturbations, SIAM J. Math. Anal. 52 2020, 6, 5467–5499] and [Rakesh and M. Salo, The fixed angle scattering problem and wave equation inverse problems with two measurements, Inverse Problems 36 2020, 3, Article ID 035005] from the Euclidean case to certain Riemannian metrics. [ABSTRACT FROM AUTHOR]