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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]

We consider the inverse scattering problem to reconstruct the defect in an infinite medium with periodicity in the upper half space from near field data. This paper has two contributions. Firstly, we mention that there is a mistake in the factorization method of the earlier paper [A. Lechleiter, The factorization method is independent of transmission eigenvalues, Inverse Probl. Imaging 3 2009, 1, 123–138] and give the correct one. Secondly, we give two reconstruction algorithms for the unknown defect by a combination of the factorization method and the monotonicity method. We also give numerical examples based on the former algorithm. [ABSTRACT FROM AUTHOR]

A seven-layers parabolic model with Stephan–Boltzmann interface conditions and Robin boundary conditions is mathematically formulated to describe the heat transfer process in environment-three layers clothing-air gap-body system. Based on this model, the solution to the corresponding inverse problem of simultaneous determination of triple fabric layers thickness is given in this paper, which satisfies the thermal safety requirements of human skin. By implementing a stable finite difference scheme, the thermal burn injuries on the skin of the body can be predicted. Then a kind of stochastic method, named as particle swarm optimization (PSO) algorithm, is developed to numerically solve the inverse problem. Numerical results indicate that the formulation of the model and proposed algorithm for solving the corresponding inverse problem are effective. Hence, the results in this paper will provide scientific supports for designing and manufacturing thermal protective clothing (TPC). [ABSTRACT FROM AUTHOR]

Two main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem to reconstruct a source term from external observations. Our method to solve this inverse source problem consists of two stages. We first establish an equation of the derivative of the solution to the parabolic equation with respect to the time variable. Then, in the second stage, we solve this equation by the quasi-reversibility method. The inverse source problem considered in this paper is the linearization of a nonlinear coefficient inverse problem. Hence, iteratively solving the inverse source problem provides the numerical solution to that coefficient inverse problem. Numerical results for the inverse source problem under consideration and the corresponding nonlinear coefficient inverse problem are presented. [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 consider an inverse backward problem for a nonlinear singularly perturbed parabolic equation of the Burgers' type. We demonstrate how a method of asymptotic analysis of the direct problem allows developing a rather simple algorithm for solving the inverse problem in comparison with minimization of the cost functional. Numerical experiments demonstrate the effectiveness of this approach. [ABSTRACT FROM AUTHOR]

In this paper, an inverse problem of determining the unknown coefficient of a multidimensional nonlinear time-dependent Schrödinger equation that has a complex number at nonlinear part is considered. The inverse problem is reformulated as a variational one which aims to minimize the observation functional. This paper presents existence and uniqueness theorems of solutions of the constituted variational problem, the gradient of the observation functional and a necessary condition for the solution of the variational problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with the convergence of a series associated with a certain version of the convexification method.That version has been recently developed by the research group of the first author for solving coefficient inverse problems.The convexification method aims to construct a globally convex Tikhonov-like functional with a Carleman weight function in it.In the previous works, the construction of the strictly convex weighted Tikhonov-like functional assumes a truncated Fourier series (i.e. a finite series instead of an infinite one) for a function generated by the total wave field.In this paper, we prove a convergence property for this truncated Fourier series approximation.More precisely, we show that the residual of the approximate PDE obtained by using the truncated Fourier series tends to zero in L 2 L^{2} as the truncation index in the truncated Fourier series tends to infinity.The proof relies on a convergence result in the H 1 H^{1} -norm for a sequence of L 2 L^{2} -orthogonal projections on finite-dimensional subspaces spanned by elements of a special Fourier basis.However, due to the ill-posed nature of coefficient inverse problems, we cannot prove that the solution of that approximate PDE, which results from the minimization of that Tikhonov-like functional, converges to the correct solution. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse problem for determining the space-dependent source and the initial value simultaneously in a parabolic equation from two over-specified measurements. By means of transforming information of the initial value into the source term and obtaining a combined source term, the parabolic equation problem is converted into a parabolic problem with homogeneous conditions. Then the considered inverse problem is formulated into a regularized minimization problem, which is implemented by the finite element method based on solving a sequence of well-posed direct problems. The uniqueness of inverse solutions are proved by the solvability of the corresponding variational problem, and the conditional stability as well as the convergence rate of regularized solutions are also provided. Then the error estimate of approximate regularization solutions is presented in the finite-dimensional space. The proposed method is a very fast non-iterative algorithm, and it can successfully solve the multi-dimensional inverse problem for recovering the space-dependent source and the initial value simultaneously. Numerical results of five examples including one- and two-dimensional cases show that the proposed method is efficient and robust with respect to data noise. [ABSTRACT FROM AUTHOR]

The paper aims a logarithmic stability estimate for the inverse source problem of the one-dimensional Helmholtz equation with attenuation factor in a two layer medium. We establish a stability by using multiple frequencies at the two end points of the domain which contains the compact support of the source functions. [ABSTRACT FROM AUTHOR]

Degenerate parabolic partial differential equations (PDEs) with vanishing or unbounded leading coefficient make the PDE non-uniformly parabolic, and new theories need to be developed in the context of practical applications of such rather unstudied mathematical models arising in porous media, population dynamics, financial mathematics, etc. With this new challenge in mind, this paper considers investigating newly formulated direct and inverse problems associated with non-uniform parabolic PDEs where the leading space- and time-dependent coefficient is allowed to vanish on a non-empty, but zero measure, kernel set. In the context of inverse analysis, we consider the linear but ill-posed identification of a space-dependent source from a time-integral observation of the weighted main dependent variable. For both, this inverse source problem as well as its corresponding direct formulation, we rigorously investigate the question of well-posedness. We also give examples of inverse problems for which sufficient conditions guaranteeing the unique solvability are fulfilled, and present the results of numerical simulations. It is hoped that the analysis initiated in this study will open up new avenues for research in the field of direct and inverse problems for degenerate parabolic equations with applications. [ABSTRACT FROM AUTHOR]

This paper deals with an inverse problem of the determination of the fractional order in time-fractional diffusion equations from one interior point observation. We give a representation of the solution via the Mittag-Leffler function and eigenfunction expansion, from which the Lipschitz stability of the fractional order with respect to the measured data at the interior point is established. [ABSTRACT FROM AUTHOR]

Recently, Nair and Roy (2017) considered a linear regularization method for a parameter identification problem in an elliptic PDE. In this paper, we consider similar procedure for identifying the diffusion coefficient in the heat equation, modifying the Sobolev spaces involved appropriately. We derive error estimates under appropriate conditions and also consider the finite-dimensional realization of the method, which is essential for practical application. In the analysis of finite-dimensional realization, we give a procedure to obtain finite-dimensional subspaces of an infinite-dimensional Hilbert space L 2 ⁢ (0 , T ; H 1 ⁢ (Ω)) {L^{2}(0,T;H^{1}(\Omega))} by doing double discretization, that is, discretization corresponding to both the space and time domain. Also, we analyze the parameter choice strategy and obtain an a posteriori parameter which is order optimal. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel dynamical system with time delay to describe the outbreak of 2019-nCoV in China. One typical feature of this epidemic is that it can spread in the latent period, which can therefore be described by time delay process in the differential equations. The accumulated numbers of classified populations are employed as variables, which is consistent with the official data and facilitates the parameter identification. The numerical methods for the prediction of the outbreak of 2019-nCoV and parameter identification are provided, and the numerical results show that the novel dynamic system can well predict the outbreak trend so far. Based on the numerical simulations, we suggest that the transmission of individuals should be greatly controlled with high isolation rate by the government. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of a quasi solution to a time fractional diffusion equation related to D t α C ⁢ u - ∇ ⋅ (k ⁢ (x) ⁢ ∇ ⁡ u) = f {{}^{C}D_{t}^{\alpha}u-\nabla\cdot(k(x)\nabla u)=f} , where the function k = k ⁢ (x) {k=k(x)} is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function k. At the first step of the methodology, we give a stability result corresponding to connectivity of k and u which leads to the continuity of the cost functional. We next construct an appropriate class of admissible functions and show that a solution of the minimization problem exists for the continuous cost functional. At the end, convexity of the introduced cost functional and subsequently the uniqueness theorem of the quasi solution are given. [ABSTRACT FROM AUTHOR]

This paper deals with numerical solution and parameters inversion for a one-dimensional non-symmetric two-sided fractional advection-dispersion equation (FADE) with zero Neumann boundary condition in a finite domain. A fully discretized finite difference scheme is set forth based on Grünwald-Letnikov's definition of the fractional derivative, and its stability and convergence are proved by estimating the spectral radius of the coefficient matrix of the scheme. Furthermore, an inverse problem of simultaneously determining the fractional order and the dispersion coefficients is investigated, and numerical inversions are carried out by using the optimal perturbation regularization algorithm. The inversion results show that the fractional order and the dispersion coefficients in the FADE can be determined successfully by the final observations. [ABSTRACT FROM AUTHOR]

In this paper, the adaptive multigrid conjugate gradient method is applied to the inversion of the space-dependent permeability function in a nonlinear diffusion equation, which plays an important role in promoting the permeability estimation within multiphase porous media flow. The forward problem is discretized using finite-difference methods and the estimation is formulated as a least-square minimization problem with a regularization term. Firstly, the conjugate gradient method for optimization is used to solve this parameter identification problem. Secondly, to reduce the amount of calculation, the idea of multigrid inversion and the necessary condition of 'the optimal solution should be the fixed point of multigrid inversion' is considered. Then, by combining the multigrid multiscale method with the conjugate gradient method, a new inversion method called the adaptive multigrid conjugate gradient method is designed. The results of numerical simulations indicate that the proposed method can effectively reduce the required computation burden and have good ability in anti-noise. [ABSTRACT FROM AUTHOR]

The problem of the recovery of a real-valued potential in the two-dimensional Schrödinger equation at positive energy from the Dirichlet-to-Neumann map is considered. It is know that this problem is severely ill-posed and the reconstruction of the potential is only logarithmic stable in general. In this paper a new stability estimate is proved, which is explicitly dependent on the regularity of the potentials and on the energy. Its main feature is an efficient increasing stability phenomenon at sufficiently high energies: in some sense, the stability rapidly changes from logarithmic type to Hölder type. The paper develops also several estimates for a non-local Riemann-Hilbert problem which could be of independent interest. [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]