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In this paper, we consider the inverse coefficients problem of recovering a shear modulus μ and density ρ of a medium from the Neumann-to-Dirichlet map. This inverse problem is motivated by the reconstruction of mechanical properties of tissues in non-destructive testing. We prove Lipschitz stability results for any dimension $ d \geq 2 $ d ≥ 2 , provided that the parameters μ and ρ have upper and lower bounds and belong to a known finite dimensional subspace. The proofs rely on monotonicity relations between the parameters and the Neumann-to-Dirichlet operator, combined with the techniques of localized potentials. [ABSTRACT FROM AUTHOR]

It is well known that the backward heat conduction problem of recovering the temperature u ⁢ (⋅ , t) at a time t ≥ 0 from the knowledge of the temperature at a later time, namely g := u ⁢ (⋅ , τ) for τ > t , is ill-posed, in the sense that small error in g can lead to large deviation in u ⁢ (⋅ , t) . However, in the case of a time fractional backward heat conduction problem (TFBHCP), the above problem is well-posed for t > 0 and ill-posed for t = 0 . We use this observation to obtain stable approximate solutions as solutions for t ∈ (0 , τ ] with t as regularization parameter for approximating the solution at t = 0 , and derive error estimates under suitable source conditions. We shall also provide some numerical examples to illustrate the approximation properties of the regularized solutions. [ABSTRACT FROM AUTHOR]

We propose novel reverse time migration (RTM) methods for the imaging of periodic obstacles using only measurements from the lower or upper side of the obstacle arrays at a fixed frequency. We analyze the resolution of the lower side and upper side RTM methods in terms of propagating modes of the Rayleigh expansion, Helmholtz–Kirchhoff equation and the distance of the measurement surface to the obstacle arrays, where the periodic structure leads to novel analysis. We give some numerical experiments to justify the competitive efficiency of our imaging functionals and the robustness against noises. Further, numerical experiments show sharp images especially for the vertical part of the periodic obstacle in the lower-RTM case, which is not shared by results for imaging bounded compactly supported obstacles. [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 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]

Let (M, g) and (M ′ , g ′) be non-orientable Riemannian surfaces with fixed boundary Γ and fixed Euler characterictic m, and Λ and Λ ′ be their Dirichlet-to-Neumann maps, respectively. We prove that the closeness of Λ ′ to Λ in the operator norm implies the existence of the near-conformal diffeomorphism β between (M, g) and (M ′ , g ′) which does not move the points of Γ . Thereby we establish the continuity of the determination Λ ↦ [ (M , g) ] , where [(M, g)] is the conformal class of (M, g) and the set of such conformal classes is endowed with the natural Teichmüller-type metric d T . In both orientable and non-orientable case we provide quantitative estimates of d T ([ (M , g) ] , [ (M ′ , g ′) ]) via the operator norm of the difference Λ ′ - Λ . We also obtain generalizations of the results above to the case in which the Dirichlet-to-Neumann map is given only on a segment of the boundary. [ABSTRACT FROM AUTHOR]

In this paper, we introduce quadrature domains for the Helmholtz equation. We show existence results for such domains and implement the so-called partial balayage procedure. We also give an application to inverse scattering problems, and show that there are non-scattering domains for the Helmholtz equation at any positive frequency that have inward cusps. [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]

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]

We introduce a class of fractional Dirac type operators with time variable coefficients by means of a Witt basis, the Djrbashian–Caputo fractional derivative and the fractional Laplacian, both operators defined with respect to some given functions. Direct and inverse fractional Cauchy type problems are studied for the introduced operators. We give explicit solutions of the considered fractional Cauchy type problems. We also use a recent method to recover a variable coefficient solution of some inverse fractional wave and heat type equations. Illustrative examples are provided. [ABSTRACT FROM AUTHOR]

In this article, we discuss an inverse problem of determining unknown coefficients of the first-order derivatives in an ultrahyperbolic equation. By a finite set of measurements, we prove the uniqueness of solution of the problem in semi-geodesic coordinates under some conditions on the principal coefficients of the equation. Our main tool is a Carleman estimate. [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]

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]

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]

For the equation of a mixed parabolic-hyperbolic type with a power degeneration on the type change line, the inverse problems to determine the time-dependent factors of right-hand sides are studied. Based on the formula for solving a direct problem, the solution of inverse problems is equivalently reduced to the solvability of loaded integral equations. Using the theory of integral equations, the corresponding theorems for the existence and uniqueness of the solutions of the stated inverse problems are proved and the explicit formulas for the solution have been given. [ABSTRACT FROM AUTHOR]

In this paper, we propose and study several inverse problems of identifying/determining unknown coefficients for a class of coupled PDE systems by measuring the average flux data on part of the underlying boundary. In these coupled systems, we mainly consider the non-negative solutions of the coupled equations, which are consistent with realistic settings in biology and ecology. There are several salient features of our inverse problem study: the drastic reduction of the measurement/observation data due to averaging effects, the nonlinear coupling of multiple equations, and the non-negative constraints on the solutions, which pose significant challenges to the inverse problems. We develop a new and effective scheme to tackle the inverse problems and achieve unique identifiability results by properly controlling the injection of different source terms to obtain multiple sets of mean flux data. The approach relies on certain monotonicity properties which are related to the intrinsic structures of the coupled PDE system. We also connect our study to biological applications of practical interest. [ABSTRACT FROM AUTHOR]

Consider a fractional operator P s , 0 < s < 1 , for connection Laplacian P : = ∇ ∗ ∇ + A on a smooth Hermitian vector bundle over a closed, connected Riemannian manifold of dimension n ≥ 2 . We show that local knowledge of the metric, Hermitian bundle, connection, potential, and source-to-solution map associated with P s determines these structures globally. This extends a result known for the fractional Laplace-Beltrami operator. [ABSTRACT FROM AUTHOR]

Inspired by some interesting equations modeling anomalous diffusion and nonlinear phenomena, we will study the inverse problems of uniquely identifying coefficients in nonlinear terms from over-determined data. Precisely, we consider a semilinear fractional Schrödinger operator (- Δ) s u + Q (x , u) = 0 in Ω with 0 < s < 1 . The fractional Laplacian arises due to the anomalous diffusion, e.g., the motion of particles described by Lévy flights. Here, we consider the semilinear term Q (x , u) = q (x , | u |) u , which appears naturally in the study of nonlinear optics with cubic Kerr-type nonlinearity, the complex Ginzburg–Landau equation with cubic-quintic nonlinearity, or even the Hartree equation with convolution-type nonlinearity, etc. In this article, we consider the time-independent and the time-evolution semilinear fractional Schrödinger equations with "Dirichlet" condition given on the complement of Ω . We prove both the well-posedness of the forward problems and the unique determination of the inverse problems with measurements taken on the complement of Ω . [ABSTRACT FROM AUTHOR]

We investigate an inverse boundary value problem of determination of a nonlinear law for reaction-diffusion processes, which are modeled by general form semilinear parabolic equations. We do not assume that any solutions to these equations are known a priori, in which case the problem has a well-known gauge symmetry. We determine, under additional assumptions, the semilinear term up to this symmetry in a time-dependent anisotropic case modeled on Riemannian manifolds, and for partial data measurements on ${\mathbb R}^n$ .