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The paper is devoted to a problem of acquiring elastic properties of a composite material from the vibration testing data with a simplified experimental acquisition scheme. The specimen is considered to abide by the linear elasticity laws and subject to viscoelastic damping. The boundary value problem for transverse movement of such a specimen in the frequency domain is formulated and solved with finite-element method. The correction method is suggested for the finite element matrices to account for the mass of the accelerometer. The problem of acquiring the elastic parameters is then formulated as a nonlinear least-square optimization problem. The usage of the automatic differentiation technique for stable and efficient computation of the gradient and hessian allows to use well-studied first and second order local optimization methods. We also explore the possibility of generating initial guesses for local minimization by heuristic global methods. The results of the numerical experiments on simulated data are analyzed in order to provide insights for the following real life experiments. [ABSTRACT FROM AUTHOR]

In this paper we study a partial inverse spectral problem for non-self-adjoint Sturm–Liouville operators with a constant delay and show that subspectra of two boundary value problems with one common boundary condition are sufficient to determine the complex potential. We developed the Horváth's method in [M. Horváth, On the inverse spectral theory of Schrödinger and Dirac operators, Trans. Amer. Math. Soc. 353 2001, 10, 4155–4171] for the self-adjoint Sturm–Liouville operator without delay into the non-self-adjoint Sturm–Liouville differential operator with a constant delay. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem for the modified Helmholtz equation. We consider the data completion problem in a bounded cylindrical domain on which the Neumann and the Dirichlet conditions are given in a part of the boundary. Since this problem is ill-posed, we reformulate it as an optimal control problem with an appropriate cost function. The method of factorization of boundary value problems is used to immediately obtain an approximation of the missing boundary data. In order to regularize this problem, we firstly scrutinize two classical regularizations for the cost function. Then we propose a new numerical regularization named "adaptive Runge–Kutta regularization", which does not require any penalization term. Finally, we compare them numerically. [ABSTRACT FROM AUTHOR]

In this paper, a Sturm–Liouville boundary value problem which includes conformable fractional derivatives of order α, 0 < α ≤ 1 {0<\alpha\leq 1} is considered. We give some uniqueness theorems for the solutions of inverse problems according to the Weyl function, two given spectra and classical spectral data. We also study the half-inverse problem and prove a Hochstadt–Lieberman-type theorem. [ABSTRACT FROM AUTHOR]

This paper deals with non-self-adjoint differential operators with two constant delays generated by - y ′′ + q 1 ⁢ (x) ⁢ y ⁢ (x - τ 1) + (- 1) i ⁢ q 2 ⁢ (x) ⁢ y ⁢ (x - τ 2) {-y^{\prime\prime}+q_{1}(x)y(x-\tau_{1})+(-1)^{i}q_{2}(x)y(x-\tau_{2})} , where π 3 ≤ τ 2 < π 2 < 2 ⁢ τ 2 ≤ τ 1 < π {\frac{\pi}{3}\leq\tau_{2}<\frac{\pi}{2}<2\tau_{2}\leq\tau_{1}<\pi} and potentials q j {q_{j}} are real-valued functions, q j ∈ L 2 ⁢ [ 0 , π ] {q_{j}\in L^{2}[0,\pi]}. We will prove that the delays and the potentials are uniquely determined from the spectra of four boundary value problems: two of them under boundary conditions y ⁢ (0) = y ⁢ (π) = 0 {y(0)=y(\pi)=0} and the remaining two under boundary conditions y ⁢ (0) = y ′ ⁢ (π) = 0 {y(0)=y^{\prime}(\pi)=0}. [ABSTRACT FROM AUTHOR]

We consider the boundary value problem R ⁢ (a , q) {R(a,q)} : - y ′′ ⁢ (x) + q ⁢ (x) ⁢ y ⁢ (x) = λ ⁢ y ⁢ (x) {-y^{\prime\prime}(x)+q(x)y(x)=\lambda y(x)} with y ⁢ (0) = 0 {y(0)=0} and y ⁢ (1) ⁢ cos ⁡ (a ⁢ λ) = y ′ ⁢ (1) ⁢ sin ⁡ (a ⁢ λ) λ {y(1)\cos(a\sqrt{\lambda})=y^{\prime}(1)\frac{\sin(a\sqrt{\lambda})}{\sqrt{% \lambda}}}. Motivated by the previous work [T. Aktosun and V. G. Papanicolaou, Reconstruction of the wave speed from transmission eigenvalues for the spherically symmetric variable-speed wave equation, Inverse Problems 29 2013, 6, Article ID 065007], it is natural to consider the following interesting question: how does one characterize isospectral sets corresponding to problem R ⁢ (1 , q) {R(1,q)} ? In this paper applying constructive methods we answer the above question. [ABSTRACT FROM AUTHOR]

Consider the interior transmission problem arising in inverse boundary value problems for the diffusion equation with discontinuous diffusion coefficients. We prove the unique solvability of the interior transmission problem by constructing its Green function. First, we construct a local parametrix for the interior transmission problem near the boundary in the Laplace domain, by using the theory of pseudo-differential operators with a large parameter. Second, by carefully analyzing the analyticity of the local parametrix in the Laplace domain and estimating it there, a local parametrix for the original parabolic interior transmission problem is obtained via the inverse Laplace transform. Finally, using a partition of unity, we patch all the local parametrices and the fundamental solution of the diffusion equation to generate a global parametrix for the parabolic interior transmission problem and then compensate it to get the Green function by the Levi method. The uniqueness of the Green function is justified by using the duality argument, and then the unique solvability of the interior transmission problem is concluded. We would like to emphasize that the Green function for the parabolic interior transmission problem is constructed for the first time in this paper. It can be applied for active thermography and diffuse optical tomography modeled by diffusion equations to identify an unknown inclusion and its physical property. [ABSTRACT FROM AUTHOR]

In this paper, we study the theoretical analysis of a frictionless contact between two general elastic bodies in a stationary regime in a three-dimensional thin domain Ω ε {\Omega^{\varepsilon}} with Tresca friction law. Firstly, the problem statement and variational formulation are presented. We then obtain the estimates on displacement independently of the parameter ε. Finally, we obtain the main results concerning the limit of a weak problem and its uniqueness. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach; that is, to determine the space-dependent source term from a noisy final data. Based on the series expression of the solution for the direct problem, we improve the regularity of the weak solution for the direct problem under strong conditions, and we provide the existence and uniqueness for the adjoint problem. Further, we use the Tikhonov regularization method to solve the inverse source problem and provide a conjugate gradient algorithm to find an approximation to the minimizer of the Tikhonov regularization functional. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, we investigate the backward heat problem (BHP) which is a classical ill-posed problem. Although there are many papers relating to the BHP in many domains, considering this problem in polar coordinates is still scarce. Therefore, we wish to deal with this problem associated with a space and time-dependent heat source in polar coordinates. By modifying the quasi-boundary value method, we propose the stable solution for the problem. Furthermore, under some initial assumptions, we get the Hölder type of error estimates between the exact solution and the approximated solution. Eventually, a numerical experiment is provided to prove the effectiveness and feasibility of our method. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse source problem for elliptic partial differential equations with both Dirichlet and Neumann boundary conditions. The unknown source term is to be determined by additional boundary data. This problem is ill-posed since the dimensionality of the boundary is lower than the dimensionality of the inner domain. To overcome the ill-posed nature, using the a priori information (sourcewise representation), and based on the coupled complex boundary method, we propose a coupled complex boundary expanding compacts method (CCBECM). A finite element method is used for the discretization of CCBECM. The regularization properties of CCBECM for both the continuous and discrete versions are proved. Moreover, an a posteriori error estimate of the obtained finite element approximate solution is given and calculated by a projected gradient algorithm. Finally, numerical results show that the proposed method is stable and effective. [ABSTRACT FROM AUTHOR]

In this paper we present a study of optimal control problem for the unsteady Navier–Stokes equations. We discuss the existence of the solution, adopt a new numerical resolution for this problem and combine Euler explicit scheme in time and both of methods spectral and finite elements in space. Finally, we give some numerical results proving the effectiveness of our approach. [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]

Adsorption isotherms are the most important parameters in rigorous models of chromatographic processes. In this paper, in order to recover adsorption isotherms, we consider a coupled complex boundary method (CCBM), which was previously proposed for solving an inverse source problem [2]. With CCBM, the original boundary fitting problem is transferred to a domain fitting problem. Thus, this method has advantages regarding robustness and computation in reconstruction. In contrast to the traditional CCBM, for the sake of the reduction of computational complexity and computational cost, the recovered adsorption isotherm only corresponds to the real part of the solution of a forward complex initial boundary value problem. Furthermore, we take into account the position of the profiles and apply the momentum criterion to improve the optimization progress. Using Tikhonov regularization, the well-posedness, convergence properties and regularization parameter selection methods are studied. Based on an adjoint technique, we derive the exact Jacobian of the objective function and give an algorithm to reconstruct the adsorption isotherm. Finally, numerical simulations are given to show the feasibility and efficiency of the proposed regularization method. [ABSTRACT FROM AUTHOR]

This paper concerns the imaging of a complex-valued anisotropic tensor γ = σ + ιωε from knowledge of several inter magnetic fields H, where H satisfies the anisotropic Maxwell system on a bounded domain X ⊂ R2 with prescribed boundary conditions on ∂X. We show that can be uniquely reconstructed with a loss of two derivatives from errors in the acquisition H. A minimum number of five well-chosen functionals guaranties a local reconstruction of γ in dimension two. The explicit inversion procedure is presented in several numerical simulations, which demonstrate the influence of the choice of boundary conditions on the stability of the reconstruction. This problem finds applications in the medical imaging modalities Current Density Imaging and Magnetic Resonance Electrical Impedance Tomography. [ABSTRACT FROM AUTHOR]

This study deals with the numerical implementation of a formula in the enclosure method as applied to a prototype inverse initial boundary value problem for thermal imaging in a one-space dimension. A precise error estimate of the formula is given and the effect on the discretization of the used integral of the measured data in the formula is studied. The formula requires a large frequency to converge; however, the number of time interval divisions grows exponentially as the frequency increases. Therefore, for a given number of divisions, we fixed the trusted frequency region of convergence with some given error bound. The trusted frequency region is computed theoretically using the theorems provided in this paper and is numerically implemented for various cases. [ABSTRACT FROM AUTHOR]

In this paper a boundary value problem is considered generated by the Sturm-Liouville equation and non-separated boundary conditions, one of which contains a spectral parameter. We give a uniqueness theorem, develop an algorithm for solving the inverse problem of reconstruction of boundary value problems with spectral data. We use the spectra of two boundary value problems and some sequence of signs as a spectral data. [ABSTRACT FROM AUTHOR]

In this paper we present an algorithm to bound parameters estimated from very small data sets and use this method to suggest optimal times for data collection. The method is deterministic in that we assume that one has error bars on the data. The basic idea is based on bounded error parameter identification methods and we use this framework to assign a measure of quality to the estimated parameter. Exploiting the dynamics of the underlying mathematical model allows us to describe the quality of the estimated parameter in a way that, due to the small number of data points, is not appropriate for traditional statistical techniques. The algorithm in this paper differs from traditional bounded error parameter identification techniques in that one computes the membership set by solving an inverse problem rather than a forward interval problem. [ABSTRACT FROM AUTHOR]

In the recent paper [Colombo, Physica D 236: 81–89, 2007] the author investigates an inverse problem arising in the theory of combustion. The problem studied is: determine the temperature u and the convolution memory kernel k in the evolution equation given suitable initial-boundary conditions and the following additional restriction on u: ∫Ω φ( x) u( t, x) dx = g( t), where φ and g are given functions. The main results are a local in time existence theorem and a global in time uniqueness result. In this paper we complete the study considering the linearized version of the model. We prove that, if F( u( t, x), ∇ u( t, x)) is sublinear, then the inverse problem has a unique global in time solution. [ABSTRACT FROM AUTHOR]

In this paper, a simple and convenient new regularization method which is called modified quasi-boundary value method for solving nonlinear backward heat equation is given. Some new quite sharp error estimates between the approximate solution are provided and generalize the results in our paper [Trong, Quan, Khanh, and Tuan, Zeitschrift Analysis und ihre Anwendungen 26: 231–245, 2007, Trong and Tuan, Electron. J. Diff. Eqns. 4: 1–10, 2006, Trong and Tuan, Electron. J. Diff. Eqns. 84: 1–12, 2008]. The approximation solution is calculated by the contraction principle. A numerical experiment is given. [ABSTRACT FROM AUTHOR]

We consider a general non-conservative Schrödinger equation defined on an open bounded domain Ω in Rn, with C²-boundary ..., subject to (Dirichlet and, as a main focus, to) Neumann boundary conditions on the entire boundary Γ. Here, Γ0 and Γ1 are the unobserved (or uncontrolled) and observed (or controlled) parts of the boundary, respectively, both being relatively open in Γ. The Schrödinger equation includes energy-level (H¹(Ω)-level) terms, which accordingly may be viewed as unbounded perturbations. The first goal of the paper is to provide Carleman-type inequalities at the H¹-level, which do not contain lower-order terms; this is a distinguishing feature over most of the literature. This goal is accomplished in a few steps: the paper obtains first pointwise Carleman estimates for C²-solutions; and next, it turns these pointwise estimates into integral-type Carleman estimates with no lower-order terms, originally for H²-solutions, and ultimately for H¹-solutions, The passage from H² to H¹-solutions is readily accomplished in the case of Dirichlet B.C., but it requires a delicate regularization argument in the case of Neumann B.C. This is so since finite energy solutions are known to have L2-normal traces in the case of Dirichlet B.C., but by contrast do not produce H¹-traces in the case of Neumann B.C. From Carleman-type inequalities with no lower-order terms, one then obtains the sought-after benefits. These consist of deducing, in one shot, as a part of the same flow of arguments, two important implications: (i) global uniqueness results for H1-solutions satisfying over-determined boundary conditions, and--above all--(ii) continuous observability (or stabilization) inequalities with an explicit constant. The more demanding purely Neumann boundary conditions requires the same geometrical conditions... [ABSTRACT FROM AUTHOR]

We consider a general non-conservative Schrödinger equation defined on an open bounded domain ω in Rn, with C²-boundary Γ = ∂ Ω = Γ0 ∪ Γ 1, Γ0 ∩ Γ 1 = 0, subject to (Dirichlet and, as a main focus, to) Neumann boundary conditions on the entire boundary Γ. Here, Γ0 and Γ1 are the unobserved (or uncontrolled) and observed (or controlled) part of the boundary, respectively, both being relatively open in Γ. The Schrödinger equation includes energy-level (H¹ (&Omega)-level) terms, which according may be viewed as unbounded perturbations. The first goal of the paper is to provided Carleman-type inequalities at the H¹-level, which do not contain lower-order terms; this is a distinguishing feature over most of the literature. This goal is accomplished in a few steps: the paper obtains first pointwise Carleman estimates for C²-solutions; and next, in turns these pointwise estimates into integral-type Carleman estimates with no lower-order terms, originally for H²-solutions, and ultimately for H¹-solutions. The passage from H²- to H¹-solutions is readily accomplished in the case of Dirichlet B.C., but it requires a delicate regularization argument in the case of Neumann B.C. This is so since finite energy solutions are known to have L2-normal traces in the case of Dirichlet B.C., but by contrast do not produce H¹-traces in the case of Neumann B.C. From Carleman-type inequalities with no lower-order terms, one then obtains the sought-after benefits. These consist of deducing, in one shot, as a part of the same flow of arguments, two important implications: (i) global uniqueness results for H¹-solutions satisfying over-determined boundary conditions, and--above all--(ii) continuous observability (or stabilization) inequalities with an explicit... [ABSTRACT FROM AUTHOR]

This paper concerns the inverse problem of retrieving the principal coefficient in a Korteweg-de Vries (KdV) equation from boundary measurements of a single solution. The Lipschitz stability of this inverse problem is obtained using a new global Carleman estimate for the linearized KdV equation. The proof is based on the Bukhgeĭm-Klibanov method. [ABSTRACT FROM AUTHOR]

This paper investigates an adaptive algorithm for identification of source terms in the linear parabolic equation and Robin boundary condition from the measured final data and the measurement of the temperature in a subregion. This is based on the minimization of Tikhonov functional. We pursue the approach iteratively regularized gradient method. Results with the adaptive algorithm are presented. [ABSTRACT FROM AUTHOR]

In this paper, we want to estimate the true resistivity of permeable bed using the spontaneous potential on the measuring electrode in heterogeneous formation, where the resistivity of the objective layer varies continuously as a function of position. The problem is a generalization of Peng's work for the traditional three-parameter invasion model, where the resistivity of formation is piecewise constant. The mathematical model of the problem is described by a variational problem of elliptic type, where the formation resistivity appears as a nonlinear coefficient. We study an extended problem and then apply its results to prove the existence of solutions to the parameter identification problem arising from SP logging. Finally, a sufficient condition is given to ensure the uniqueness of solutions. [ABSTRACT FROM AUTHOR]

Diffuse optical tomography (DOT) is an optical imaging modality, which provides the spatial distribution of the optical parameters inside a random medium. A propagation back-propagation method named EM-like reconstruction method for stationary DOT problem has been proposed yet. This method is really time consuming. Hence the ordered-subsets (OS) technique for this reconstruction method is studied in this paper. The boundary measurements of DOT are grouped into nonoverlapping and overlapping ordered sequence of subsets with random partition, sequential partition and periodic partition, respectively. The performance of OS methods is compared with the standard EM-like reconstruction method with two-dimensional and three-dimensional numerical experiments. The numerical experiments indicate that reconstruction of nonoverlapping subsets with periodic partition, overlapping subsets with periodic partition and standard EM-like method provide very similar acceptable reconstruction results. However, reconstruction of nonoverlapping subsets with periodic partition spends a minimum of time to get proper results. [ABSTRACT FROM AUTHOR]

We prove uniqueness and continuous dependence results for a severely ill-posed linear integrodifferential boundary-value parabolic problem with no initial condition. This latter condition is replaced with an additional boundary information prescribing the temperature on an open subset of the geometric domain . The integral operators entering the equation are defined by integrals of Volterra type with respect to time. In particular, the class of integrodifferential equations dealt with in this paper include those occurring in the linear theory of heat flow in a rigid body consisting of a material with thermal memory. [ABSTRACT FROM AUTHOR]

In this paper we review some recent results concerning inverse problems for thin elastic plates. The plate is assumed to be made by non-homogeneous linearly elastic material belonging to a general class of anisotropy. A first group of results concerns uniqueness and stability for the determination of unknown boundaries, including the cases of cavities and rigid inclusions. In the second group of results, we consider upper and lower estimates of the area of unknown inclusions given in terms of the work exerted by a couple field applied at the boundary of the plate. In particular, we extend previous size estimates for elastic inclusions to the case of cavities and rigid inclusions. [ABSTRACT FROM AUTHOR]

In this paper we develop a convergence analysis in an infinite dimensional setting of the Levenberg-Marquardt iteration for the solution of a hybrid conductivity imaging problem. The problem consists in determining the spatially varying conductivity σ from a series of measurements of power densities for various voltage inductions. Although this problem has been very well studied in the literature, convergence and regularizing properties of iterative algorithms in an infinite dimensional setting are still rudimentary. We provide a partial result under the assumptions that the derivative of the operator, mapping conductivities to power densities, is injective and the data is noise-free. Moreover, we implemented the Levenberg-Marquardt algorithm and tested it on simulated data. [ABSTRACT FROM AUTHOR]

In the paper the differential identity was obtained for quantum kinetic equation. On the basis of this identity the theorem of uniqueness of inverse problem of determining the solution and right-hand was proved given the boundary and initial information. The problem of recovery of the solution and potential was considered. Also, the solution in the quasipolynomial form was constructed. [ABSTRACT FROM AUTHOR]

In this paper, we show the uniqueness and local stability of an inverse source problem for the quasi-static Maxwell equation in a layered domain, where the source consists of multiple point dipoles. Also, an algebraic algorithm is proposed to identify the number, locations, and moments of the dipoles from boundary measurements of tangential components of the electric and magnetic fields. The proposed algorithm is numerically verified. [ABSTRACT FROM AUTHOR]

A Cauchy problem for elliptic equations with nonhomogeneous Neumann data in a cylindrical domain is investigated in this paper. For the theoretical aspect the a-priori and a-posteriori parameter choice rules are suggested and the corresponding error estimates are obtained. About the numerical aspect, for a simple case results given by two methods based on the discrete Sine transform and the finite difference method are presented; an idea of left-preconditioned GMRES (Generalized Minimum Residual) method is proposed to deal with the high dimensional case to save the time; a view of dealing with a general domain is suggested. Some ill-posed problems regularized by the quasi-boundary-value method are listed and some rules of this method are suggested. [ABSTRACT FROM AUTHOR]

In this paper, the criterion of resolvability of the inverse problem of heat conduction is established. [ABSTRACT FROM AUTHOR]

In this paper we investigate the identification of the time-dependent perfusion coefficient in the transient bio-heat conduction equation. Apart from the usual initial and Dirichlet/Neumann boundary conditions, in this inverse coefficient identification problem the additional measurement information to render a unique solution is represented by the time history of a heat flux/boundary temperature. Several necessary and sufficient conditions for the existence and uniqueness of the solution are provided. [ABSTRACT FROM AUTHOR]

The paper is concerned with properties of an ill-posed problem for the Helmholtz equation when Dirichlet and Neumann conditions are given only on a part Γ of the boundary ∂Ω. We present an equivalent formulation of this problem in terms of a moment problem defined on the part of the boundary where no boundary conditions are imposed. Using a weak definition of the normal derivative, we prove the equivalence between these two problems for an arbitrary Lipschitz domain in . Moreover, uniqueness of the solution is proved for the general case when Γ is a non-empty open subset of the Lipschitz boundary. [ABSTRACT FROM AUTHOR]

In this paper the author consider uniqueness of solution of determining attenuation coefficient of X-ray radiation inside absorbing and scattering medium. The known data are densities of incoming and outgoing flows of radiation in the boundary of the medium. The specific character of the problem is that the density of external sources of radiation depending on energy is disconnected in the finite number of points which corresponds to the radiation resonance. This assumption is sufficient to successful research of the problem which may be considered as the problem of X-ray tomography. The generality of the mathematical model used in research makes possible to apply the result for other problems of radiation tomography. [ABSTRACT FROM AUTHOR]

A new second order Newton method for reconstructing the shape of a sound soft scatterer from the measured far-field pattern for scattering of time harmonic plane waves is presented. This method extends a hybrid between regularized Newton iterations and decomposition methods that has been suggested and analyzed in a number of papers by Kress and Serranho [Inverse Problems 19: 91–104, 2003, Inverse Problems 21: 773–784, 2005, J. Comput. Appl. Math. 204: 418–427, 2007, Inverse Problems 22: 663–680, 2006, Inverse Problems and Imaging 1: 691–712, 2007] and has some features in common with the second degree method for ill-posed nonlinear problems as considered by Hettlich and Rundell [SIAM J.Numer. Anal. 37: 587–620, 2000]. The main idea of our iterative method is to use Huygen's principle, i.e., represent the scattered field as a single-layer potential. Given an approximation for the boundary of the scatterer, this leads to an ill-posed integral equation of the first kind that is solved via Tikhonov regularization. Then, in a second order Taylor expansion, the sound soft boundary condition is employed to update the boundary approximation. In an iterative procedure, these two steps are alternated until some stopping criterium is satisfied. We describe the method in detail and illustrate its feasibility through examples with exact and noisy data. [ABSTRACT FROM AUTHOR]

For solving linear ill-posed problems regularization methods are required when the right hand side and the operator is with some noise. In the present paper regularized approximations are obtained by Tikhonov regularization, by regularized total least squares (RTLS) and by dual regularized total least squares (DRTLS). We discuss computational aspects and provide order optimal error bounds under two assumptions characterizing the smoothness of the unknown solution and the smoothing properties of the forward operator. The derived error bounds extend results from [S. Lu, S.V. Pereverzev, and U. Tautenhahn, Regularized total least squares: computational aspects and error bounds, Johann Radon Institute for Computational and Applied Mathematics, Report no. 2007-30, 2007.] where the above regularization methods are studied for finitely smoothing forward operators. We illustrate our theory by a special inverse heat conduction problem. [ABSTRACT FROM AUTHOR]

In the present paper, the inverse problem with unknown composite right-hand side for second-order linear hyperbolic equations is examined. Solution existence, uniqueness, and stability for the boundary-value problem are established. [ABSTRACT FROM AUTHOR]

In this paper electromagnetic Herglotz dyadics are used to develop a superposition method and apply it for the study of an inversion scheme for electromagnetic scattering in chiral media. The direct scattering problem for the perfect conductor is formulated and Beltrami Herglotz dyadics and dyadic electromagnetic Herglotz pairs are being used. Assuming that the incident electromagnetic field is produced by a superposition of plane incident waves, the scattered field and the corresponding far-field pattern are expressed as the superposition of the scattered fields and the corresponding to them far-field patterns respectively. Far-field operators are defined and studied and an integral equation is posed. The solvability of this equation is related to the solution of the interior perfect conductor boundary value problem. Finally, an inversion scheme is posed and a theorem for it's solvability is proved. [ABSTRACT FROM AUTHOR]

This paper deals with a two or three dimensional inverse scattering problem for identifying a sound hard obstacle in a homogeneous medium. We provide a reconstruction scheme for identifying the obstacle from far field patterns of scattered waves associated with incident waves in all directions when the obstacle is not necessarily convex. [ABSTRACT FROM AUTHOR]

Linear integral equations of the third kind usually lead to ill-posed inverse problems if the normed solution space X and the normed data space Y are required to be equal. In the present paper we develop a two-step regularization method—called RPMO method—to regularize such inverse problems. This method does not require Hilbert space properties. Convergence results are presented indicating that there is no general theoretical upper bound less than one for the convergence rates if the corresponding exact data solutions are sufficiently smooth. Moreover, we illustrate the RPMO method by applying its discretized version to an implicit linear boundary value problem which can be transformed into an equivalent third-kind integral equation. [ABSTRACT FROM AUTHOR]

In this paper we consider an inverse boundary value problem of determining three dimensional unknown inclusions in an elliptic equation in a bounded domain Ω ⊂ R3 from finite boundary measurements on ∂Ω. We will show that poly-hedral inclusions in Ω can be uniquely determined up to their convex edges from a single boundary measurement on ∂Ω. [ABSTRACT FROM AUTHOR]

Akduman, Haddar and Kress [1, 12, 15] proposed a conformal mapping approach for solving inverse boundary value problems for the two-dimensional Laplace equation in a doubly connected domain D with interior boundary curve Γ0 and exterior boundary curve Γ1. The inverse problem consists of reconstructing the interior boundary Γ0 from the Cauchy data on Γ1 of a harmonic function satisfying a homogeneous Dirichlet or Neumann boundary condition on the unknown interior boundary curve Γ0. The reconstruction method consists of two parts: In the first step, by successive approximations a nonlocal and nonlinear ordinary differential equation is solved to determine the boundary values of a holomorphic function Ψ on the outer boundary circle C1 of an annulus B. Then in the second step via regularizing a Laurent expansion in the sense of Tikhonov an ill-posed Cauchy problem is solved to determine Ψ in the annulus and the unknown Γ0 as the image Ψ(C0) of the interior boundary circle C0 of B. The present paper extends this approach to the case of a homogeneous impedance boundary condition. The analysis and the numerical implementation of the method differ from the limiting cases of the Dirichlet and Neumann conditions since the impedance problem in the annulus B that is associated with the impedance problem in the original domain D depends on the conformal map Ψ. [ABSTRACT FROM AUTHOR]

A planar graph consisting of strings of variable densities is considered. The spectrum of the Dirichlet problem on the graph and the values of derivatives of the normalized eigenfunctions at the boundary vertices constitute the spectral data. The inverse problem is to recover the structure of the graph and the densities from the spectral data. If the graph doesn't contain cycles (is a tree), it is determined by the spectral data up to a natural isometry on the plane (Belishev, 2004). In the paper this uniqueness result is supplied with an efficient procedure of recovering the tree. The numerical illustration is presented. [ABSTRACT FROM AUTHOR]

In this paper the inverse problem of finding the minor coefficient and the inverse problem of finding the right-hand side member of special type in a second order parabolic equation with changing time direction are considered. The existence and uniqueness of solutions to these problems are investigated. [ABSTRACT FROM AUTHOR]

We consider the problem of detecting corrosion damage on an inaccessible part of a specimen. We adopt a model in which the corrosion damage is represented by an unknown boundary coefficient appearing in a third kind boundary value problem for Laplace's equation. The associated inverse problem consists in the determination of the unknown boundary coefficient from an overdetermined data on an accessible part of the boundary. In the present paper we are interested in the derivation of stability estimates for this inverse problem. We first prove a local Lipschitz stability estimate for an arbitrary smooth domain. Next, we establish a log-log stability estimate in the case of a rectangular domain. [ABSTRACT FROM AUTHOR]

In this paper we present a new formulation for the nonlinear Cauchy problem for elliptic equations. This formulation allows us to give a method for solving this class of problems. The nonlinear problem is reduced to a linear Cauchy problem for the Laplace equation coupled with a sequence of nonlinear scalar equations. We solve the linear problem using the iterative method introduced in [7]. The linear dense systems obtained from a boundary element approximation are solved using an efficient implementation which accelerate the iterative process. Various types of convergence, comparison results and effects of small perturbations on boundary data are investigated. The numerical results show that the method produces a stable good approximate solution. [ABSTRACT FROM AUTHOR]

The uniqueness of reconstruction of a connected compact analytic em- bedded in Rn manifold without boundary from some projection and section functionals of its polar of the convex hull is discussed in the paper. [ABSTRACT FROM AUTHOR]

Modeling of large lakes or oceans for environmental studies or deformable blood vessels for mini-invasive surgery involve dynamical free boundaries governed by their interaction with their immediate environment. The object of this paper is to further Shape Analysis techniques in the formulation and analysis of dynamical free boundary problems defined over a time-varying domain (noncylindrical problems). The generic example of a potential incompressible fluid flow with a pressure condition (Bernouilli type) on the dynamical free boundary is presented along with a discrete time algorithm to follow the evolution of the time-space differential equation for the normal component of the velocity field on the time-varying free boundary. These ideas and constructions can be extended to more complex and realistic models. [ABSTRACT FROM AUTHOR]