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In the framework of stationary anisotropic heat conduction (the Laplace-Beltrami equation) without heat sources, we investigate both theoretically and numerically the acceleration of the two iterative algorithms of Kozlov et al. (U.S.S.R. Computational Mathematics and Mathematical Physics 31:45–52, 1991) for the accurate, convergent and stable reconstruction of the missing temperature and normal heat flux on an inaccessible boundary of the domain occupied by a solid from the knowledge of Cauchy data on the remaining and accessible boundary. For each of the two algorithms with relaxation studied, this inverse Cauchy problem for the Laplace-Beltrami equation with exact data is transformed into an equivalent fixed point problem for an associated operator that is defined on and takes values in a suitable function space, and also accounts for the relaxation parameter. Consequently, the convergence of each relaxation algorithm reduces to analysing the properties of the corresponding operator and this enables us to determine the admissible range for the relaxation parameter, as well as a criterion for selecting its optimal value at each iteration, for each iterative procedure and exact Cauchy data. In case of perturbed Cauchy data, regularisation is achieved by terminating the iteration according to the discrepancy principle. The numerical implementation is realised for two-dimensional homogeneous anisotropic solids using the finite element method and confirms a significant reduction in the number of iterations and hence CPU time required for the two relaxation algorithms proposed to achieve convergence, for both exact and perturbed Cauchy data, provided that the dynamical selection of the optimal value for the relaxation parameter is employed. [ABSTRACT FROM AUTHOR]

We study the reconstruction of an unknown/inaccessible smooth inner boundary from the knowledge of the Dirichlet condition (temperature) on the entire boundary of a doubly connected domain occupied by a two-dimensional homogeneous anisotropic solid and an additional Neumann condition (normal heat flux) on the known, accessible, and smooth outer boundary in the framework of steady-state heat conduction with heat sources. This inverse geometric problem is approached through an operator that maps an admissible inner boundary belonging to the space of 2π-\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$2\pi -$$\end{document}periodic and twice continuously differentiable functions into the Neumann data on the outer boundary which is assumed to be continuous. We prove that this operator is differentiable, and hence, a gradient-based method that employs the anisotropic single layer representation of the solution to an appropriate Dirichlet problem for the two-dimensional anisotropic heat conduction is developed for approximating the unknown inner boundary. Numerical results are presented for both exact and perturbed Neumann data on the outer boundary and show the convergence, stability, and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

The isotropic thermoelasticity system of type-III, describing both the mechanical and the thermal behaviours of a body occupying a bounded domain with a Lipschitz boundary, is considered. The displacement vector and either the normal heat flux or the temperature are prescribed on the boundary. Both the theoretical and the numerical reconstructions of a time-dependent heat source from the knowledge of an additional weighted integral measurement of the temperature are investigated. It is shown that the appropriate type of measurement depends on the thermal boundary condition available, whilst the existence and uniqueness of a weak solution for exact data are also proved. For each of the two inverse source problems investigated herein, a numerical algorithm is also proposed and the convergence of these numerical schemes for exact data is proved. Four numerical examples with noisy measurements are implemented using the finite element method and thoroughly investigated to validate the convergence and stability of the proposed algorithms. [ABSTRACT FROM AUTHOR]

We investigate the numerical reconstruction of the missing thermal boundary data on a part of the boundary for the steady-state heat conduction equation in anisotropic solids from the knowledge of exact or noisy Cauchy data on the remaining and accessible boundary. This inverse boundary value problem is tackled by applying and adapting to the anisotropic case the algorithm based on the fading regularization method, originally proposed by Cimetière, Delvare, and Pons (Comptes Rendus de l'Académie des Sciences - Série IIb - Mécanique, 328 639–644 2000), and Cimetière, Delvare, et al. (Inverse Probl., 17 553–570 2001) for the isotropic heat conduction equation. The numerical implementation is realised for 2D homogeneous solids by using the boundary element method, whilst the numerical solution is stabilized/regularized by stopping the iterative process based on an L-curve type criterion (Hansen 1998). [ABSTRACT FROM AUTHOR]

The Cauchy problem in 2D and 3D steady-state anisotropic heat conduction is investigated for both exact and perturbed data, i.e. the numerical reconstruction of the missing temperature and normal heat flux on a part of the boundary from the knowledge of exact or noisy Cauchy data on the remaining and accessible boundary. This inverse Cauchy problem is solved by applying and adapting the fading regularization method, proposed by Cimetière et al. [7-8] for the steady-state isotropic heat conduction, to the anisotropic case. An appropriate stabilizing/regularizing stopping criterion for the resulting iterative algorithm is provided for each type of Cauchy data considered. The numerical implementation is realized for 2D and 3D homogeneous solids by using the meshless method of fundamental solutions. [ABSTRACT FROM AUTHOR]

We investigate the numerical reconstruction of the missing thermal boundary conditions on an inaccessible part of the boundary in the case of steady-state heat conduction in anisotropic solids from the knowledge of over-prescribed noisy data on the remaining accessible boundary. This inverse problem is tackled by employing a variational formulation that transforms it into an equivalent control problem; four such approaches are discussed thoroughly. The numerical implementation is realised for the 2D case via the boundary element method for perturbed Cauchy data, whilst the numerical solution is stabilised/regularised by stopping the iterative procedure according to Morozov's discrepancy principle (Morozov, VA. On the solution of functional equations by the method of regularization. Doklady Mathematics 1966; 7: 414–417). [ABSTRACT FROM AUTHOR]

• Inverse problem of determining a time-dependent load source in a simply supported beam. • Missing information is recovered from an additional "local" integral measurement • The existence and uniqueness of a solution to the inverse source problem is proved. • A convergent and stable algorithm is proposed. • Numerical experiments validate the convergence and stability of the proposed procedure. In this paper, the theoretical and numerical determination of a solely time-dependent load distribution is investigated for a simply supported non-homogeneous Euler–Bernoulli beam. The missing source is recovered from an additional "local" integral measurement. The existence and uniqueness of a solution to the corresponding variational problem is proved by employing Rothe's method. This method also reveals a time-discrete numerical scheme based on the backward Euler method to approximate the solution. Corresponding error estimates are proved and assessed by two numerical experiments. [ABSTRACT FROM AUTHOR]

We investigate the Cauchy problem associated with the Helmholtz equation in three dimensions, namely the numerical reconstruction of the primary field (Dirichlet data) and its normal derivative (Neumann data) on a part of the boundary from the knowledge of overprescribed noisy measurements taken on the remaining boundary part. This inverse problem is solved by combining the fading regularization method with the method of fundamental solutions (MFS). A stopping regularizing/stabilizing criterion is also proposed. Two numerical examples are investigated in order to validate the proposed method in terms of its accuracy, convergence, stability and efficiency. [ABSTRACT FROM AUTHOR]

In the framework of steady-state inhomogeneous anisotropic heat conduction with heat sources, we study the convergent and stable numerical reconstruction of the missing boundary conditions (temperatures and normal heat fluxes) on an inaccessible boundary and the thermal field in a 2D solid from the knowledge of over-prescribed data on an accessible portion of the boundary and temperature data on the remaining boundary. This inverse boundary value problem is approached by employing the alternating iterative algorithms of Kozlov et al. (1991), in conjunction with the standard finite-difference method and a novel modified one, for both exact and perturbed data. For noisy data available on the over-prescribed boundary, the numerical solution is retrieved by using three regularising/stabilising stopping criteria. Numerical results are presented for 2D simply and doubly connected domains bounded by a (piecewise) smooth curve, as well as exact and noisy boundary data, whilst appropriate errors in the fractional Sobolev spaces corresponding to the boundary temperatures and normal heat fluxes, respectively, are computed efficiently. [ABSTRACT FROM AUTHOR]

We investigate the theoretical and numerical determination of a space-dependent vector source (load) in an anisotropic thermoelastic system of type-III from the knowledge of an additional final time measurement. The uniqueness of a solution to this inverse source problem is proved for various assumptions made on the convolution kernel. A convergent and stable iterative algorithm is proposed for the recovery of the unknown vector source in the linear case and, at the same time, a stopping criterion is also given. Three numerical experiments are considered to validate the properties of the proposed iterative procedure and the regularizing/stabilizing character of the corresponding stopping criterion. The numerical experiments carried out showed that it exists a certain limitation of the method with respect to the recovery of non-symmetric sources. [ABSTRACT FROM AUTHOR]

In this paper, we propose an invariant method of fundamental solutions for solving boundary value problems in two-dimensional isotropic linear elasticity. The invariant method of fundamental solutions keeps a very basic natural property the so-called invariance property under trivial coordinate changes in the problem description, e.g. dilations and/or contractions. The problems are solved by the Tikhonov regularization method in conjunction with Morozov’s discrepancy principle. Finally, five examples of boundary value problems are investigated to show the efficiency of this method. The numerical convergence, accuracy, and stability of the proposed method with respect to the number of source points, the distance between the sources and the boundary, and the amount of noise added into the input data, respectively, are also analyzed. [ABSTRACT FROM AUTHOR]

The numerical reconstruction of the missing Dirichlet and Neumann data on an inaccessible part of the boundary in the case of two- and three-dimensional linear isotropic elastic materials from the knowledge of over-prescribed noisy measurements taken on the remaining accessible boundary part is investigated. This inverse problem is solved using the method of fundamental solutions (MFS), whilst its stabilization is achieved through several singular value decomposition (SVD)-based regularization methods, such as the Tikhonov regularization method [48], the damped SVD and the truncated SVD [18]. The regularization parameter is selected according to the discrepancy principle [40], generalized cross-validation criterion [14] and Hansen’s L-curve method [20]. [ABSTRACT FROM AUTHOR]

We investigate the numerical reconstruction of the missing thermal and mechanical boundary conditions on an inaccessible part of the boundary in the case of three-dimensional linear isotropic thermoelastic materials from the knowledge of over-prescribed noisy data on the remaining accessible boundary. We employ the method of fundamental solutions (MFS) and several singular value decomposition (SVD)-based regularization methods, e.g. the Tikhonov regularization method (Tikhonov and Arsenin, 1986), the damped SVD and the truncated SVD (Hansen, 1998), whilst the regularization parameter is selected according to the discrepancy principle (Morozov, 1966),generalized cross-validation criterion (Golub et al., 1979) and Hansen’s L-curve method (Hansen and O’Leary, 1993). [ABSTRACT FROM AUTHOR]

We investigate both theoretically and numerically the so-called invariance property, see e.g. Sun and Ma (2015a,b), of the solution of boundary value problems associated with the anisotropic heat conduction equation (or Laplace–Beltrami’s equation) in two dimensions with respect to elementary transformations of the solution domain, e.g. dilations or contractions. We also show that the standard method of fundamental solutions (MFS) does not satisfy the invariance property. Motivated by these reasons, we introduce, in a natural manner, a modified version of the MFS that remains invariant under elementary transformations of the solution domain and is referred to as the invariant MFS (IMFS). Five two-dimensional examples are thoroughly investigated to assess the numerical accuracy, convergence and stability of the proposed IMFS, in conjunction with the Tikhonov regularization method (Tikhonov and Arsenin, 1986) and Morozov’s discrepancy principle (Morozov, 1966), for Laplace–Beltrami’s equation with perturbed boundary conditions. [ABSTRACT FROM AUTHOR]

We investigate the numerical reconstruction of the missing displacements (Dirichlet data) and tractions (Neumann data) on an inaccessible part of the boundary in the case of two-dimensional linear isotropic elastic materials from the knowledge of over-prescribed noisy measurements taken on the remaining accessible boundary part. This inverse problem is solved using the fading regularization method, originally proposed by Cimetière et al. (2000, 2001) for the Laplace equation, in conjunction with a meshless method, namely the method of fundamental solutions (MFS). The stabilisation of the numerical method proposed herein is achieved by stopping the iterative procedure according to Morozov’s discrepancy principle (Morozov, 1966). [ABSTRACT FROM AUTHOR]

We study the reconstruction of the missing thermal and mechanical data on an inaccessible part of the boundary in the case of two-dimensional linear isotropic thermoelastic materials from overprescribed noisy measurements taken on the remaining accessible boundary part. This inverse problem is solved by using the method of fundamental solutions together with the method of particular solutions. The stabilization of this inverse problem is achieved using several singular value decomposition (SVD)-based regularization methods, such as the Tikhonov regularization method (Tikhonov and Arsenin, Methods for solving ill-posed problems, Nauka, Moscow, 1986), the damped SVD and the truncated SVD (Hansen, Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion, SIAM, Philadelphia, 1998), whilst the optimal regularization parameter is selected according to the discrepancy principle (Morozov, Sov Math Doklady 7 (1966), 414-417), generalized cross-validation criterion (Golub et al. Technometrics 22 (1979), 1-35) and Hansen's L-curve method (Hansen and O'Leary, SIAM J Sci Comput 14 (1993), 1487-503). © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 168-201, 2015 [ABSTRACT FROM AUTHOR]

We study the recovery of the missing discontinuous/non-smooth thermal boundary conditions on an inaccessible portion of the boundary of the domain occupied by a solid from Cauchy data prescribed on the remaining boundary assumed to be accessible, in case of stationary anisotropic heat conduction with non-smooth/discontinuous conductivity coefficients. This inverse boundary value problem is ill-posed and, therefore, should be regularized. Consequently, a stabilising method is developed based on a priori knowledge on the solution to this inverse problem and the smoothing feature of the direct problems involved. The original problem is transformed into a control one which reduces to solving an appropriate minimisation problem in a suitable function space. The latter problem is tackled by employing an appropriate variational method which yields a gradient-type iterative algorithm that consists of two direct problems and their corresponding adjoint ones. This approach yields an algorithm designed to approximate specifically merely L2–boundary data in the context of a non-smooth/discontinuous anisotropic conductivity tensor, hence both the notion of solution to the direct problems involved and the convergence analysis of the approximate solutions generated by the algorithm proposed require special attention. The numerical implementation is realised for two-dimensional homogeneous anisotropic solids using the finite element method, whilst regularization is achieved by terminating the iteration according to two stopping criteria. [ABSTRACT FROM AUTHOR]

Abstract: We study the reconstruction of the missing thermal and mechanical fields on an inaccessible part of the boundary for two-dimensional linear isotropic thermoelastic materials from over-prescribed noisy (Cauchy) data on the remaining accessible boundary. This problem is solved with the method of fundamental solutions (MFS) together with the method of particular solutions (MPS) via the MFS-based particular solution for two-dimensional problems in uncoupled thermoelasticity developed in Marin and Karageorghis (2012a, 2013). The stabilisation/regularization of this inverse problem is achieved by using the Tikhonov regularization method (Tikhonov and Arsenin, 1986), whilst the optimal value of the regularization parameter is selected by employing Hansen’s L-curve method (Hansen, 1998). [Copyright &y& Elsevier]

Abstract: We investigate two numerical procedures for the Cauchy problem in linear elasticity, involving the relaxation of either the given boundary displacements (Dirichlet data) or the prescribed boundary tractions (Neumann data) on the over-specified boundary, in the alternating iterative algorithm of . The two mixed direct (well-posed) problems associated with each iteration are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method, while the optimal value of the regularization parameter is chosen via the generalized cross-validation (GCV) criterion. An efficient regularizing stopping criterion which ceases the iterative procedure at the point where the accumulation of noise becomes dominant and the errors in predicting the exact solutions increase, is also presented. The MFS-based iterative algorithms with relaxation are tested for Cauchy problems for isotropic linear elastic materials in various geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: We propose two algorithms involving the relaxation of either the given Dirichlet data (boundary displacements) or the prescribed Neumann data (boundary tractions) on the over-specified boundary in the case of the alternating iterative algorithm of Kozlov et al. applied to Cauchy problems in linear elasticity. A convergence proof of these relaxation methods is given, along with a stopping criterion. The numerical results obtained using these procedures, in conjunction with the boundary element method (BEM), show the numerical stability, convergence, consistency and computational efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the stable numerical reconstruction of an unknown portion of the boundary of a two-dimensional domain occupied by an isotropic linear elastic material from a prescribed boundary condition on this part of the boundary and additional displacement and traction measurements (i.e. Cauchy data) on the remaining known portion of the boundary. This inverse geometric problem is approached by combining the method of fundamental solutions (MFS) and the Tikhonov regularization method, whilst the optimal value of the regularization parameter is chosen according to the discrepancy principle. Various geometries are considered, i.e. convex and non-convex domains with a smooth or piecewise smooth boundary, in order to show the numerical stability, convergence, consistency and computational efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: We study the stable numerical identification of an unknown portion of the boundary on which either a Dirichlet or a Robin boundary condition is provided, while additional Cauchy data are given on the remaining known part of the boundary of a two-dimensional domain, in the case of steady state anisotropic heat conduction problems. This inverse geometric problem is solved using the method of fundamental solutions (MFS) in conjunction with the Tikhonov regularization method . The optimal value for the regularization parameter is chosen according to Hansen’s L-curve criterion . The stability, convergence, accuracy and efficiency of the proposed method are investigated by considering several examples in both smooth and piecewise smooth geometries. [Copyright &y& Elsevier]

Abstract: We investigate a meshless method for the accurate and non-oscillatory solution of problems associated with two-dimensional Helmholtz-type equations in the presence of boundary singularities. The governing equation and boundary conditions are approximated by the method of fundamental solutions (MFS). It is well known that the existence of boundary singularities affects adversely the accuracy and convergence of standard numerical methods. The solutions to such problems and/or their corresponding derivatives may have unbounded values in the vicinity of the singularity. This difficulty is overcome by subtracting from the original MFS solution the corresponding singular functions, without an appreciable increase in the computational effort and at the same time keeping the same MFS approximation. Four examples for both the Helmholtz and the modified Helmholtz equations are carefully investigated and the numerical results presented show an excellent performance of the approach developed. [Copyright &y& Elsevier]

We investigate the numerical implementation of the alternating iterative algorithm originally proposed by Kozlov et al. (Comput Math Math Phys 31:45–52) for the Cauchy problem associated with the two-dimensional modified Helmholtz equation using a meshless method. The two mixed, well-posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is chosen according to the generalized cross-validation criterion. An efficient regularizing stopping criterion which ceases the iterative procedure at the point where the accumulation of noise becomes dominant and the errors in predicting the exact solutions increase, is also presented. The iterative MFS algorithm is tested for Cauchy problems for the two-dimensional modified Helmholtz operator to confirm the numerical convergence, stability and accuracy of the method. [ABSTRACT FROM AUTHOR]

An iterative procedure, namely the minimal error method, for solving stably the Cauchy problem associated with Helmholtz-type equations is introduced and investigated in this paper. This method is compared with another two iterative algorithms previously proposed by Marin et al. (Comput Mech 31:367–377, 2003; Eng Anal Bound Elem 28:1025–1034, 2004), i.e. the conjugate gradient and Landweber–Fridman methods, respectively. The inverse problem analysed in this study is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions. The numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method for both two-dimensional Helmholtz and modified Helmholtz equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, yet another iterative procedure, namely the minimal error method (MEM), for solving stably the Cauchy problem in linear elasticity is introduced and investigated. Furthermore, this method is compared with another two iterative algorithms, i.e. the conjugate gradient (CGM) and Landweber–Fridman methods (LFM), previously proposed by Marin et al. [Marin, L., Háo, D.N., Lesnic, D., 2002b. Conjugate gradient-boundary element method for the Cauchy problem in elasticity. Quarterly Journal of Mechanics and Applied Mathematics 55, 227–247] and Marin and Lesnic [Marin, L., Lesnic, D., 2005. Boundary element-Landweber method for the Cauchy problem in linear elasticity. IMA Journal of Applied Mathematics 18, 817–825], respectively, in the case of two-dimensional homogeneous isotropic linear elasticity. The inverse problem analysed in this paper is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions. The numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method (BEM) for two-dimensional homogeneous isotropic linear elastic materials. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the application of the method of fundamental solutions (MFS) to two-dimensional steady-state heat conduction problems for both isotropic and anisotropic, single and composite (bi-materials), nonlinear functionally graded materials (FGMs). In the composite case, the interface continuity conditions are approximated in the same manner as the boundary conditions. The method is tested on several examples and its relative merits and disadvantages are discussed. [Copyright &y& Elsevier]

Abstract: Helmholtz-type equations arise naturally in many physical applications related to wave propagation, vibration phenomena and heat transfer. These equations are often used to describe the vibration of a structure, the acoustic cavity problem, the radiation wave, the scattering of a wave and heat conduction in fins. In this paper, the numerical recovery of a single and two circular cavities in Helmholtz-type equations from boundary data is investigated. The boundary element method (BEM), in conjunction with a constrained least-squares minimisation, is used to solve this inverse geometric problem. The accuracy and stability of the proposed numerical method with respect to the distance between the cavities and the outer boundary of the solution domain, the location and size of the cavities, and the distance between the cavities are also analysed. Unique and stable numerical solutions are obtained. [Copyright &y& Elsevier]

Abstract: The application of the method of fundamental solutions to the Cauchy problem for steady-state heat conduction in two-dimensional functionally graded materials (FGMs) is investigated. The resulting system of linear algebraic equations is ill-conditioned and, therefore, regularization is required in order to solve this system of equations in a stable manner. This is achieved by employing the zeroth-order Tikhonov functional, while the choice of the regularization parameter is based on the L-curve method. Numerical results are presented for both smooth and piecewise smooth geometries. The convergence and the stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain, and decreasing the amount of noise added into the input data, respectively, are analysed. [Copyright &y& Elsevier]

Abstract: In this paper, the application of the method of fundamental solutions to the Cauchy problem associated with three-dimensional Helmholtz-type equations is investigated. The resulting system of linear algebraic equations is ill-conditioned and therefore its solution is regularized by employing the zeroth-order Tikhonov functional, while the choice of the regularization parameter is based on the L-curve method. Numerical results are presented for under-, equally- and over-determined Cauchy problems in a piecewise smooth geometry. The convergence, accuracy and stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain, and decreasing the amount of noise added into the input data, respectively, are analysed. [Copyright &y& Elsevier]

In this paper, an iterative algorithm based on the Landweber method in combination with the boundary element method is developed for solving the Cauchy problem in isotropic linear elasticity. An efficient regularizing stopping criterion is also employed. The numerical results obtained confirm that the iterative method produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data, respectively. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, the application of the method of fundamental solutions to the Cauchy problem associated with two-dimensional Helmholtz-type equations is investigated. The resulting system of linear algebraic equations is ill-conditioned and therefore its solution is regularized by employing the first-order Tikhonov functional, while the choice of the regularization parameter is based on the L-curve method. Numerical results are presented for both smooth and piecewise smooth geometries. The convergence and the stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain, and decreasing the amount of noise added into the input data, respectively, are analysed. [Copyright &y& Elsevier]

In this paper, the application of the method of fundamental solutions to the Cauchy problem in two-dimensional isotropic linear elasticity is investigated. The resulting system of linear algebraic equations is ill-conditioned and therefore its solution is regularised by employing the first-order Tikhonov functional, while the choice of the regularisation parameter is based on the L-curve method. Numerical results are presented for both smooth and piecewise smooth geometries, as well as for constant and linear stress states. The convergence and the stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain, and decreasing the amount of noise added into the input data, respectively, are analysed. [Copyright &y& Elsevier]

We study the numerical reconstruction of the missing thermal boundary data on a portion of the boundary occupied by an anisotropic solid in the case of the steady-state heat conduction equation from the knowledge of both the temperature and the normal heat flux (i.e. Cauchy data) on the remaining and accessible part of the boundary. This inverse problem is known to be ill-posed and hence a regularization procedure is required. Herein we develop a solver for this problem by exploiting two sources of regularization, namely the smoothing nature of the corresponding direct problems and a priori knowledge on the solution to the inverse problem investigated. Consequently, this inverse problem is reformulated as a control one which reduces to minimising a corresponding functional defined on a fractional Sobolev space on the inaccessible part of the boundary. This approach yields a gradient-based iterative algorithm that consists, at each step, of the resolution of two direct problems and three corresponding adjoint problems in accordance with the function space where the control is sought. The theoretical convergence of the algorithm is studied by deriving an iteration-dependent admissible range for the parameter. Numerical experiments are realized for the two-dimensional case by employing the finite-difference method, whilst the numerical solution is stabilised/regularized by stopping the iterative process based on three criteria. [ABSTRACT FROM AUTHOR]

In this paper, we combine the fading regularization method with the method of fundamental solutions (MFS) and investigate its application to the Cauchy problem for the two-dimensional Helmholtz equation. We present a numerical reconstruction of the missing data on an inaccessible part of the boundary from the knowledge of overprescribed noisy data taken on the remaining accessible boundary part for both smooth and piecewise smooth two-dimensional geometries. The accuracy, convergence, stability and efficiency of the proposed numerical algorithm, as well as its capability to deblur the noisy data, are validated by three numerical examples. [ABSTRACT FROM AUTHOR]

We propose a new moving pseudo-boundary method of fundamental solutions (MFS) for the determination of the boundary of a void. This problem can be modeled as an inverse boundary value problem for harmonic functions. The algorithm for imaging the interior of the medium also makes use of radial polar parametrization of the unknown void shape in two dimensions. The center of this radial polar parametrization is considered to be unknown. We also include the contraction and dilation factors to be part of the unknowns in the resulting nonlinear least-squares problem. This approach addresses the major problem of locating the pseudo-boundary in the MFS in a natural way, because the inverse problem in question is nonlinear anyway. The feasibility of this new method is illustrated by several numerical examples. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 [ABSTRACT FROM AUTHOR]

Abstract: We investigate the numerical reconstruction of smooth star-shaped voids (rigid inclusions and cavities) which are compactly contained in a two-dimensional isotropic linear elastic body from a single non-destructive measurement of both the displacement and traction vectors (Cauchy data) on the external boundary. The displacement vector satisfying the Lamé system in linear elasticity is approximated using the meshless method of fundamental solutions (MFS). The fictitious source points are located both outside the (known) outer boundary of the body and inside the (unknown) void. The inverse geometric problem is then reduced to finding the minimum of a nonlinear least-squares functional that measures the gap between the given and computed data, penalized with respect to both the MFS constants and the derivative of the radial polar coordinates describing the position of the star-shaped void. The interior source points are anchored and move with the void during the iterative reconstruction procedure. The stability of the numerical method is investigated by inverting measurements contaminated with noise. [Copyright &y& Elsevier]