An Alternating Iterative MFS Algorithm for the Cauchy Problem in Two-Dimensional Anisotropic Heat Conduction

In this paper, the alternating iterative algorithm originally proposed by Kozlov, Maz'ya and Fomin (1991) is numerically implemented for the Cauchy problem in anisotropic heat conduction using a meshless method. Every iteration of the numerical procedure consists of two mixed, well-posed and direct problems which 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 (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 iterative MFS algorithm is tested for Cauchy problems related to heat conduction in two-dimensional anisotropic solids to confirm the numerical convergence, stability and accuracy of the method.     

Relaxation procedures for an iterative MFS algorithm for two-dimensional steady-state isotropic heat conduction Cauchy problems

We investigate two algorithms involving the relaxation of either the given Dirichlet data (boundary temperatures) or the prescribed Neumann data (normal heat fluxes) on the over-specified boundary in the case of the alternating iterative algorithm of Kozlov et al. [26] applied to two-dimensional steady-state heat conduction Cauchy problems, i.e. Cauchy problems for the Laplace equation. The two mixed, well-posed and direct problems corresponding to each iteration of the numerical procedure are solved using a meshless method, namely 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 (GCV) criterion. The iterative MFS algorithms with relaxation are tested for Cauchy problems associated with the Laplace operator in various two-dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method.     

Stable Boundary and Internal Data Reconstruction in Two-Dimensional Anisotropic Heat Conduction Cauchy Problems Using Relaxation Procedures for an Iterative MFS Algorithm

We investigate two algorithms involving the relaxation of either the given boundary temperatures (Dirichlet data) or the prescribed normal heat fluxes (Neumann data) on the over-specified boundary in the case of the iterative algorithm of Kozlov91 applied to Cauchy problems for two-dimensional steady-state anisotropic heat conduction (the Laplace-Beltrami equation). 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 (GCV) criterion. The iterative MFS algorithms with relaxation are tested for over-, equally and under-determined Cauchy problems associated with the steady-state anisotropic heat conduction in various two-dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method.     

Regularized MFS-Based Boundary Identification in Two-Dimensional Helmholtz-Type Equations

We study the stable numerical identification of an unknown portion of the boundary on which a given boundary condition is provided and additional Cauchy data are given on the remaining known portion of the boundary of a two-dimensional domain for problems governed by either the Helmholtz or the modified Helmholtz equation. 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.     

Boundary element–minimal error method for the Cauchy problem associated with Helmholtz-type equations

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.     

A meshless method for the stable solution of singular inverse problems for two-dimensional Helmholtz-type equations

We investigate a meshless method for the stable and accurate solution of inverse problems associated with two-dimensional Helmholtz-type equations in the presence of boundary singularities. The governing equation and boundary conditions are discretized by the method of fundamental solutions (MFS). The existence of boundary singularities affects adversely the accuracy and convergence of standard numerical methods. Solutions to such problems and/or their corresponding derivatives may have unbounded values in the vicinity of the singularity. Moreover, when dealing with inverse problems, the stability of solutions is a key issue and this is usually taken into account by employing a regularization method. These difficulties are overcome by combining the Tikhonov regularization method (TRM) with the subtraction from the original MFS solution of the corresponding singular solutions, without an appreciable increase in the computational effort and at the same time keeping the same MFS discretization. Three examples for both the Helmholtz and the modified Helmholtz equations are carefully investigated.     

An alternating iterative algorithm for the Cauchy problem in anisotropic elasticity

The alternating iterative algorithm proposed by Kozlov et al. [An iterative method for solving the Cauchy problem for elliptic equations. USSR Comput Math Math Phys 1991;31:45–52] for obtaining approximate solutions to the Cauchy problem in two-dimensional anisotropic elasticity is analysed and numerically implemented using the boundary element method (BEM). The ill-posedness of this inverse boundary value problem is overcome by employing an efficient regularising stopping. The numerical results confirm that the iterative BEM 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.     

Regularized solutions with a singular point for the inverse biharmonic boundary value problem by the method of fundamental solutions

Two numerical methods for the Cauchy problem of the biharmonic equation are proposed. The solution of the problem does not continuously depend on given Cauchy data since the problem is ill-posed. A small noise contained in the Cauchy data sensitively affects on the accuracy of the solution. Our problem is directly discretized by the method of fundamental solutions (MFS) to derive an ill-conditioned matrix equation. As another method, our problem is decomposed into two Cauchy problems of the Laplace and the Poisson equations, which are discretized by the MFS and the method of particular solutions (MPS), respectively. The Tikhonov regularization and the truncated singular value decomposition are applied to the matrix equation to stabilize a numerical solution of the problem for the given Cauchy data with high noises. The L-curve and the generalized cross-validation determine a suitable regularization parameter for obtaining an accurate solution. Based on numerical experiments, it is concluded that the numerical method proposed in this paper is effective for the problem that has an irregular domain and the Cauchy data with high noises. Furthermore, our latter method can successfully solve the problem whose solution has a singular point outside the computational domain.     

Relaxation of Alternating Iterative Algorithms for the Cauchy Problem Associated with the Modified Helmholtz Equation

We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over-specified boundary, in the case of the alternating iterative algorithm of Kozlov, Maz'ya and Fomin(1991) applied to Cauchy problems for the modified Helmholtz equation. 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 methods.     

The Spring-Damping Regularization Method and the Lie-Group Shooting Method for Inverse Cauchy Problems

The inverse Cauchy problems for elliptic equations, such as the Laplace equation, the Poisson equation, the Helmholtz equation and the modified Helmholtz equation, defined in annular domains are investigated. The outer boundary of the annulus is imposed by overspecified boundary data, and we seek unknown data on the inner boundary through the numerical solution by a spring-damping regularization method and its Lie-group shooting method (LGSM). Several numerical examples are examined to show that the LGSM can overcome the ill-posed behavior of inverse Cauchy problem against the disturbance from random noise, and the computational cost is very cheap.     

Cauchy problems of Laplace's equation by the methods of fundamental solutions and particular solutions

The Cauchy problems of Laplace's equation are ill-posed with severe instability. In this paper, numerical solutions are solicited by the method of fundamental solutions (MFS) and the method of particular solutions (MPS). We focus on the analysis of the MFS, and derive the bounds of errors and condition numbers. The analysis for the MPS can also be obtained similarly. Numerical experiments and comparisons are reported for the Cauchy and Dirichlet problems by the MPS and the MFS. The Cauchy noise data and the regularization are also adopted in numerical experiments. Both the MFS and the MPS are effective to Cauchy problems. The MPS is superior in accuracy and stability; but the MFS owns simplicity of algorithms, and earns flexibility for a wide range of applications, such as Cauchy problems. These conclusions also coincide with [37]. The basic analysis of error and stability is explored in this paper, and applied to the Cauchy data. There are many reports on numerical Cauchy problems, see the survey paper in [12]; most of them are of computational aspects. The strict analysis of this paper may, to a certain degree, fill up the existing gap between theory and computation of Cauchy problems by the MFS and the MPS. Moreover, comprehensive analysis and compatible computation are two major characteristics of this paper, which may enhance the study of numerical Cauchy problems forward to a higher and advanced level.     

Boundary element method for the Cauchy problem in linear elasticity

In this paper, the iterative algorithm proposed by Kozlov et al. [Comput Maths Math Phys 32 (1991) 45] for obtaining approximate solutions to ill-posed boundary value problems in linear elasticity is analysed. The technique is then numerically implemented using the boundary element method (BEM). The numerical results obtained confirm that the iterative BEM 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. An efficient stopping regularizing criterion is given and in addition, the accuracy of the iterative algorithm is improved by using a variable relaxation procedure. Analytical formulae for the integration constants resulting from the direct application of the BEM for an isotropic linear elastic medium are also presented.     

Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation

In this paper, several boundary element regularization methods, such as iterative, conjugate gradient, Tikhonov regularization and singular value decomposition methods, for solving the Cauchy problem associated to the Helmholtz equation are developed and compared. Regularizing stopping criteria are developed and the convergence, as well as the stability, of the numerical methods proposed are analysed. The Cauchy problem for the Helmholtz equation can be regularized by various methods, such as the general regularization methods presented in this paper, but more accurate results are obtained by classical methods, such as the singular value decomposition and the Tikhonov regularization methods. Copyright © 2004 John Wiley & Sons, Ltd.     

An iterative algorithm for singular Cauchy problems for the steady state anisotropic heat conduction equation

In this paper, we propose an alternating iterative algorithm to solve a singular Cauchy problem for the anisotropic heat conduction equation. The numerical algorithm is based on the boundary element method (BEM), modified to take into account the form of the singularity, without substantially increasing the amount of computation involved. Two test examples, the first with a singularity caused by an abrupt change in the boundary conditions and the second with a singularity caused by a sharp re-entrant corner, are investigated. The numerical results obtained confirm that provided an appropriate stopping regularization criterion is imposed, the iterative BEM is efficient in dealing with the difficulties arising from both the instabilities produced by the boundary condition formulation and the slow rate of convergence of standard numerical methods around the singular point.     

Numerical boundary identification for Helmholtz-type equations

We study the identification of an unknown portion of the boundary of a two-dimensional domain occupied by a material satisfying Helmholtz-type equations from additional Cauchy data on the remaining known portion of the boundary. This inverse geometric problem is approached using the boundary element method (BEM) in conjunction with the Tikhonov first-order regularization procedure, whilst the choice of the regularization parameter is based on the L-curve criterion. The numerical results obtained show that the proposed method produces a convergent and stable solution     

The method of fundamental solutions for a time-dependent two-dimensional Cauchy heat conduction problem

We investigate an application of the method of fundamental solutions (MFS) to the time-dependent two-dimensional Cauchy heat conduction problem, which is an inverse ill-posed problem. Data in the form of the solution and its normal derivative is given on a part of the boundary and no data is prescribed on the remaining part of the boundary of the solution domain. To generate a numerical approximation we generalize the work for the stationary case in Marin (2011) [23] to the time-dependent setting building on the MFS proposed in Johansson and Lesnic (2008) [15], for the one-dimensional heat conduction problem. We incorporate Tikhonov regularization to obtain stable results. The proposed approach is flexible and can be adjusted rather easily to various solution domains and data. An additional advantage is that the initial data does not need to be known a priori, but can be reconstructed as well.     

A meshless method for solving nonhomogeneous Cauchy problems

In this paper the method of fundamental solutions (MFS) and the method of particular solution (MPS) are combined as a one-stage approach to solve the Cauchy problem for Poisson's equation. The main idea is to approximate the solution of Poisson's equation using a linear combination of fundamental solutions and radial basis functions. As a result, we provide a direct and effective meshless method for solving inverse problems with inhomogeneous terms. Numerical results in 2D and 3D show that our proposed method is effective for Cauchy problems.     

Modified boundary Tikhonov-type regularization method for the Cauchy problem of a semi-linear elliptic equation

In this paper, we consider a Cauchy problem for the semi-linear elliptic equation and use a modified boundary Tikhonov-type regularization method to overcome its ill-posedness. The existence, uniqueness and stability of the regularization solution are proven. Under an a-priori bound assumption for the exact solution, a convergence estimate of Hölder type for this method is obtained. Finally, an iterative scheme is proposed to calculate the regularization solution, numerical results show that this method works well.     

Time-dependent fundamental solutions for homogeneous diffusion problems

This paper describes the applications of the method of fundamental solutions (MFS) for 1-, 2- and 3-D diffusion equations. The time-dependent fundamental solutions for diffusion equations are used directly to obtain the solution as a linear combination of the fundamental solution of the diffusion operator. The proposed scheme is free from the conventionally used Laplace transform or the finite difference scheme to deal with the time derivative of the governing equation. By properly placing the field points and the source points at a given time level, the solution is advanced in time until steady state solutions are reached. Test results obtained for 1-, 2- and 3-D diffusion problems show good comparisons with the analytical solutions and some with the MFS based on the modified Helmholtz fundamental solutions, thus the demonstration present numerical scheme of MFS with the space–time unification has been demonstrated as a promising mesh-free numerical tool to solve homogeneous diffusion problem.     

Numerical solution of two backward parabolic problems using method of fundamental solutions

This work is devoted to a numerical algorithm based on the method of fundamental solutions (MFS) for solving two backward parabolic problems with different boundary conditions, one with nonlocal Dirichlet boundary conditions, and second one with Robin type boundary conditions. The initial temperature distribution will be identified from the final temperature distribution, which appear in some applied subjects. The Tikhonov regularization method with the L-curve criterion for choosing the regularization parameter is adopted for solving the resulting matrix equation which is highly ill-conditioned. Two numerical examples are provided to show the high efficiency of the suggested method.