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.     

Note on using radial basis functions and Tikhonov regularization method to solve an inverse heat conduction problem

In this note, a meshless numerical scheme for solving an inverse heat conduction problem (IHCP) is considered. The fundamental solution of heat equation is used as a radial basis function. Applying this radial basis function results in a badly ill-condition system of equations. The Tikhonov regularization method is employed for solving this system of equations. Determination of regularization parameter is based on L-curve technique. The numerical results demonstrate the accuracy and ability of this method.     

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.     

Multivariate numerical derivative by solving an inverse heat source problem

A method for approximating multivariate numerical derivatives is presented from multidimensional noise data in this paper. Starting from solving a direct heat conduction problem using the multidimensional noise data as an initial condition, we conclude estimations of the partial derivatives by solving an inverse heat source problem with an over-specified condition, which is the difference of the solution to the direct problem and the given noise data. Then, solvability and conditional stability of the proposed method are discussed for multivariate numerical derivatives, and a regularized optimization is adopted for overcoming instability of the inverse heat source problem. For achieving partial derivatives successfully and saving amount of computation, we reduce the multidimensional problem to a one-dimensional case, and give a corresponding algorithm with a posterior strategy for choosing regularization parameters. Finally, numerical examples show that the proposed method is feasible and stable to noise data.     

A meshless method for solving 1D time-dependent heat source problem

A novel meshless numerical procedure based on the method of fundamental solutions (MFS) and the heat polynomials is proposed for recovering a time-dependent heat source and the boundary data simultaneously in an inverse heat conduction problem (IHCP). We will transform the problem into a homogeneous IHCP and initial value problems for the first-order ordinary differential equation. An improved method of MFS is used to solve the IHCP and a finite difference method is applied for solving the initial value problems. The advantage of applying the proposed meshless numerical scheme is producing the shape functions which provide the important delta function property to ensure that the essential conditions are fulfilled. Numerical experiments for some examples are provided to show the effectiveness of the proposed algorithm.     

The method of fundamental solutions for inverse source problems associated with the steady‐state heat conduction

This paper presents the use of the method of fundamental solutions (MFS) for recovering the heat source in steady‐state heat conduction problems from boundary temperature and heat flux measurements. It is well known that boundary data alone do not determine uniquely a general heat source and hence some a priori knowledge is assumed in order to guarantee the uniqueness of the solution. In the present study, the heat source is assumed to satisfy a second‐order partial differential equation on a physical basis, thereby transforming the problem into a fourth‐order partial differential equation, which can be conveniently solved using the MFS. Since the matrix arising from the MFS discretization is severely ill‐conditioned, a regularized solution is obtained by employing the truncated singular value decomposition, whilst the optimal regularization parameter is determined by the L‐curve criterion. Numerical results are presented for several two‐dimensional problems with both exact and noisy data. The sensitivity analysis with respect to two solution parameters, i.e. the number of source points and the distance between the fictitious and physical boundaries, and one problem parameter, i.e. the measure of the accessible part of the boundary, is also performed. The stability of the scheme with respect to the amount of noise added into the data is analysed. The numerical results obtained show that the proposed numerical algorithm is accurate, convergent, stable and computationally efficient for solving inverse source problems in steady‐state heat conduction. Copyright © 2006 John Wiley & Sons, Ltd.     

Simultaneous reconstruction of the time-dependent Robin coefficient and heat flux in heat conduction problems

This paper aims to solve an inverse heat conduction problem in two-dimensional space under transient regime, which consists of the estimation of multiple time-dependent heat sources placed at the boundaries. Robin boundary condition (third type boundary condition) is considered at the working domain boundary. The simultaneous identification problem is formulated as a constrained minimization problem using the output least squares method with Tikhonov regularization. The properties of the continuous and discrete optimization problem are studied. Differentiability results and the adjoint problems are established. The numerical estimation is investigated using a modified conjugate gradient method. Furthermore, to verify the performance of the proposed algorithm, obtained results are compared with results obtained from the well-known finite-element software COMSOL Multiphysics under the same conditions. The numerical results show that the proposed algorithm is accurate, robust and capable of simultaneously representing the time effects on reconstructing the time-dependent Robin coefficient and heat flux.     

不适定热传导反问题的两种求解方法及其比较

本文以不适定热传导反问题为对象，采用两种方法进行了求解。一种方法基于对具有测量误差的边界条件进行适当的微扰，使之化为适定问题；另一种方法基于Ｔｉｋｈｏｎｏｖ的正则平滑思想，对反问题中的输入数据进行平滑处理，以便使函数及其一阶导数均实现一致逼近。通过计算与求解表明，两种方法均能得到具有一定精度与稳定性的结果，其中以正则化法更为理想     

Regularization of the inverse heat conduction problem by the discrete Fourier transform

Two methods of solving the inverse heat conduction problem with employment of the discrete Fournier transform are presented in this article. The first one operates similarly to the SVD algorithm and consists in reducing the number of components of the discrete Fournier transform which are taken into account to determine the solution to the inverse problem. The second method is related to the regularization of the solution to the inverse problem in the discrete Fournier transform domain. Those methods were illustrated by numerical examples. In the first example, an influence of the boundary conditions disturbance by a random error on the solution to the inverse problem (its stability) was examined. In the second example, the temperature distribution on the inner boundary of the multiply connected domain was determined. Results of calculations made in both ways brought very good outcomes and confirm the usefulness of applying the discrete Fournier transform to solving inverse problems.     

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.     

A hybrid regularization method for inverse heat conduction problems

This paper presents a hybrid regularization method for solving inverse heat conduction problems. The method uses future temperatures and past fluxes to reduce the sensitivity to temperature noise. A straightforward comparison technique is suggested to find the optimal number of the future temperatures. Also, an eigenvalue reduction technique is used to further improve the accuracy of the inverse solution. The method provides a physical insight into the inverse problems under study. The insight indicates that the inverse algorithm is a general purpose algorithm and applicable to various numerical methods (although our development was based on FEM), and that the inverse solutions can be obtained by directly extending Stolz's equation in the least‐squares error (LSE) sense. Direct extension of the present method to the inverse internal heat generation problems is made. Four numerical examples are given to validate the method. The effects of the future temperatures, the past fluxes, the eigenvalue reduction, the varying number of future temperatures and local iterations for non‐linear problems are studied. Copyright © 2005 John Wiley & Sons, Ltd.     

多宗量一维瞬态非线性热传导反问题的正则解

基于一种时域正演精细算法,引入Bregman距离函数作为Tikhonov函数的正则项,建立了求解多宗量一维瞬态非线性热传导反问题的数学模型,可对非线性内热源强度、导温系数和边界条件等多个热学参数进行组合识别。对信息测量误差作了初步探讨,数值验证给出令人满意的结果。     

The method of fundamental solutions for inverse boundary value problems associated with the steady‐state heat conduction in anisotropic media

In this paper, the method of fundamental solutions is applied to solve some inverse boundary value problems associated with the steady‐state heat conduction in an anisotropic medium. Since the resulting matrix equation is severely ill‐conditioned, a regularized solution is obtained by employing the truncated singular value decomposition, while the optimal regularization parameter is chosen according to the L‐curve criterion. Numerical results are presented for both two‐ and three‐dimensional problems, as well as exact and noisy data. The convergence and stability of the proposed numerical scheme with respect to increasing the number of source points and the distance between the fictitious and physical boundaries, and decreasing the amount of noise added into the input data, respectively, are analysed. A sensitivity analysis with respect to the measure of the accessible part of the boundary and the distance between the internal measurement points and the boundary is also performed. The numerical results obtained show that the proposed numerical method is accurate, convergent, stable and computationally efficient, and hence it could be considered as a competitive alternative to existing methods for solving inverse problems in anisotropic steady‐state heat conduction. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

The MFS for numerical boundary identification in two-dimensional harmonic problems

In this study, we briefly review the applications of the method of fundamental solutions to inverse problems over the last decade. Subsequently, we consider the inverse geometric problem of identifying an unknown part of the boundary of a domain in which the Laplace equation is satisfied. Additional Cauchy data are provided on the known part of the boundary. The method of fundamental solutions is employed in conjunction with regularization in order to obtain a stable solution. Numerical results are presented and discussed.     

The method of fundamental solutions for a biharmonic inverse boundary determination problem

In this paper, a nonlinear inverse boundary value problem associated to the biharmonic equation is investigated. This problem consists of determining an unknown boundary portion of a solution domain by using additional data on the remaining known part of the boundary. The method of fundamental solutions (MFS), in combination with the Tikhonov zeroth order regularization technique, are employed. It is shown that the MFS regularization numerical technique produces a stable and accurate numerical solution for an optimal choice of the regularization parameter. A. Zeb on study leave visiting the University of Leeds.     

Simultaneous inversion for the diffusion and source coefficients in the multi-term TFDE

This article deals with an inverse problem of determining the space-dependent diffusion coefficient and the source coefficient simultaneously in the multi-term time fractional diffusion equation (TFDE in short) using measurements at one inner point. From a view point of optimality, solving the inverse problem is transformed to minimize an error functional with the help of the solution operator from the unknown to the additional observation. The solution operator is nonlinear but it is of Lipschitz continuity by which existence of a minimum to the error functional is obtained using Sobolev embedding theorems. The homotopy regularization algorithm is introduced to solve the simultaneous inversion problem based on the minimization problem, and numerical examples are presented. The inversion solutions give good approximations to the exact solutions demonstrating that the homotopy regularization algorithm is efficient for the simultaneous inversion problem arising in the multi-term TFDE.     

Automated algorithm for the nondestructive detection of subsurface cavities by the IR-CAT method

An algorithm is presented to automate the detection of irregular-shaped subsurface cavities within irregular shaped bodies by the IR-CAT method. The algorithm is based on the solution of an inverse geometric steady state heat conduction problem. Cauchy boundary conditions are prescribed at the exposed surface. An inverse heat conduction problem is formulated by specifying the thermal boundary condition along the inner cavities whose unknown geometries are to be determined. An initial guess is made for the location of the inner cavities. The domain boundaries are discretized, and an Anchored Grid Pattern (AGP) is established. The nodes of the inner cavities are constrained to move along the AGP at each iterative step. The location of inner cavities is determined by using the Newton Raphson method with a Broyden update to drive the error between the imposed boundary conditions and computed boundary conditions to zero. During the iterative procedure, the movement of the inner cavity walls is restricted to physically realistic intermediate solutions. A dynamic relocation of the AGP is introduced in the Traveling Hole Method to adaptively refine the detection of inner cavities. The proposed algorithm is general and can be used to detect multiple cavities. Results are presented for the detection of single and multiple irregular shaped cavities. Convergence under grid refinement is demonstrated.     

The boundary-element method for the determination of a heat source dependent on one variable

This paper investigates the inverse problem of determining a heat source in the parabolic heat equation using the usual conditions of the direct problem and a supplementary condition, called an overdetermination. In this problem, if the heat source is taken to be space-dependent only, then the overdetermination is the temperature measurement at a given single instant, whilst if the heat source is time-dependent only, then the overdetermination is the transient temperature measurement recorded by a single thermocouple installed in the interior of the heat conductor. These measurements ensure that the inverse problem has a unique solution, but this solution is unstable, hence the problem is ill-posed. This instability is overcome using the Tikhonov regularization method with the discrepancy principle or the L-curve criterion for the choice of the regularization parameter. The boundary-element method (BEM) is developed for solving numerically the inverse problem and numerical results for some benchmark test examples are obtained and discussed     

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.