Boundary Element Method for Solving Inverse Heat Source Problems

In this thesis, the boundary element method (BEM) is applied for solving inverse source problems for the heat equation. Through the employment of the Green’s formula and fundamental solution, the BEM naturally reduces the dimensionality of the problem by one although domain integrals are still present due to the initial condition and the heat source. We mainly consider the identification of time-dependent source for heat equation with several types of conditions such as non-local, non-classical, periodic, fixed point, time-average and integral which are considered as boundary or overdetermination conditions. Moreover, the more challenging cases of finding the space- and time-dependent heat source functions for additive and multiplicative cases are also considered. Under the above additional conditions a unique solution is known to exist, however, the inverse problems are still ill-posed since small errors in the input measurements result in large errors in the output heat source solution. Then some type of regularisation method is required to stabilise the solution. We utilise regularisation methods such as the Tikhonov regularisation with order zero, one, two, or the truncated singular value decomposition (TSVD) together with various choices of the regularisation parameter. The numerical results obtained from several benchmark test examples are presented in order to verify the efficiency of adopted computational methodology. The retrieved numerical solutions are compared with their analytical solutions, if available, or with the corresponding direct numerical solution, otherwise. Accurate and stable numerical solutions have been obtained throughout for all the inverse heat source problems considered

Identification of a multi-dimensional space-dependent heat source from boundary data

We investigate the linear but ill-posed inverse problem of determining a multi-dimensional space-dependent heat source in the parabolic heat equation from Cauchy boundary data. This model is important in practical applications where the distribution of internal sources is to be monitored and controlled with care and accuracy from non-invasive and non-intrusive boundary measurements only. The mathematical formulation ensures that a solution of the inverse problem is unique but the existence and stability are still issues to be dealt with. Even if a solution exists it is not stable with respect to small noise in the measured boundary data hence the inverse problem is still ill-posed. The Landweber method is developed in order to restore stability through iterative regularization. Furthermore, the conjugate gradient method is also developed in order to speed up the convergence. An alternating direction explicit finite-difference method is employed for discretising the well-posed problems resulting from these iterative procedures. Numerical results in two-dimensions are illustrated and discussed

A Finite Integration Method for A Time-Dependent Heat Source Identification of Inverse Problem

We investigate an inverse problem of reconstructing a timewise-dependent source for the heat equation. The solution of this problem is uniquely solvable, yet unstable. The inverse source problem two unknowns is reformulated to be a new form of forward problem one unknown. Furthermore, we propose that the finite integration method combined with the backward finite difference method can be used to solve the reformulated heat equation. The Tikhonov regularization method is employed to stabilize the noisy data. The proposed algorithm is not only easy to use but also can give an accurate and stable solution. Numerical result is presented and discussed.

Inverse time-dependent source problem for the heat equation with nonlocal boundary conditions

In this paper, we consider inverse problems of finding the time-dependent source function for the population model with population density nonlocal boundary conditions and an integral over-determination measurement. These problems arise in mathematical biology and have never been investigated in the literature in the forms proposed, although related studies do exist. The unique solvability of the inverse problems are rigorously proved using generalized Fourier series and the theory of Volterra integral equations. Continuous dependence on smooth input data also holds but, as in reality noisy errors are random and non-smooth, the inverse problems are still practically ill-posed. The degree of ill-posedness is characterised by the numerical differentiation of a noisy function. In the numerical process, the boundary element method together with either a smoothing spline regularization or the first-order Tikhonov regularization are employed with various choices of regularization parameter. One is based on the discrepancy principle and another one is the generalized cross-validation criterion. Numerical results for some benchmark test examples are presented and discussed in order to illustrate the accuracy and stability of the numerical inversion

Simultaneous determination of time-dependent coefficients and heat source

This article presents a numerical solution to the inverse problems of simultaneous determination of the time-dependent coefficients and the source term in the parabolic heat equation subject to overspecified conditions of integral type. The ill-posed problems are numerically discretized using the finite-difference method. The resulting system of nonlinear equations is solved numerically using the MATLAB toolbox routine lsqnonlin applied to minimizing the nonlinear Tikhonov regularization functional subject to simple physical bounds on the variables. Numerical examples are presented to illustrate the accuracy and stability of the solution

Simultaneous determination of time and space-dependent coefficients in a parabolic equation

This paper investigates a couple of inverse problems of simultaneously determining time and space dependent coefficients in the parabolic heat equation using initial and boundary conditions of the direct problem and overdetermination conditions. The measurement data represented by these overdetermination conditions ensure that these inverse problems have unique solutions. However, the problems are still ill-posed since small errors in the input data cause large errors in the output solution. To overcome this instability we employ the Tikhonov regularization method. The finite-difference method (FDM) is employed as a direct solver which is fed iteratively in a nonlinear minimization routine. Both exact and noisy data are inverted. Numerical results for a few benchmark test examples are presented, discussed and assessed with respect to the FDM mesh size discretisation, the level of noise with which the input data is contaminated, and the chosen regularization parameters

Reconstruction of an additive space- and time-dependent heat source

In this paper, we consider the inverse problem of simultaneous determination of an additive space- and time-dependent heat source together with the temperature in the heat equation, with Dirichlet boundary conditions and two over-determination conditions. These latter ones consist of a specified temperature measurement at an internal point and a time-average temperature condition. The mathematical problem is linear but ill-posed since the continuous dependence on the input data is violated. In discretised form, the problem reduces to solving an ill-conditioned system of linear equations. We investigate the performances of several regularisation methods and examine their stability with respect to noise in the input data. The boundary element method combined with either the truncated singular value decomposition, or the Tikhonov regularisation, using various methods for choosing regularisation parameters, e.g. The L-curve method, the generalised cross-validation criterion, the discrepancy principle and the L-surface method, are utilised in order to obtain accurate and stable numerical solutions

An inverse time-dependent source problem for the heat equation with a non-classical boundary condition

This paper investigates the inverse problem of determining the time-dependent heat source and the temperature for the heat equation with a non-classical boundary and an integral over-determination conditions. The existence, uniqueness and continuous dependence upon the data of the classical solution of the inverse problem is shown by using the generalised Fourier method. Furthermore in the numerical process, the boundary element method (BEM) together with the second-order Tikhonov regularization is employed with the choice of regularization parameter based on the generalised cross-validation (GCV) criterion. Numerical results are presented and discussed