D. Lesnic

A survey of applications of the MFS to inverse problems

The method of fundamental solutions (MFS) is a relatively new method for the numerical solution of boundary value problems and initial/boundary value problems governed by certain partial differential equations. The ease with which it can be implemented and its effectiveness have made it a very popular tool for the solution of a large variety of problems arising in science and engineering. In recent years, it has been used extensively for a particular class of such problems, namely inverse problems. In this study, in view of the growing interest in this area, we review the applications of the MFS to inverse and related problems, over the last decade.

An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation

In this paper, the iterative algorithm proposed by Kozlov et al. [Comput. Maths. Math. Phys. 31 (1991) 45] for obtaining approximate solutions to the ill-posed Cauchy problem for the Helmholtz equation is analysed. The technique is then numerically implemented using the boundary element method (BEM). 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. An efficient stopping regularising criterion is also proposed.

Free convection boundary-layer flow along a vertical surface in a porous medium with Newtonian heating

Abstract In this paper the steady free convection boundary-layer flow along a vertical surface embedded in a porous medium with Newtonian heating is investigated. The mathematical problem reduces to a pair of coupled partial differential equations for the temperature and the streamfunction, and full numerical, asymptotic and matching solutions are obtained for a wide range of values of the coordinate along the plate. The results for the temperature profiles on the plate and in the convective fluid are presented. A comparison between the full finite-difference solution and the small and large series expansion solutions illustrates that the full numerical solution is accurate. Furthermore, a matching closed form of solution for the scaled temperature on the wall is fitted to the theoretical results and this will be useful in numerous engineering practical applications.

Application of the boundary element method to inverse heat conduction problems

Abstract The solution of the one-dimensional, linear, inverse, unsteady heat conduction problem (IHCP) in a slab geometry is analysed. The initial temperature is known, together with a condition on an accessible part of the boundary of the body under investigation. Additional temperature measurements in time are taken with a sensor positioned at an arbitrary location within the solid material, and it is required to determine the temperature and the heat flux on the remaining part of the unspecified boundary. As the problem is improperly posed the direct method of solution cannot be used and hence the least squares, regularization and energy method have been introduced into the boundary element method (BEM) formulation. When noise is present in the measured data some of the numerical results obtained using the least squares method exhibit oscillatory behaviour, but these large oscillations are substantially reduced on the introduction of the minimal energy technique based on minimizing the kinetic energy functional subject to certain constraints. Furthermore, the numerical results obtained using this technique compare well with the results obtained using regularization procedures, showing a good stable estimation of the available test solutions. Further, the constraints, subject to which the minimization is performed, depend on a small parameter of which selection is more natural and easier to implement than the choice of the regularization parameter, which is always a difficult task when using the regularization procedures.

A procedure for determining a spacewise dependent heat source and the initial temperature

The inverse problem of determining a spacewise dependent heat source, together with the initial temperature for the parabolic heat equation, using the usual conditions of the direct problem and information from two supplementary temperature measurements at different instants of time is studied. These spacewise dependent temperature measurements ensure that this inverse problem has a unique solution, despite the solution being unstable, hence the problem is ill-posed. We propose an iterative algorithm for the stable reconstruction of both the initial data and the source based on a sequence of well-posed direct problems for the parabolic heat equation, which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for a typical benchmark test example, which has the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure gives accurate numerical approximations in relatively few iterations.

The method of fundamental solutions for inverse boundary value problems associated with the two-dimensional biharmonic equation

In this paper, the application of the method of fundamental solutions to inverse problems associated with the two-dimensional biharmonic equation is investigated. The resulting system of linear algebraic equations is ill-conditioned and, therefore, its solution is regularized by employing the 0^t^h-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.

Boundary Element Solution For The Cauchy Problem In Linear Elasticity

In this paper we investigate the solution of the Cauchy problem in linear elasticity using the iterative algorithm proposed by Kozlov et al. [1] for obtaining approximate solutions to ill-posed boundary value problems. 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.

Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition

Abstract In this paper the singular value decomposition (SVD), truncated at an optimal number, is analysed for obtaining approximate solutions to ill-conditioned linear algebraic systems of equations which arise from the boundary element method (BEM) discretisation of an ill-posed boundary value problem in linear elasticity. The regularisation parameter, namely the optimal truncation number, is chosen according to the discrepancy principle. The numerical results obtained confirm that the SVD+BEM produces a convergent and stable numerical solution with respect to decreasing the mesh size discretisation and the amount of noise added into the input data.

A non-local boundary value problem method for the Cauchy problem for elliptic equations

Let H be a Hilbert space with norm || ||, A:D(A) ⊂ H → H a positive definite, self-adjoint operator with compact inverse on H, and T and given positive numbers. The ill-posed Cauchy problem for elliptic equations is regularized by the well-posed non-local boundary value problem with a ≥ 1 being given and α > 0 the regularization parameter. A priori and a posteriori parameter choice rules are suggested which yield order-optimal regularization methods. Numerical results based on the boundary element method are presented and discussed.

An iterative boundary element method for solving the backward heat conduction problem using an elliptic approximation

In this paper a new numerical method which does not require a regularization parameter is developed for solving the backward heat conduction problem (BHCP). The inverse and ill-posed BHCP for the heat equation is approximated with a convergent sequence of elliptic Cauchy problems for which the stable algorithm of the Kozlov et al. [1] type is accommodated. Further, a boundary element method (BEM) is developed for obtaining a stable solution of the BHCP.