We present a modification of the alternating iterative method, which was introduced by V.A. Kozlov and V. Maz’ya in for solving the Cauchy problem for the Helmholtz equation in a Lipschitz domain. The method is implemented numerically using the finite difference method.

The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Mazya does not converge for large wavenumbers k in the Helmholtz equation. Here, we present some simple modifications of the algorithm which may restore the convergence. They consist of the replacement of the Neumann-Dirichlet iterations by the Robin-Dirichlet ones which repairs the convergence for less than the first Dirichlet-Laplacian eigenvalue. In order to treat large wavenumbers, we present an algorithm based on iterative solution of Robin-Dirichlet boundary value problems in a sufficiently narrow border strip. Numerical implementations obtained using the finite difference method are presented. The numerical results illustrate that the algorithms suggested in this paper, produce convergent iterative sequences.

We consider the Cauchy problem for the Laplace equation, i.e. the reconstruction of a harmonic function from knowledge of the value of the function and its normal derivative given on a part of the boundary of the solution domain. The solution domain considered is a bounded smooth doubly connected planar domain bounded by two simple disjoint closed curves. Since the Cauchy problem is ill-posed, as a regularizing method we generalize the novel direct integral equation approach in [1], originally proposed for a circular outer boundary curve, to a more general simply connected domain. The solution is represented as a sum of two single-layer potentials defined on each of the two boundary curves and in which both densities are unknown. To identify these densities, the representation is matched with the given Cauchy data to generate a system to solve for the densities. It is shown that the operator corresponding to this system is injective and has dense range, thus Tikhonov regularization is applied to solve it in a stable way. For the discretisation, the Nystrom method is employed generating a linear system to solve, and via Tikhonov regularization a stable discrete approximation to these densities are obtained. Using these one can then find an approximation to the solution of the Cauchy problem. A numerical example is included and we compare with other regularizing methods as well (implemented via integral methods). These results show that the proposed direct method gives accurate reconstructions with little computational effort (the computational time is of order of seconds). Moreover, the obtained approximation can be used as an initial guess in more involved regularizing methods to further improve the accuracy.

We propose a nonlinear Landweber method for the inverse problem of locating the brain tumour source (origin where the tumour formed) based on well-established models of reaction-diffusion type for brain tumour growth. The approach consists of recovering the initial density of the tumour cells starting from a later state, which can be given by a medical image, by running the model backwards. Moreover, full three-dimensional simulations are given of the tumour source localization on two types of data, the three-dimensional Shepp-Logan phantom and an MRI T1-weighted brain scan. These simulations are obtained using standard finite difference discretizations of the space and time derivatives, generating a simple approach that performs well.

We propose an application of the method of fundamental solutions (MFS) for the two-dimensional inverse Stefan problem, where data are to be reconstructed from knowledge of the moving surface and the given Stefan conditions on this surface. We present numerical results for several examples both when the initial data are given but also when it is not specified. These results show good accuracy with low computational cost and are compared with results obtained by other methods.

The mathematical and numerical properties of an ill-posed Cauchy problem for a convection - diffusion equation are investigated in this study. The problem is reformulated as a Volterra integral equation of the first kind with a smooth kernel. The rate of decay of the singular values of the integral operator determines the degree of ill-posedness. The purpose of this article is to study how the convection term influences the degree of ill-posedness by computing numerically the singular values. It is also shown that the sign of the coefficient in the convection term determines the rate of decay of the singular values. Some numerical examples are also given to illustrate the theory.