Regularization methods for solving Cauchy problems for elliptic and degenerate elliptic equations

Sammanfattning: In this thesis, we study Cauchy problems for the elliptic and degenerate elliptic equations. These problems are ill-posed. We split the boundary of the domain into two parts. On one of them, say Γ0, we have available Cauchy data and on remaining part Γ1 we introduce unknown Robin data. To construct the operator equation which replaces our Cauchy problem we use two boundary value problems (BVP). The first one is the mixed BVP with Robin condition on Γ1 and with Dirichlet condition on Γ0 and the second BVP with Dirichlet Data on Γ1 and with Robin data on Γ0. The well–posedness of these problems is achieved by an appropriate choice of parameters in Robin boundary conditions. The first Dirichlet–Robin BVP is used to construct the operator equation replacing the Cauchy problem and the second Robin–Dirichlet problem for adjoint operator. Using these problems we can apply various regularization methods for stable reconstruction of the solution. In Paper I, the Cauchy problem for the elliptic equation with variable coefficients, which includes Helmholtz type equations, is analyzed. A proof showing that the Dirichlet–Robin alternating algorithm is convergent is given, provided that the parameters in the Robin conditions are chosen appropriately. Numerical experiments that shows the behaviour of the algorithm are given. In particular, we show how the speed of convergence depends on the choice of Robin parameters. In Paper II, the Cauchy problem for the Helmholtz equation, for moderate wave numbers k2, is considered. The Cauchy problem is reformulated as an operator equation and iterative method based on Krylov subspaces are implemented. The aim is to achieve faster convergence in comparison to the Alternating algorithm from the previous paper. Methods such as the Landweber iteration, the Conjugate gradient method and the generalized minimal residual method are considered. We also discuss how the algorithms can be adapted to also cover the case of non–symmetric differential operators. In Paper III, we look at a steady state heat conduction problem in a thin plate. The plate connects two cylindrical containers and fix their relative positions. A two dimensional mathematical model of heat conduction in the plate is derived. Since the plate has sharp edges on the sides we obtained a degenerate elliptic equation. We seek to find the temperature on the interior cylinder by using data on the exterior cylinder. We reformulate the Cauchy problem as an operator equation, with a compact operator. The operator equation is solved using the Landweber method and the convergence is investigated. In Paper IV, the Cauchy problem for a more general degenerate elliptic equation is considered. We stabilize the computations using Tikhonov regularization. The normal equation, in the Tikhonov algorithm, is solved using the Conjugate gradient method. The regularization parameter is picked using either the L–curve or the Discrepancy principle. In all papers, numerical examples are given where we solve the various boundary value problems using a finite difference scheme. The results show that the suggested methods work quite well. 

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