An iterative method for the Cauchy problem for second-order elliptic equations

Abstract

The problem of reconstructing the solution to a second-order elliptic equation in a doubly-connected domain from knowledge of the solution and its normal derivative on the outer part of the boundary of the solution domain, that is from Cauchy data, is considered. An iterative method is given to generate a stable numerical approximation to this inverse ill-posed problem. The procedure is physically feasible in that boundary data is updated with data of the same type in the iterations, meaning that Dirichlet values is updated with Dirichlet values from the previous step and Neumann values by Neumann data. Proof of convergence and stability are given by showing that the proposed method is an extension of the Landweber method for an operator equation reformulation of the Cauchy problem. Connection with the alternating method is discussed. Numerical examples are included confirming the feasibility of the suggested approach.

An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation

“…The regularizing parameter can be chosen as = 1. This was shown for the similar procedure given for second-order elliptic equations in [9]. Starting from…”

“…The regularizing parameter γ can be chosen as . This was shown for the similar procedure given for second‐order elliptic equations in []. Starting from and then following the steps for the similar result in [], we obtain with the norm induced by the inner product , and u 0 and u 2 as in the previous section.…”

“…Comparing the above scheme with the alternating method, it is similar in the sense that the type of boundary condition alternates on the two boundary parts during the iterations but the alternating method only updates data on the inner boundary Γ 1 , see further [] for explanation of similarities and differences between the methods and in the case of the Cauchy problem for second‐order elliptic equations.…”

“…From a physical point of view, the alternating method is natural in that it updates traces on the boundary with corresponding traces from the previous iteration step. In [], an iterative method was proposed (also mentioned in []) having the similar feature as the alternating method but being of Landweber‐type. The method we propose builds on [].…”

“…The novelty of the present work is then to extend the method, originally given for second‐order Cauchy problems, to an incomplete data problem – for a higher order elliptic equation, and to show convergence of this method.…”