Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques

Abstract

We consider the problem of stable determination of a harmonic function from knowledge of the solution and its normal derivative on a part of the boundary of the (bounded) solution domain. The alternating method is a procedure to generate an approximation to the harmonic function from such Cauchy data and we investigate a numerical implementation of this procedure based on Fredholm integral equations and Nyström discretization schemes, which makes it possible to perform a large number of iterations (millions) with minor computational cost (seconds) and high accuracy. Moreover, the original problem is rewritten as a fixed point equation on the boundary, and various other direct regularization techniques are discussed to solve that equation. We also discuss how knowledge of the smoothness of the data can be used to further improve the accuracy. Numerical examples are presented showing that accurate approximations of both the solution and its normal derivative can be obtained with much less computational time than in previous works.Mathematics Subject Classification (2000): 35R25.

Fast reconstruction of harmonic functions from Cauchy data using the Dirichlet-to-Neumann map and integral equations

“…Inspired and encouraged by those results, we reformulate the Cauchy problem as another operator equation on the boundary. In the literature, the most straightforward reformulation based on the Dirichletto-Neumann map seems to have been overlooked and we therefore present this reformulation in this paper and shall compare the obtained results with those in [17]. Note that, however, a method in this direction was given in [33], where the Cauchy problem (1) was discretized using the boundary element method, and the corresponding linear system of equations was regularized using various techniques including Tikhonov regularization.…”

“…Recently, in [17], the authors of the present paper investigated ways of implementing the alternating method to speed up convergence and minimize the computational cost. The authors took advantage of the reformulation of the alternating method in terms of an operator equation on the boundary, together with a recent integral equation method [15,16].…”

“…We outline how to adjust the method to our case. Then, for stability reasons due to the ill-posedness of the Cauchy problem (1), to solve the discretized integral equation we use the method of smoothing projection introduced in [17,Section 7], which makes it possible to solve the discretized operator equation in a stable way with minor computational cost and high accuracy.…”

A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source

“…Nevertheless, the literature devoted to the Cauchy problem for linear homogeneous elliptic equations is very rich, see e.g. [4,5,7,9,12,13,16,21,23,29,33,35] and the references therein. Recently, a linear inhomogeneous version of Helmholtz equation (i.e.…”

A Robust and Accurate Solver of Laplace's Equation with General Boundary Conditions on General Domains in the Plane

“…One might argue that an accuracy of twelve digits in the solution, an accuracy which is well within the reach of the methods of this paper, is a bit over the top in many cases, but for some problems, due to illposedness, the end result may be meaningless unless the underlying solver is accurate enough. As an example, when reconstructing harmonic functions from Cauchy boundary data [10], one can easily lose ten digits of accuracy even on smooth domains, demanding very accurate underlying solvers. Such problems, but with corners, can potentially be addressed using the methods of this paper.…”