Optimal Stable Approximation for the Cauchy Problem for Laplace Equation

Hongfang Li,Feng Zhou

doi:10.1155/2016/1468634

Abstract

Cauchy problem for Laplace equation in a strip is considered. The optimal error bounds between the exact solution and its regularized approximation are given, which depend on the noise level either in a Hölder continuous way or in a logarithmic continuous way. We also provide two special regularization methods, that is, the generalized Tikhonov regularization and the generalized singular value decomposition, which realize the optimal error bounds.

Highlights

The Cauchy problem for the Laplace equation in particular, and for other elliptic equations in general, occurs in the study of many practical problems in areas such as plasma physics [1], electrocardiology [2, 3], bioelectric field problems [4], nondestructive testing [5], magnetic recording [6], and the Cauchy problem for elliptic equations [7, 8]
We provide two special regularization methods, that is, the generalized Tikhonov regularization and the generalized singular value decomposition, which realize the optimal error bounds
These problems are known to be severely ill-posed [9], in the sense that the solution, if it exists, does not depend continuously on the Cauchy data in some natural norm. This is because the Cauchy problem is an initial value problem which represents a transient phenomenon in a time-like variable while elliptic equation describes steady-state processes in physical fields

Summary

Introduction

The Cauchy problem for the Laplace equation in particular, and for other elliptic equations in general, occurs in the study of many practical problems in areas such as plasma physics [1], electrocardiology [2, 3], bioelectric field problems [4], nondestructive testing [5], magnetic recording [6], and the Cauchy problem for elliptic equations [7, 8] These problems are known to be severely ill-posed [9], in the sense that the solution, if it exists, does not depend continuously on the Cauchy data in some natural norm (see, e.g., [5] and references therein).

Preliminary Result

Optimal Regularization Methods