%0 Journal Article
%T A Posteriori Regularization Parameter Choice Rule for Truncation Method for Identifying the Unknown Source of the Poisson Equation
%A Xiao-Xiao Li
%A Dun-Gang Li
%J International Journal of Partial Differential Equations
%D 2013
%R 10.1155/2013/590737
%X We consider the problem of determining an unknown source which depends only on one variable in two-dimensional Poisson equation. We prove a conditional stability for this problem. Moreover, we propose a truncation regularization method combined with an a posteriori regularization parameter choice rule to deal with this problem and give the corresponding convergence estimate. Numerical results are presented to illustrate the accuracy and efficiency of this method. 1. Introduction Inverse source problems arise in many branches of science and engineering, for example, heat conduction, crack identification, electromagnetic theory, geophysical prospecting, and pollutant detection. In this paper, we consider the following inverse problem: to find a pair of functions satisfying where is the unknown source depending only on one spatial variable and is the supplementary condition. In applications, input data can only be measured, and there will be measured data function which is merely in and satisfies where the constant represents a bound on the measurement error. For the heat equation, there has been a large number of research results for the different forms of heat source [1¨C6]. In [7], the authors identified the unknown source of the Poisson equation using the modified regularization method. In [8], the authors identified the unknown source of the Poisson equation using the truncation method. In [7, 8], the regularization parameters which depend on the noise level and the a priori bound are selected by the a priori rule. Generally speaking, there is a defect for any a priori method; that is, the a priori choice of the regularization parameter depends seriously on the a priori bound of the unknown solution. However, in general, the a priori bound cannot be known exactly in practice, and working with a wrong constant may lead to the bad regularized solution. In the present paper, a posteriori choice of the regularization parameter will be given. To the authorsĄŻ knowledge, there are few papers for choosing the regularization parameter by the a posteriori rule for this problem. The truncation regularization methods have been studied for solving various types of inverse problems. Eld¨Śn et al. [9] used the truncation method to analyze and compute one-dimensional IHCP, Xiong et al. [10] used it to consider the surface heat flux for the sideways heat equation, Fu et al. [11] used it to solve the BHCP, Qian et al. [12] used it to consider the numerical differentiation, and Regi¨˝ska and Regi¨˝ski [13] applied the idea of truncation to a Cauchy problem for the Helmholtz
%U http://www.hindawi.com/journals/ijpde/2013/590737/