In this paper, we consider the Cauchy problem for the Laplace equation, which is severely ill-posed in the sense that the solution does not depend continuously on the data. A modified Tikhonov regularization method is...In this paper, we consider the Cauchy problem for the Laplace equation, which is severely ill-posed in the sense that the solution does not depend continuously on the data. A modified Tikhonov regularization method is proposed to solve this problem. An error estimate for the a priori parameter choice between the exact solution and its regularized approximation is obtained. Moreover, an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained. Numerical examples illustrate the validity and effectiveness of this method.展开更多
基金supported by the National Natural Science Foundation of China(1117113611261032)+2 种基金the Distinguished Young Scholars Fund of Lan Zhou University of Technology(Q201015)the basic scientific research business expenses of Gansu province collegethe Natural Science Foundation of Gansu province(1310RJYA021)
文摘In this paper, we consider the Cauchy problem for the Laplace equation, which is severely ill-posed in the sense that the solution does not depend continuously on the data. A modified Tikhonov regularization method is proposed to solve this problem. An error estimate for the a priori parameter choice between the exact solution and its regularized approximation is obtained. Moreover, an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained. Numerical examples illustrate the validity and effectiveness of this method.
