A Modified Tikhonov Regularization for Linear Operator Equations

ABSTRACT

We construct with the aid of regularizing filters a new class of improved regularization methods, called modified Tikhonov regularization (MTR), for solving ill-posed linear operator equations. Regularizing properties and asymptotic order of the regularized solutions are analyzed in the presence of noisy data and perturbation error in the operator. With some accurate estimates in the solution errors, optimal convergence order of the regularized solutions is obtained by a priori choice of the regularization parameter. Furthermore, numerical results are given for several ill-posed integral equations, which not only roughly coincide with the theoretical results but also show that MTR can be more accurate than ordinary Tikhonov regularization (OTR).

Keywords:

• Convergence and asymptotic order of the regularized solutions
• Error analysis
• First kind operator equation
• Ill-posed problem
• Modified Tikhonov regularization

Mathematics Subject Classification 2000:

• 65J20

ACKNOWLEDGMENTS

This project was supported by NSF of China (10471080) and NSF of Shandong Province (Y2001E03). The first author was also supported by the Visiting Funds from the Ministry of Education of China; he thanks the Institute of Mathematics at Fudan University for its hospitality during his visit.