A posteriori parameter choice with an efficient discretization scheme for solving ill-posed problems

摘要

Recently, Rajan [M.P. Rajan, An efficient discretization scheme for solving the ill-posed problems, Journal of Mathematical Analysis and Applications 313 (2) (2006) 654–677] considered a discretization scheme for solving ill-posed problems and obtained optimal error estimates under an a priori parameter choice strategy with a particular smoothness assumption on the solution. It is shown that the computational information for solving the system is far less than the traditional projection schemes. Neubauer [A. Neubauer, An a posteriori parameter choice for Tikhonov regularization in the presence of modeling error, Applied Numerical Mathematics 4 (1988) 507–519] and Engl and Gfrerer [H.W. Engl, H. Gfrerer, A posteriori parameter choice for general regularization methods for solving ill-posed problems, Applied Numerical Mathematics 4 (1988) 395–417] suggested an a posteriori method for choosing the regularization parameter which does not require any knowledge about the exact solution, respectively under finite dimensional as well as general setting. In this paper, we consider an a posteriori parameter choice for finite dimensional approximation which does not require any information about the smoothness of the exact solution and apply to the discretization scheme considered by Rajan. The computational efficiency of the scheme is illustrated through numerical examples.