Using Lagrange principle for solving linear ill-posed problems with a priori information

Linear ill-posed problems with a priori information on the exact solution are considered. Using the method of extending compacts, the Lagrange principle and the optimal recovery theory, we propose a method for constructing an optimal regularization algorithm for solving linear ill-posed problems with sourcewise representable solutions and a method of calculating the corresponding optimal worst a posteriori error estimate of the proposed method. A numerical simulation of a heat equation is also considered. This work was partially supported by the Russian Foundation for Basic Research (projects 11-01-00040, 12-01-00524 and 12-01-91153-NFSCa).