We propose a new discretization scheme for solving ill-posed integral equations of the third kind. Combining this scheme with Morozov's discrepancy principle for Landweber iteration we show that for some classes of equations in such method a number of arithmetic operations of smaller order than in collocation method is required to appoximately solve an equation with the same accuracy.

An adaptive discretization scheme of ill-posed problems is used for nonstationary iterated Tikhonov regularization. It is shown that for some classes of operator equations of the first kind the proposed algorithm is more efficient compared with standard methods.