Heuristic two-parameter regularization based on second discrepancy principle for Tikhonov’s method

Abstract

Second discrepancy principle was found as an effective rule for choosing the regularization parameter in Tikhonov’s method applied for solving linear operator equations of the first kind. The rule is formulated as an additional non-linear equation involving second discrepancy to be satisfied by the regularized solution and regularization parameter together. Two constant parameters appearing in the definition of this criterion may be chosen freely within some limits, preserving asymptotic convergence properties of the method when errors in data tend to zero. This work presents a heuristic approach to determining both constants: they are found on the basis of numerical simulations exploiting realistic model representations with inexact data of problems which are to be solved and accounting for appropriate error levels. The approach is expressed as an approximate optimization problem defined on a class of inverse problems studied. The formulation of the method is given together with a numerical illustration. A set of Fredholm’s integral equations of the first kind with convolution-type operators is taken as an example. Inexact data imitate peaks from X-ray diffraction patterns realistically. The effectiveness of Tikhonov’s regularization with this version of second discrepancy principle is demonstrated on further examples. Weak sensitivity of the method to over- or under-estimation of errors in data is observed. It is concluded that only random errors in data (while discretization errors are omitted) can be accounted for solving the analysed inverse problems effectively.