Descriptive Regularization Algorithms on the basis of the Conjugate Gradient Projection method

Abstract

In most methods of solving ill-posed problems (the Tikhonov regularization method [172, 173, 174], the residual principle [127, 128, 175, 176]) the traditional way to regularize them, i.e., to convert them into related well-posed problems, is applied. This way is based on utilization of quantitative information about the level of errors in the input data and on greatly general a priori information pertaining to smoothness of the solution. This ensures asymptotic (with respect to the level of errors) stability of the approximate solutions but can be insufficient to preserve the main qualitative characteristics of the functions sought. On the other hand, in addition to conditions of smoothness, shape constraints on the solution (i.e., nonnegativity, monotonicity, convexity etc.) may exist. It is known that shape constraints have the stabilizing properties [80, 160] which are utilized in descriptive regularization of ill-posed problems for obtaining stable approximate solutions with the desired qualitative behavior.