The use of monotonicity for choosing the regularization parameter in ill-posed problems

Abstract

We discuss a new a posteriori rule for choosing the regularization parameter in regularization methods for solving linear ill-posed problems Ax = y with minimum-norm solution x. Main emphasis is devoted to the methods of ordinary and iterated Tikhonov regularization. We suppose that instead of y there are given noisy data y satisfying |y-y| with known noise level . The approach of the new rule consists of finding the smallest = ME for which it can be proved that the error |x-x| of the regularized solution xisstrictlymonotonicallyincreasingforincreasing-values, that is, there holds {d}{{d\\alpha}}|x-x|2>0 for all (ME,). Due to this property we call our rule of choosing the monotone error rule (ME rule). Our ME rule leads to smaller errors than the a posteriori rule of Raus (1985 Acta Comment. Univ. Tartuensis 715 12-20 (in Russian)) and Gfrerer (1987 Math. Comput. 49 507-22). Furthermore, we discuss discretization issues and illustrate some of our theoretical results by numerical experiments.