An optimal order yielding discrepancy principle for simplified regularization of ill‐posed problems in Hilbert scales

Abstract

have considered a monotone rule as a parameter choice strategy for choosing the regularization parameter while considering approximate solution of an ill-posed operator equation T x = y, where T is a bounded linear operator between Hilbert spaces. Motivated by this, we propose a new discrepancy principle for the simplified regularization, in the setting of Hilbert scales, when T is a positive and selfadjoint operator. When the data y is known only approximately, our method provides optimal order under certain natural assumptions on the ill-posedness of the equation and smoothness of the solution. The result, in fact, improves an earlier work of the authors (1997).

On The Analog Of The Monotone Error Rule For Parameter Choice In The (Iterated) Lavrentiev Regularization

“…Note also that in [1,2] the Lavrentiev method u α = (αL s + A) −1 f δ with a strictly positive operator L : D(L) → H and s 0 in Hilbert scales was considered. Assuming c 1 L a u Au c 2 L a u for some positive a, c 1 , c 2 and choosing α from the equation…”