The Ivanov regularized Gauss–Newton method in Banach space with an a posteriori choice of the regularization radius

Abstract

In this paper we consider the iteratively regularized Gauss-Newton method, where regularization is achieved by Ivanov regularization, i.e., by imposing a priori constraints on the solution. We propose an a posteriori choice of the regularization radius, based on an inexact Newton / discrepancy principle approach, prove convergence and convergence rates under a variational source condition as the noise level tends to zero, and provide an analysis of the discretization error. Our results are valid in general, possibly nonreflexive Banach spaces, including, e.g., L ∞ as a preimage space. The theoretical findings are illustrated by numerical experiments.

Iterative regularization for constrained minimization formulations of nonlinear inverse problems

“…(which for = 0 follows from convexity of J , i.e., monotonicity of ∇J ) and assuming approximate stationarity Using (18), (19), we get from (17), that for all k ≤ k * − 1 with k * defined by the estimate…”

“…Alternatively, we can further estimate (17) under a condition following from ( 18), (19) and comprising both convexity and approximate stationarity which for k ≤ k * − 1 implies as well as Using ( 23)-( 24) we get from ( 17)…”

“…where G and H should be viewed as (approximations to) the gradient and Hessian of J, G (x k ) ≈ J � (x k ) , H (x k ) ≈ J �� (x k ) , and R is a regularization functional. Since we do not necessarily neglect terms in J �� (x k ) , this differs from the iteratively regularized Gauss-Newton method IRGNM studied, e.g., in [2,17,22], cf. (35) below.…”

“…A special case of this with (note that H in general does not coincide with the Hessian of J , since the term containing F ′′ is skipped ) in Hilbert space is the iteratively regularized Gauss-Newton method for the operator equation formulation (3) of the inverse problem, see, e.g., [2,17,22].…”

“…The advantages of method (32) lie in its versatility: Besides the various options of choosing H (x k ) , it also works in general Banach spaces and with quite general regularization functionals R . In case of a quadratic functional R , note that as opposed to the IRGNM considered, e.g., in [17,22,29], where depending on the choice of the data misfit term, the cost function can become nonlinear, we always deal with a quadratic overall cost function here.…”