On the choice of Lagrange multipliers in the iterated Tikhonov method for linear ill-posed equations in Banach spaces

ABSTRACT

This article is devoted to the study of nonstationary Iterated Tikhonov (nIT) type methods (Hanke M, Groetsch CW. Nonstationary iterated Tikhonov regularization. J Optim Theory Appl. 1998;98(1):37–53; Engl HW, Hanke M, Neubauer A. Regularization of inverse problems. Vol. 375, Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers Group; 1996. MR 1408680) for obtaining stable approximations to linear ill-posed problems modelled by operators mapping between Banach spaces. Here we propose and analyse an a posteriori strategy for choosing the sequence of regularization parameters for the nIT method, aiming to obtain a pre-defined decay rate of the residual. Convergence analysis of the proposed nIT type method is provided (convergence, stability and semi-convergence results). Moreover, in order to test the method's efficiency, numerical experiments for three distinct applications are conducted: (i) a 1D convolution problem (smooth Tikhonov functional and Banach parameter-space); (ii) a 2D deblurring problem (nonsmooth Tikhonov functional and Hilbert parameter-space); (iii) a 2D elliptic inverse potential problem.

KEYWORDS:

• Ill-posed problems
• Banach spaces
• linear operators
• iterated Tikhonov method

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65J20
• 47J06

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The differentiability and convexity properties of this functional are independent of the particular choice of $p>1.$

2 This Lemma guarantees that, given a reflexive Banach space E, and a nonempty closed convex set $A\subset E$, then any convex l.s.c proper function $\varphi :A\to (-\mathrm{\infty },\mathrm{\infty }]$ achieves its minimum on A.

3 Here (1(1) $Ax=y,$(1) ) is replaced by $Ag=y^{\delta }$.

4 Notice that we are dealing with a discrete inverse problem, and discretization errors associated to the continuous model are not being considered.

5 For the purpose of comparison, the iteration error is ploted in the in $L^2$-norm for both choices of the parameter space $X=L^2$ and $X=L^{1.001}$.

Additional information

Funding

A.L. acknowledges support from CNPq [grant number 311087/2017-5], and from the AvH Foundation.