An inexact Newton regularization in Banach spaces based on the nonstationary iterated Tikhonov method

Abstract

A version of the nonstationary iterated Tikhonov method was recently introduced to regularize linear inverse problems in Banach spaces [7]. In the present work we employ this method as inner iteration of the inexact Newton regularization method REGINN [14] which stably solves nonlinear ill-posed problems. Further, we propose and analyze a Kaczmarz version of the new scheme which allows fast solution of problems which can be split into smaller subproblems. As special cases we prove strong convergence of Kaczmarz variants of the Levenberg-Marquardt and the iterated Tikhonov methods in Banach spaces.

Inexact Newton regularization combined with gradient methods in Banach spaces

“…holds for all x and y, where K p,s > 0 is a constant depending only on p and s. Similarly, manipulating the Xu-Roach inequalities, one can prove an inequality similar to (15) (but in the opposite direction) in s-convex Banach spaces. In particular, in a s-convex Banach space, for any p ∈ (1, s] , there exists a constant C > 0 depending only on p, s and ρ such that…”

“…For avoiding possible misinterpretations, notice that, if X is s-convex with p s, then X * is s * -smooth with p * s * . In this case, inequality (15) holds in X * with p and s replaced by p * and s * respectively and the constant K p,s replaced by K p * ,s * .…”

“…In our previous work [16], we extended these results for strong convergence of a Kaczmarz variation of REGINN using the Landweber method as inner iteration. In [15], a Kaczmarz version of the iterated-Tikhonov method in Banach spaces was adopted in the inner iteration of REGINN, which includes a variant of the Levenberg-Marquardt method as a particular case. A related method is proposed in [10], where the authors make use of a Tikhonov functional penalized with the Bregman distance induced by a general (possibly non-smooth) uniformly convex functional.…”

A Splitting Strategy for the Calibration of Jump-Diffusion Models

“…where J : Y → Y * is the duality map, and x δ α 1 ,α 2 = (w δ α 1 ,α 2 , z δ α 1 ,α 2 ). See Margotti and Rieder (2014) and Chapter II in Cioranescu (1990) for more details on duality maps. Applying x δ α 1 ,α 2 − x † on both sides of the above equality, we have:…”

An iterated multiplicative regularization for force reconstruction problems

“…First, the regularization term promotes distributed solutions, which is not desirable when the structure is actually excited by localized or impulsive sources. This problem can be solved by implementing the iterated Tikhonov regularization in Banach spaces instead of Hilbert spaces [15,16,17]. Practically, this means that the regularization term is express as the q -norm of the residual solution F − F…”