On convergence rates of inexact Newton regularizations

Rieder, Andreas

doi:10.1007/pl00005448

Cited by 67 publications

(65 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [11] we were able to verify (under reasonable assumptions) that REGINN with a linear regularization scheme {g r } r∈N is well defined and indeed terminates. Moreover, we proved the existence of a positive κ min < 1 such that the source condition…”

Section: Introduction Our Goal Is To Find a Stable Approximate Solutmentioning

confidence: 81%

“…Here, C Q is a positive constant. For a discussion of the nontrivial factorization (2.1) and for examples of meaningful operators satisfying (2.1), we refer to [6,10,11], [12,Kapitel 7.3], and the literature cited therein.…”

Section: General Assumptions and Termination Of The Repeat-loop Thromentioning

confidence: 99%

“…In [10,11] we proposed algorithm REGINN for solving (1.1). As a Newton-type algorithm, REGINN updates the actual iterate x n by adding a correction step s δ n obtained from solving a linearization of (1.1): Here, we have to take into account that the ill-posedness of (1.1) is passed on to (1.4).…”

Section: Introduction Our Goal Is To Find a Stable Approximate Solutmentioning

confidence: 99%

See 2 more Smart Citations

Inexact Newton Regularization Using Conjugate Gradients as Inner Iteration

Rieder¹

2005

SIAM J. Numer. Anal.

Self Cite

View full text Add to dashboard Cite

Abstract. In our papers [Inverse Problems, 15 (1999), pp. 309-327] and [Numer. Math., 88 (2001), pp. 347-365] we proposed algorithm REGINN, an inexact Newton iteration for the stable solution of nonlinear ill-posed problems. REGINN consists of two components: the outer iteration, which is a Newton iteration stopped by the discrepancy principle, and an inner iteration, which computes the Newton correction by solving the linearized system. The convergence analysis presented in both papers covers virtually any linear regularization method as inner iteration, especially Landweber iteration, ν-methods, and Tikhonov-Phillips regularization. In the present paper we prove convergence rates for REGINN when the conjugate gradient method, which is nonlinear, serves as inner iteration. Thereby we add to a convergence analysis of Hanke, who had previously investigated REGINN furnished with the conjugate gradient method [Numer. Funct. Anal. Optim., 18 (1997), pp. 971-993]. By numerical experiments we illustrate that the conjugate gradient method outperforms the ν-method as inner iteration.

show abstract

Section: Introduction Our Goal Is To Find a Stable Approximate Solutmentioning

confidence: 81%

Section: General Assumptions and Termination Of The Repeat-loop Thromentioning

confidence: 99%

Section: Introduction Our Goal Is To Find a Stable Approximate Solutmentioning

confidence: 99%

See 1 more Smart Citation

Inexact Newton Regularization Using Conjugate Gradients as Inner Iteration

Rieder¹

2005

SIAM J. Numer. Anal.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The regularising effect of such methods has been studied by Hanke [19] and Rieder [36,37]. When applying an iterative regularisation method to (3.14) the inexact Newton method for solving (3.6) can be written as follows.…”

Section: Inexact Newton Iterationsmentioning

confidence: 99%

PDE based determination of piezoelectric material tensors

Kaltenbacher¹,

Lahmer²,

Mohr³

et al. 2006

Eur. J. Appl. Math

View full text Add to dashboard Cite

The exact numerical simulation of piezoelectric transducers requires the knowledge of all material tensors that occur in the piezoelectric constitutive relations. To account for mechanical, dielectric and piezoelectric losses, the material parameters are assumed to be complex. The issue of material tensor identification is formulated as an inverse problem: As input measured impedance values for different frequency points are used, the searched-for output is the complete set of material parameters. Hence, the forward operator F mapping from the set of parameters to the set of measurements, involves solutions of the system of partial differential equations arising from application of Newton's and Gauss' law to the piezoelectric constitutive relations. This, via two or three dimensional finite element discretisation, leads to an indefinite system of equations for solving the forward problem. Well-posedness of the infinite dimensional forward problem is proven and efficient solution strategies for its discretized version are presented. Since unique solvability of the inverse problem may hardly be verified, the system of equations we have to solve for recovering the material tensor entries can be rank deficient and therefore requires application of appropriate regularisation strategies. Consequently, inversion of the (nonlinear) parameter-to-measurement map F is performed using regularised versions of Newton's method. Numerical results for different piezoelectric specimens conclude this paper.

show abstract

“…Other options for regularization are variants of inexact Newton methods [12,16,17]. A nice review is given in the recent book by Kaltenbacher et al [14].…”

Section: Introductionmentioning

confidence: 99%