Convergence and application of a modified iteratively regularized Gauss-Newton algorithm

Alexandra Smirnova, Rosemary Renaut, Taufiquar Khan

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

Theoretical convergence results for an iteratively regularized Gauss-Newton (IRGN) algorithm with a Tikhonov regularization term using a seminorm generated by a linear operator are established. The convergence theorem uses an a posteriori stopping rule and a modified source condition, without any restriction on the nonlinearity of the operator. The theoretical results are illustrated by simulations for a one-dimensional version of the exponentially ill-posed optical tomography inverse problem for the diffusion (D) and absorption (μ) coefficient spatial distributions. The modified Tikhonov regularization performs the mapping of the minimization variables, which are the coefficients of the spline expansions for D and μ, to physical space. This incorporates the inherently differing scales of these variables in the minimization, and also suggests relative weighting of the regularization terms with respect to each parameter space. The presented modification of the IRGN allows greater flexibility for implementations of IRGN solutions of ill-posed inverse problems in which differing scales in physical space hinder standard IRGN inversions.

Original languageEnglish (US)
Article number011
Pages (from-to)1547-1563
Number of pages17
JournalInverse Problems
Volume23
Issue number4
DOIs
StatePublished - Aug 1 2007

Fingerprint

Gauss-Newton
Inverse problems
newton
Optical tomography
Tikhonov Regularization
Splines
Spatial distribution
Inverse Problem
Optical Tomography
linear operators
optimization
Seminorm
Stopping Rule
Absorption Coefficient
Ill-posed Problem
splines
Term
Spatial Distribution
stopping
Convergence Theorem

ASJC Scopus subject areas

  • Applied Mathematics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Convergence and application of a modified iteratively regularized Gauss-Newton algorithm. / Smirnova, Alexandra; Renaut, Rosemary; Khan, Taufiquar.

In: Inverse Problems, Vol. 23, No. 4, 011, 01.08.2007, p. 1547-1563.

Research output: Contribution to journalArticle

Smirnova, Alexandra ; Renaut, Rosemary ; Khan, Taufiquar. / Convergence and application of a modified iteratively regularized Gauss-Newton algorithm. In: Inverse Problems. 2007 ; Vol. 23, No. 4. pp. 1547-1563.
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