TIGRA -- an iterative algorithm for regularizing nonlinear ill-posed problems. A sophisticated numerical analysis of a combination of Tikhonov regularization and the gradient method for solving nonlinear ill-posed problems is presented. The TIGRA (Tikhonov-gradient method) algorithm proposed uses steepest descent iterations in an inner loop for approximating the Tikhonov regularized solutions with a fixed regularization parameter and a parameter iteration for satisfying a discrepancy criterion in an outer loop. The method with given convergence rate results works in a Hilbert space setting whenever the nonlinear forward operator is twice continuous Fréchet differentiable with a Lipschitz-continuous first derivative and obvious source conditions are fulfilled. For applying the method the forward operator must be defined on the whole Hilbert space, which seems to be the essential restriction of the given approach. Numerical results are presented for an inverse problem occurring in single-photon-emission computed tomography, where the assumptions of the paper are satisfied.

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  1. Anzengruber, Stephan W.; Bürger, Steven; Hofmann, Bernd; Steinmeyer, Günter: Variational regularization of complex deautoconvolution and phase retrieval in ultrashort laser pulse characterization (2016)
  2. Bürger, Steven; Flemming, Jens; Hofmann, Bernd: On complex-valued deautoconvolution of compactly supported functions with sparse Fourier representation (2016)
  3. Zhong, Min; Wang, Wei: A global minimization algorithm for Tikhonov functionals with $p$-convex $(p \geqslant 2)$ penalty terms in Banach spaces (2016)
  4. Egger, Herbert; Schlottbom, Matthias: Numerical methods for parameter identification in stationary radiative transfer (2015)
  5. Brandt, C.; Niebsch, J.; Ramlau, R.; Maass, P.: Modeling the influence of unbalances for ultra-precision cutting processes (2011)
  6. Egger, Herbert; Schlottbom, Matthias: Efficient reliable image reconstruction schemes for diffuse optical tomography (2011)
  7. Klann, Esther; Ramlau, Ronny; Ring, Wolfgang: A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data (2011)
  8. Beilina, L.; Klibanov, M.V.; Kokurin, M.Yu.: Adaptivity with relaxation for ill-posed problems and global convergence for a coefficient inverse problem (2010)
  9. Kokurin, M.Yu.: Convexity of the Tikhonov functional and iteratively regularized methods for solving irregular nonlinear operator equations (2010)
  10. Kokurin, M.Yu.: The global search in the Tikhonov scheme (2010)
  11. Liu, Y.; Han, B.; Dou, Y.X.: A homotopy-projection method for the parameter estimation problems (2008)
  12. Gasparo, Maria Grazia; Papini, Alessandra; Pasquali, Aldo: A two-stage method for nonlinear inverse problems (2007)
  13. Lu, Shuai; Pereverzev, Sergei V.; Ramlau, Ronny: An analysis of Tikhonov regularization for nonlinear ill-posed problems under a general smoothness assumption (2007)
  14. Pricop, Mihaela: Tikhonov regularization in Hilbert scales for nonlinear statistical inverse problems. (2007)
  15. George, S.: Newton-Tikhonov regularization of ill-posed Hammerstein operator equation (2006)
  16. Kaltenbacher, B.; Neubauer, A.: Convergence of projected iterative regularization methods for nonlinear problems with smooth solutions (2006)
  17. Klann, E.; Maaß, P.; Ramlau, R.: Two-step regularization methods for linear inverse problems (2006)
  18. Kokurin, Mihail Yu.: Stable iteratively regularized gradient method for nonlinear irregular equations under large noise (2006)
  19. Ramlau, Ronny; Teschke, Gerd: A Tikhonov-based projection iteration for nonlinear ill-posed problems with sparsity constraints (2006)
  20. Estatico, C.; Bozza, G.; Massa, A.; Pastorino, M.; Randazzo, A.: A two-step iterative inexact-Newton method for electromagnetic imaging of dielectric structures from real data (2005)

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