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. Hinterer, Fabian; Hubmer, Simon; Ramlau, Ronny: A note on the minimization of a Tikhonov functional with (\ell^1)-penalty (2020)
  2. Kokurin, Mikhail Y.: On the clustering of stationary points of Tikhonov’s functional for conditionally well-posed inverse problems (2020)
  3. Pes, Federica; Rodriguez, Giuseppe: The minimal-norm Gauss-Newton method and some of its regularized variants (2020)
  4. Otárola, Enrique; Quyen, Tran Nhan Tam: A reaction coefficient identification problem for fractional diffusion (2019)
  5. Zhong, Min; Wang, Wei: A regularizing multilevel approach for nonlinear inverse problems (2019)
  6. Garrigos, Guillaume; Rosasco, Lorenzo; Villa, Silvia: Iterative regularization via dual diagonal descent (2018)
  7. George, Santhosh; Sabari, M.: Numerical approximation of a Tikhonov type regularizer by a discretized frozen steepest descent method (2018)
  8. Hubmer, Simon; Ramlau, Ronny: Nesterov’s accelerated gradient method for nonlinear ill-posed problems with a locally convex residual functional (2018)
  9. Sabari, M.; George, Santhosh: Modified minimal error method for nonlinear ill-posed problems (2018)
  10. George, Santhosh; Sabari, M.: Convergence rate results for steepest descent type method for nonlinear ill-posed equations (2017)
  11. Hubmer, Simon; Ramlau, Ronny: Convergence analysis of a two-point gradient method for nonlinear ill-posed problems (2017)
  12. Anzengruber, Stephan W.; Bürger, Steven; Hofmann, Bernd; Steinmeyer, Günter: Variational regularization of complex deautoconvolution and phase retrieval in ultrashort laser pulse characterization (2016)
  13. Bürger, Steven; Flemming, Jens; Hofmann, Bernd: On complex-valued deautoconvolution of compactly supported functions with sparse Fourier representation (2016)
  14. Zhong, Min; Wang, Wei: A global minimization algorithm for Tikhonov functionals with (p)-convex ((p \geqslant2)) penalty terms in Banach spaces (2016)
  15. Egger, Herbert; Schlottbom, Matthias: Numerical methods for parameter identification in stationary radiative transfer (2015)
  16. Argyros, Ioannis K.; George, Santhosh; Jidesh, P.: Inverse free iterative methods for nonlinear ill-posed operator equations (2014)
  17. Niebsch, Jenny; Ramlau, Ronny: Simultaneous estimation of mass and aerodynamic rotor imbalances for wind turbines (2014)
  18. Brandt, C.; Niebsch, J.; Ramlau, R.; Maass, P.: Modeling the influence of unbalances for ultra-precision cutting processes (2011)
  19. Egger, Herbert; Schlottbom, Matthias: Efficient reliable image reconstruction schemes for diffuse optical tomography (2011)
  20. Klann, Esther; Ramlau, Ronny; Ring, Wolfgang: A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data (2011)

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