TIGRA swMATH ID: 2333 Software Authors: Ramlau, Ronny Description: 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. Homepage: http://iopscience.iop.org/article/10.1088/0266-5611/19/2/312/meta Keywords: nonlinear ill-posed problems; Tikhonov regularization; gradient method; steepest descent iterations; convergence; Hilbert space; inverse problem; single-photon-emission computed tomography Related Software: NewtonLib; FEniCS; mngn; cosereg; corner.m; FDEMtools; Regularization tools; AIR tools; UNLocBoX; NESTA Cited in: 42 Documents Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year TIGRA – an iterative algorithm for regularizing nonlinear ill-posed problems. Zbl 1029.65059Ramlau, Ronny 2003 all top 5 Cited by 57 Authors 12 Ramlau, Ronny 5 George, Santhosh 5 Kokurin, Mikhail Yur’evich 4 Hubmer, Simon 3 Sabari, M. 2 Burger, Steven K. 2 Egger, Herbert 2 Hofmann, Bernd 2 Klann, Esther 2 Klibanov, Michael V. 2 Maass, Peter 2 Niebsch, Jenny 2 Schlottbom, Matthias 2 Teschke, Gerd 2 Wang, Wei 2 Zhong, Min 1 Anzengruber, Stephan W. 1 Argyros, Ioannis Konstantinos 1 Bakushinskĭ, Anatoliĭ Borisovich B. 1 Bal, Guillaume 1 Beilina, Larisa 1 Bell, Bradley M. 1 Bozza, Giovanni 1 Dou, Yixin 1 Estatico, Claudio 1 Flemming, Jens 1 Garrigos, Guillaume 1 Gasparo, Maria Grazia 1 Giri, Ankik Kumar 1 Han, Bo 1 Hinterer, Fabian 1 Jidesh, P. 1 Kaltenbacher, Barbara 1 Kindermann, Stefan 1 Liu, Y. 1 Lu, Shuai 1 Massa, Andrea 1 Mittal, Gaurav 1 Neubauer, Andreas 1 Otárola, Enrique 1 Papini, Alessandra 1 Pasquali, Aldo 1 Pastorino, Matteo 1 Pereverzev, Sergei V. 1 Pes, Federica 1 Pillonetto, Gianluigi 1 Pricop, Mihaela 1 Quyen, Tran Nhan Tam 1 Raik, Kemal 1 Randazzo, Andrea 1 Ring, Wolfgang 1 Rodriguez, Giuseppe 1 Rosasco, Lorenzo A. 1 Sherina, Ekaterina Sergeevna 1 Steinmeyer, Günter 1 Timonov, Alexandre 1 Villa, Silvia all top 5 Cited in 20 Serials 15 Inverse Problems 4 Journal of Inverse and Ill-Posed Problems 3 Journal of Computational and Applied Mathematics 2 ETNA. Electronic Transactions on Numerical Analysis 2 Inverse Problems in Science and Engineering 1 Applicable Analysis 1 Zhurnal Vychislitel’noĭ Matematiki i Matematicheskoĭ Fiziki 1 Applied Mathematics and Computation 1 International Journal of Mathematics and Mathematical Sciences 1 Numerische Mathematik 1 Applied Numerical Mathematics 1 Journal of Mathematical Imaging and Vision 1 Russian Mathematics 1 Computational Optimization and Applications 1 Journal of Mathematical Sciences (New York) 1 ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik 1 Computational Methods in Applied Mathematics 1 Mathematics and its Applications (Springer) 1 Inverse Problems and Imaging 1 Journal of Mathematics in Industry all top 5 Cited in 14 Fields 37 Numerical analysis (65-XX) 24 Operator theory (47-XX) 5 Partial differential equations (35-XX) 4 Biology and other natural sciences (92-XX) 3 Calculus of variations and optimal control; optimization (49-XX) 3 Optics, electromagnetic theory (78-XX) 3 Information and communication theory, circuits (94-XX) 2 Integral transforms, operational calculus (44-XX) 2 Statistics (62-XX) 1 Functions of a complex variable (30-XX) 1 Ordinary differential equations (34-XX) 1 Mechanics of deformable solids (74-XX) 1 Fluid mechanics (76-XX) 1 Operations research, mathematical programming (90-XX) Citations by Year