A projection-regularized Newton method for nonlinear ill-posed problems and its application to parameter identification problems with finite element discretization

SIAM Journal on Numerical Analysis

Volumn 37, Issue 6, 2000, Pages 1885-1908

  Kaltenbacher, Barbara ^a  

^a JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

Author keywords

Convergence rates; Iterative regularization methods; Nonlinear ill posed problems; Parameter identification; Stopping rule

Indexed keywords

EID: 0012966823     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142998347322     Document Type: Article

References (39)

1. A.B. BAKUSHINSKII, The Problem of the convergence of the iteratively regularized Gauss-Newton method, Comput. Math. Math. Phys., 32 (1992), pp. 1353-1359.
2. H.T. BANKS AND K. KUNISCH, Estimation Techniques for Distributed Parameter Systems, Birkhäuser, Basel-Boston-Berlin, 1989.
3. B. BLASCHKE(-KALTENBACHER), Some Newton Type Methods for the Solution of Nonlinear Ill-Posed Problems, Ph.D. thesis, Schriften der Johannes Kepler Universität Linz, Trauner, Linz, 1996.
4. B. BLASCHKE(-KALTENBACHER), A. NEUBAUER, AND O. SCHERZER, On convergence rates for the iteratively regularized Gauß-Newton method, IMA J. Numer. Anal., 17 (1997), pp. 421-436.
5. J.H. BRAMBLE AND S.R. HILBERT, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer Anal., 7 (1970), pp. 112-124.
6. J.H. BRAMBLE, Z. LEYK, AND J.E. PASCIAK, The analysis of multigrid algorithms for pseudo-differential operators of order minus one, Math. Comp., 63 (1994), pp. 461-478.
7. G. CHAVENT, K. KUNISCH, AND J.E. ROBERTS, Primal-dual formulations for parameter estimation problems, Comput. Appl. Math, 18 (1999).
8. G. CHAVENT AND K. KUNISCH, On weakly nonlinear ill-posed problems, SIAM J. Appl. Math., 56 (1996), pp. 542-572.
9. P.C. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
10. P. DEUFLHARD, H.W. ENGL, AND O. SCHERZER, A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions, Inverse Problems, 14 (1998), pp. 1081-1106.
11. P. DEUFLHARD AND G. HEINDL, Affine invariant convergence theorems for Newton's method and extensions to related methods, SIAM J. Numer. Anal., 16 (1979), pp. 1-10.
12. H.W. ENGL, On the convergence of regularization methods for ill-posed operator equations, in Improperly Posed Problems and Their Numerical Treatment, G. Hämmerlin and K.H. Hoffmann, eds., Birkhäuser, Basel, 1983, pp. 81-95.
13. H.W. ENGL, M. HANKE, AND A. NEUBAUER, Regularization of inverse problems, in Mathematics and its Applications 375, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
14. H.W. ENGL, K. KUNISCH, AND A. NEUBAUER, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems, Inverse Problems, 5 (1989), pp. 523-540.
15. H.W. ENGL AND A. NEUBAUER, On projection methods for solving linear ill-posed problems, in Model Optimization in Exploration Geophysics, A. Vogel, ed., Friedr. Vieweg & Sohn, Braunschweig, 1987, pp. 73-92.
16. C.W. GROETSCH, The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind, Pitman, Boston, MA, 1984.
17. C.W. GROETSCH, Inverse Problems in Mathematical Sciences, Friedr. Vieweg & Sohn, Braunschweig, 1993.
18. C.W. GROETSCH AND A. NEUBAUER, Convergence of a general projection method for an operator equation of the first kind, Houston J. Math., 14 (1988), pp. 201-207.
19. CH. GROSSMANN AND H.-G. ROOS, Numerik partieller Differentialgleichungen, Teubner, Stuttgart, 1994.
20. M. HANKE, A regularization Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems, Inverse Problems, 13 (1997), pp. 79-95.
21. M. HANKE, Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems, Numer. Funct. Anal. Optim., 18 (1997), pp. 971-993.
22. M. HANKE, A. NEUBAUER, AND O. SCHERZER, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), pp. 21-37.
23. M. HANKE AND C.R. VOGEL, Two-level preconditioners for regularized inverse problems I: Theory, Numer. Math. 83 (1999), pp. 385-402.
24. B. HOFMANN AND O. SCHERZER, Influence factors of ill-posedness for nonlinear ill-posed problems, Inverse Problems, 10 (1994), pp. 1277-1297.
25. B. KALTENBACHER, Some Newton type methods for the regularization of nonlinear ill-posed problems, Inverse Problems, 13 (1997), pp. 729-753.
26. B. KALTENBACHER, A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems, Numer. Math., 79 (1998), pp. 501-528.
27. B. KALTENBACHER AND J. SCHÖBERL, A Saddle Point Variational Formulation for Projection-Regularized Parameter Identification, in preparation.
28. J.T. KING, Multilevel algorithms for ill-posed problems, Numer. Math., 61 (1992), pp. 311-334.
29. C. KRAVARIS AND J.H. SEINFELD, Identification of parameters in distributed parameter systems by regularization, SIAM J. Control Optim., 23 (1985), pp. 217-241.
30. A. K. LOUIS, Inverse und schlecht gestellte Probleme, Teubner, Stuttgart, 1989.
31. V.A. MOROZOV, Regularization Methods for Ill-Posed Problems, CRC Press, Boca Raton, FL, 1993.
32. M.Z. NASHED AND X. CHEN, Convergence of Newton-like methods for singular operator equations using outer inverses, Numer. Math., 66 (1993), pp. 235-257.
33. F. NATTERER, Regularisierung schecht gestellter Probleme durch Projektionsverfahren, Numer. Math., 28 (1977), pp. 329-341.
34. F. NATTERER, The Mathematics of Computerized Tomography, Teubner, Stuttgart, 1986.
35. A. RIEDER, On the regularization of nonlinear ill-posed problems via inexact Newton methods, Inverse Problems, 15 (1999), pp. 309-327.
36. O. SCHERZER, H.W. ENGL, AND K. KUNISCH, Optimal a posteriori parameter choice for Tikhonov regularization for solving ill-posed problems, SIAM J. Numer. Anal., 30 (1993), pp. 1796-1838.
37. A.N. TIKHONOV AND V.A. ARSENIN, Methods for Solving Ill-Posed Problems, Nauka, Moskau, 1979 (in Russian).
38. G. VAINIKKO, Identification of filtration coefficient, in Ill-posed Problems in Natural Sciences, A. Tikhonov, ed., VSP, Utrecht, 1992, pp. 202-213.
39. G.M. VAINIKKO AND A.Y. VETERENNIKOV, Iteration Procedures in Ill-Posed Problems, Nauka, Moscow, 1986 (in Russian).