Mathematics of Computation

Author(s): Barbara Kaltenbacher.
Journal: Math. Comp. 72 (2003), 1711-1730.
MSC (2000): Primary 65J20, 65R30, 65N55
Posted: May 1, 2003
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Abstract: For ill-posed linear operator equations we consider some V-cycle multigrid approaches, that, in the framework of Bramble, Pasciak, Wang, and Xu (1991), we prove to yield level independent contraction factor estimates. Consequently, we can incorporate these multigrid operators in a full multigrid method, that, together with a discrepancy principle, is shown to act as an iterative regularization method for the underlying infinite-dimensional ill-posed problem. Numerical experiments illustrate the theoretical results.

1.: D. Braess, W. Hackbusch, A new convergence proof for the multigrid method including the -cycle, SIAM J. Numer. Anal. 20 (1983), 967-975. MR 85h:65233
2.: J.H. Bramble, Multigrid methods, Pitman Research Notes in Mathematics Series 294, Longman Scientific & Technical, Harlow, UK, 1993. MR 95b:65002
3.: J.H. Bramble, Z. Leyk, J.E. Pasciak, The analysis of multigrid algorithms for pseudo-differential operators of order minus one, Math. Comp. 63 (1994), 461-478. MR 94m:65184
4.: J.H. Bramble, J.E. Pasciak, New estimates for multigrid algorithms including the V-cycle, Math. Comp. 60 (1993), 447-471. MR 94a:65064
5.: J.H. Bramble, J.E. Pasciak, P.S. Vassilevski, Computational scales of Sobolev norms with application to preconditioning, Math. Comp. 69 (2000), 463-480. MR 2000k:65088
6.: J.H. Bramble, J.E. Pasciak, J. Wang, J. Xu, Convergence estimates for multigrid algorithms without regularity assumptions, Math. Comp. 57 (1991), 23-45. MR 91m:65158
7.: J.H. Bramble, J.E. Pasciak, J. Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1990), 1-22. MR 90k:65170
8.: A. Brandt, Multilevel adaptive solutions to boundary value problems, Math. Comp. 31 (1977), 333-390. MR 55:4714
9.: H.W. Engl, A. Neubauer, On projection methods for solving linear ill-posed problems, in: A. Vogel, ed., Model Optimization in Exploration Geophysics, Vieweg, Braunschweig, 1987, 73-92. MR 90g:65078
10.: H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996. MR 97k:65145
11.: C.W. Groetsch, Inverse Problems in Mathematical Sciences, Vieweg, Braunschweig, 1993. MR 94m:00008
12.: C.W. Groetsch, A. Neubauer, Convergence of a general projection method for an operator equation of the first kind, Houston J. Mathem. 14 (1988), 201-207. MR 90b:65109
13.: Ch. Grossmann, H.-G. Roos, Numerik partieller Differentialgleichungen, Teubner, Stuttgart, 1994. MR 95i:65003
14.: W. Hackbusch, Multi-Grid Methods and Applications, Springer, Berlin, 1985. MR 87e:65082
15.: M. Hanke, C.R. Vogel, Two-level preconditioners for regularized inverse problems I: Theory, Numer. Math. 83 (1999), 385-402. MR 2001h:65069
16.: B. Kaltenbacher, A projection-regularized Newton method for nonlinear ill-posed problems with application to parameter identification problems with finite element discretization, SIAM J.Numer.Anal. 37 (2000), 1885-1908. MR 2001f:65070
17.: B. Kaltenbacher, Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems, Inverse Problems 16 (2000), 1523-1539. MR 2001h:65070
18.: B. Kaltenbacher, On the regularizing properties of a full multigrid method for ill-posed problems, Inverse Problems 17 (2001), 767-788. MR 2002h:65094
19.: B. Kaltenbacher, V-cycle convergence of some multigrid methods for ill-posed problems, SFB013-report, University of Linz, 2000.
20.: B. Kaltenbacher, M. Kaltenbacher, S. Reitzinger, Identification of nonlinear curves based on magnetic field computations and using multigrid preconditioners for ill-posed problems, to appear in Europ. J. Appl. Math., 2002.
21.: B. Kaltenbacher, J. Schicho, A multigrid method with a priori and a posteriori level choice for the regularization of nonlinear ill-posed problems, Numer. Math., DOI 10.1007/s002110100375, January 2002.
22.: J.T. King, Multilevel algorithms for ill-posed problems, Numer. Math. 61 (1992), 311-334. MR 92k:65090
23.: A.K. Louis, Inverse und schlecht gestellte Probleme, Teubner, Stuttgart, 1989. MR 90g:65075
24.: V.A. Morozov, Regularization Methods for Ill-Posed Problems, CRC Press, Boca Raton, 1993. MR 94g:65002
25.: F. Natterer, Regularisierung schecht gestellter Probleme durch Projektionsverfahren, Numer. Math. 28 (1977), 329-341. MR 58:8238
26.: F. Natterer, The Mathematics of Computerized Tomography, Teubner, Stuttgart, 1986. MR 88m:44008
27.: P. Oswald, Multilevel Finite Element Approximations. Theory and Applications, Teubner, Stuttgart, 1994. MR 95k:65110
28.: S.V. Pereverzev, S. Prössdorf, On the characterization of self-regularization properties of a fully discrete projection method for Symm's integral equation, J. Integral Equations Appl. 12 (2000), no. 2, 113-130. MR 2001j:65198
29.: A. Rieder, A wavelet multilevel method for ill-posed problems stabilized by Tikhonov regularization, Numer. Math. 75 (1997), 501-522. MR 97k:65299
30.: A.N. Tikhonov, V.A. Arsenin, Methods for Solving Ill-Posed Problems, Nauka, Moskau, 1979. MR 82e:65002
31.: G.M. Vainikko, U. Hämarik, Projection methods and self-regularization in ill-posed problems, Sov. Math. 29 (1985), 1-20. MR 87m:65096
32.: G.M. Vainikko, A.Y. Veterennikov, Iteration Procedures in Ill-Posed Problems, Nauka, Moscow, 1986. In Russian. MR 88c:47019

Retrieve articles in Mathematics of Computation with MSC (2000): 65J20, 65R30, 65N55

Barbara Kaltenbacher
Affiliation: SFB013 Numerical and Symbolic Scientific Computing, University of Linz, Freitaedterstrasse 313, A-4040 Linz, Austria
Email: barbara.kaltenbacher@sfb013.uni-linz.ac.at

DOI: 10.1090/S0025-5718-03-01533-3
PII: S 0025-5718(03)01533-3
Keywords: Ill-posed problem, multigrid methods
Received by editor(s): November 21, 2000
Received by editor(s) in revised form: April 11, 2002
Posted: May 1, 2003
Additional Notes: The author was supported by the Fonds zur Förderung der wissenschaftlichen Forschung under grant T 7-TEC and project F1308 within Spezialforschungsbereich F013
Copyright of article: Copyright 2003, American Mathematical Society

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