Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
BIT Numerical Mathematics
Erschienen in: BIT Numerical Mathematics 4/2015

01.12.2015

Tikhonov regularization via flexible Arnoldi reduction

verfasst von: Lothar Reichel, Xuebo Yu

Erschienen in: BIT Numerical Mathematics | Ausgabe 4/2015

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Flexible GMRES, introduced by Saad, is a generalization of the standard GMRES method for the solution of large linear systems of equations. It is based on the flexible Arnoldi process for reducing a large square matrix to a small matrix. We describe how the flexible Arnoldi process can be applied to implement one-parameter and multi-parameter Tikhonov regularization of linear discrete ill-posed problems. The method proposed is well suited for large-scale problems. Moreover, computed examples show that our method can give approximate solutions of higher accuracy than available direct methods for small-scale problems.
Vorheriger Artikel Multilevel Monte Carlo for the Feynman–Kac formula for the Laplace equation
Nächster Artikel An optimal a priori error estimate in the maximum norm for the Il’in scheme in 2D

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren
Literatur
1.
Baart, M.L.: The use of auto-correlation for pseudo-rank determination in noisy ill-conditioned least-squares problems. IMA J. Numer. Anal. 2, 241–247 (1982)CrossRefMathSciNetMATH
2.
Baglama, J., Reichel, L.: Decomposition methods for large linear discrete ill-posed problems. J. Comput. Appl. Math. 198, 332–342 (2007)CrossRefMathSciNetMATH
3.
Bouhamidi, A., Jbilou, K.: A Sylvester–Tikhonov regularization method for image restoration. J. Comput. Appl. Math. 206, 86–98 (2007)
4.
Brezinski, C., Redivo-Zaglia, M., Rodriguez, G., Seatzu, S.: Multi-parameter regularization techniques for ill-conditioned linear systems. Numer. Math. 94, 203–224 (2003)CrossRefMathSciNetMATH
5.
6.
Chan, T.F., Jackson, K.R.: Nonlinearly preconditioned Krylov subspace methods for discrete Newton algorithms. SIAM J. Sci. Stat. Comput. 5, 533–542 (1984)CrossRefMathSciNetMATH
7.
Daniel, J.W., Gragg, W.B., Kaufman, L., Stewart, G.W.: Reorthogonalization and stable algorithms for updating the Gram–Schmidt QR factorization. Math. Comput. 30, 772–795 (1976)MathSciNetMATH
8.
Donatelli, M., Neuman, A., Reichel, L.: Square regularization matrices for large linear discrete ill-posed problems. Numer. Linear Algebra Appl. 19, 896–913 (2012)CrossRefMathSciNetMATH
9.
Donatelli, M., Reichel, L.: Square smoothing regularization matrices with accurate boundary conditions. J. Comput. Appl. Math. 272, 334–349 (2014)CrossRefMathSciNetMATH
10.
Dykes, L., Reichel, L.: On the reduction of Tikhonov minimization problems and the construction of regularization matrices. Numer. Algorithms 60, 683–696 (2012)CrossRefMathSciNetMATH
11.
Eldén, L.: Algorithms for the regularization of ill-conditioned least squares problems. BIT Numer. Math. 17, 134–145 (1977)CrossRefMathSciNetMATH
12.
Eldén, L.: A weighted pseudoinverse, generalized singular values, and constrained least squares problems. BIT Numer. Math. 22, 487–501 (1982)CrossRefMathSciNetMATH
13.
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)CrossRefMATH
14.
Gazzola, S.: Regularization techniques based on Krylov subspace methods for ill-posed linear systems, Ph.D. thesis, Department of Mathematics, University of Padova, Padova, Italy, March 2014
15.
Gazzola, S., Novati, P.: Muli-parameter Arnoldi–Tikhonov methods. Electron. Trans. Numer. Anal. 40, 452–475 (2013)MathSciNetMATH
16.
Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. Johns Hopkins University Press, Baltimore (2013)MATH
17.
Handlovicová, A., Mikula, K., Sgallari, F.: Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution. Numer. Math. 93, 675–695 (2003)CrossRefMathSciNetMATH
18.
Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia (1998)CrossRef
19.
20.
Hearn, T.A., Reichel, L.: Application of denoising methods to regularization of ill-posed problems. Numer. Algorithms 66, 761–777 (2014)CrossRefMathSciNetMATH
21.
Hnětynková, I., Plešinger, M., Strakoš, Z.: The regularizing effect of the Golub–Kahan iterative bidiagonalization and revealing the noise level in the data. BIT Numer. Math. 49, 669–696 (2009)CrossRefMathSciNetMATH
22.
Hochstenbach, M.E., Reichel, L.: An iterative method for Tikhonov regularization with a general linear regularization operator. J. Integral Equ. Appl. 22, 463–480 (2010)CrossRefMathSciNetMATH
23.
Hochstenbach, M.E., Reichel, L., Yu, X.: A Golub–Kahan-type reduction method for matrix pairs (submitted for publication)
24.
Kilmer, M.E., Hansen, P.C., Español, M.I.: A projection-based approach to general-form Tikhonov regularization. SIAM J. Sci. Comput. 29, 315–330 (2007)CrossRefMathSciNetMATH
25.
Kindermann, S.: Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems. Electron. Trans. Numer. Anal. 38, 233–257 (2011)MathSciNetMATH
26.
Li, R.-C., Ye, Q.: A Krylov subspace method for quadratic matrix polynomials with application to constrained least squares problems. SIAM J. Matrix Anal. Appl. 25, 405–428 (2003)CrossRefMathSciNetMATH
27.
Lu, S., Pereverzyev, S.V.: Multi-parameter regularization and its numerical realization. Numer. Math. 118, 1–31 (2011)CrossRefMathSciNet
28.
29.
Morikuni, K., Reichel, L., Hayami, K.: FGMRES for linear discrete ill-posed problems. Appl. Numer. Math. 75, 175–187 (2014)CrossRefMathSciNetMATH
30.
Neuman, A., Reichel, L., Sadok, H.: Implementations of range restricted iterative methods for linear discrete ill-posed problems. Linear Algebra Appl. 436, 3974–3990 (2012)CrossRefMathSciNetMATH
31.
Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)CrossRef
32.
Reichel, L., Rodriguez, G.: Old and new parameter choice rules for discrete ill-posed problems. Numer. Algorithms 63, 65–87 (2013)CrossRefMathSciNetMATH
33.
Reichel, L., Sgallari, F., Ye, Q.: Tikhonov regularization based on generalized Krylov subspace methods. Appl. Numer. Math. 62, 1215–1228 (2012)CrossRefMathSciNetMATH
34.
Reichel, L., Ye, Q.: Simple square smoothing regularization operators. Electron. Trans. Numer. Anal. 33, 63–83 (2009)MathSciNetMATH
35.
36.
Weickert, J., Romeny, B.M.H., Viergever, M.A.: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans. Image Process. 7, 398–410 (1998)CrossRef
37.
Yu, X.: Generalized Krylov subspace methods with applications, Ph.D. thesis, Department of Mathematics, Kent State University, May 2014
38.
Zha, H.: Computing the generalized singular values/vectors of large sparse or structured matrix pairs. Numer. Math. 72, 391–417 (1996)CrossRefMathSciNetMATH
Metadaten
Titel
Tikhonov regularization via flexible Arnoldi reduction
verfasst von
Lothar Reichel
Xuebo Yu
Publikationsdatum
01.12.2015
Verlag
Springer Netherlands
Erschienen in
BIT Numerical Mathematics / Ausgabe 4/2015
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-014-0542-9

Weitere Artikel der Ausgabe 4/2015

BIT Numerical Mathematics 4/2015 Zur Ausgabe

OriginalPaper

Multilevel Monte Carlo for the Feynman–Kac formula for the Laplace equation

OriginalPaper

A type of multi-level correction scheme for eigenvalue problems by nonconforming finite element methods

OriginalPaper

Weak backward error analysis for Langevin process

OriginalPaper

The multivariate Horner scheme revisited

OriginalPaper

An optimal a priori error estimate in the maximum norm for the Il’in scheme in 2D

OriginalPaper

A posteriori error analysis for finite element methods with projection operators as applied to explicit time integration techniques