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The Grand Four: Affine Invariant Globalizations of Newton’s Method

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Abstract

This paper gives a concise synopsis and some new insights concerning four affine invariant globalizations of the local Newton method. The invariance classes include affine covariance, affine contravariance, affine conjugacy, and affine similarity. In view of algorithmic robustness, each of these classes of algorithms is particularly suitable for some corresponding problem class.

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References

  1. Bock, H.G.: Numerical treatment of inverse problems in chemical reaction kinetics. In: Ebert, K.H., Deuflhard, P., Jäger, W (eds.) Modelling of Chemical Reaction Systems, pp 102–125. Springer, Berlin (1981)

    Google Scholar 

  2. Bock, H.G.: Recent advances in parameteridentification techniques for ODE. In: Deuflhard, P., Hairer, E (eds.) Numerical Treatment of Inverse Problems in Differential and Integral Equations. Progress in Scientific Computing, vol. 2, pp 95–121. Birkhäuser, Boston (1983)

    Chapter  Google Scholar 

  3. Bock, H.G.: Randwertproblemmethoden Zur Parameteridentifizierung in Systemen Nichtlinearer Differentialgleichungen. PhD thesis, Universität Bonn (1985)

    Google Scholar 

  4. Bock, H.G., Kostina, E.A., Schlöder, J.P.: On the role of natural level functions to achieve global convergence for damped Newton methods. In: Powell, M.J.D., Scholtes, S. (eds.) System Modelling and Optimization: Methods, Theory and Applications, pp 51–74. Kluwer Academic Publishers, Dordrecht (2000)

    Chapter  Google Scholar 

  5. Deuflhard, P.: A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with applications to multiple shooting. Numer. Math. 22, 289–315 (1974)

    Article  MathSciNet  Google Scholar 

  6. Deuflhard, P.: A stepsize control for continuation methods and its special application to multiple shooting methods. Numer. Math. 33, 115–146 (1979)

    Article  MathSciNet  Google Scholar 

  7. Deuflhard, P.: Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms Springer Series in Computational Mathematics, vol. 35. Springer, Berlin (2004)

    Google Scholar 

  8. Deuflhard, P., Fiedler, B., Kunkel, P.: Efficient numerical pathfollowing beyond critical points. SIAM J. Numer. Anal. 24, 912–927 (1987)

    Article  MathSciNet  Google Scholar 

  9. Deuflhard, P., Freund, R., Walter, A.: Fast secant methods for the iterative solution of large nonsymmetric linear systems. IMPACT Comp. Sci. Eng. 2, 244–276 (1990)

    Article  Google Scholar 

  10. Deuflhard, P., Heindl, G.: Affine invariant convergence theorems for Newton’s method and extensions to related methods. SIAM J. Numer. Anal. 16, 1–10 (1979)

    Article  MathSciNet  Google Scholar 

  11. Deuflhard, P., Pesch, H. -J., Rentrop, P.: A modified continuation method for the numerical solution of nonlinear two-point boundary value problems by shooting techniques. Numer. Math. 26, 327–343 (1976)

    Article  MathSciNet  Google Scholar 

  12. Deuflhard, P., Weiser, M.: Global inexact newton multilevel FEM for nonlinear elliptic problems. In: Hackbusch, W., Wittum, G (eds.) Multigrid Methods V. Lecture Notes in Computational Science and Engineering, vol. 3, pp 71–89. Springer, Berlin (1998)

    Google Scholar 

  13. Hohmann, A.: Inexact gauss newton methods for parameter dependent nonlinear problems. PhD thesis, Freie Universität, Berlin (1994)

    MATH  Google Scholar 

  14. Kantorovich, L.: On Newton’s method for functional equations. (Russian) Dokl. Akad. Nauk SSSR 59, 1237–1249 (1948)

    MathSciNet  Google Scholar 

  15. Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations Frontiers in Applied Mathematics, vol. 16. SIAM, Philadelphia (1995)

    Book  Google Scholar 

  16. Kelley, C.T., Keyes, D.E.: Convergence analysis of pseudo-transient continuation. SIAM J. Numer. Anal. 35, 508–523 (1998)

    Article  MathSciNet  Google Scholar 

  17. Krämer-Eis, P.: Ein Mehrzielverfahren Zur Numerischen Berechnung Optimaler Feedback-Steuerungen Bei Beschränkten Nichtlinearen Problemen. PhD thesis, Universität Bonn (1985)

    Google Scholar 

  18. Mysovskikh, I.: On convergence of Newton’s method. (Russian) Trudy Mat. Inst. Steklov 28, 145–147 (1949)

    Google Scholar 

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Acknowledgements

The author remembers with joy the early company of Georg Bock on the road to affine invariant (covariant) Newton methods.

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Authors and Affiliations

  1. Zuse Institute Berlin, Berlin, Germany

    Peter Deuflhard

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  1. Peter Deuflhard
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Correspondence to Peter Deuflhard.

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Dedicated to Hans-Georg Bock on the occasion of his 70th birthday.

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Deuflhard, P. The Grand Four: Affine Invariant Globalizations of Newton’s Method. Vietnam J. Math. 46, 761–777 (2018). https://doi.org/10.1007/s10013-018-0301-3

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