Skip to main content

Non-Generic Case of Leap-Frog for Optimal Knots Selection in Fitting Reduced Data

  • Conference paper
  • First Online:
Computational Science – ICCS 2022 (ICCS 2022)

Abstract

The problem of fitting multidimensional reduced data is analyzed here . The missing interpolation knots \(\mathcal{T}\) are substituted by \(\hat{\mathcal{T}}\) which minimize a non-linear multivariate function \(\mathcal{J}_0\). One of numerical schemes designed to compute such optimal knots relies on iterative scheme called Leap-Frog Algorithm. The latter is based on merging the respective generic and non-generic univariate overlapping optimizations of \(\mathcal{J}_0^{(k,i)}\). The discussion to follow establishes the sufficient conditions enforcing unimodality of the non-generic case of \(\mathcal{J}_0^{(k,i)}\) (for special data set-up and its perturbation). Illustrative example supplements the analysis in question. This work complements already existing analysis on generic case of Leap-Frog Algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    This work is a part of Polish National Centre of Research and Development research project POIR.01.02.00-00-0160/20.

References

  1. de Boor, C.: A Practical Guide to Splines. 2nd edn. Springer-Verlag, New York (2001). https://www.springer.com/gp/book/9780387953663

  2. Kozera, R., Noakes, L.: Optimal knots selection for sparse reduced data. In: Huang, F., Sugimoto, A. (eds.) PSIVT 2015. LNCS, vol. 9555, pp. 3–14. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-30285-0_1

    Chapter  Google Scholar 

  3. Kozera, R., Noakes, L.: Non-linearity and Non-convexity in optimal knots selection for sparse reduced data. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2017. LNCS, vol. 10490, pp. 257–271. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66320-3_19

    Chapter  MATH  Google Scholar 

  4. Kozera, R., Noakes, L., Wilkołazka, M.: Parameterizations and Lagrange cubics for fitting multidimensional data. In: Krzhizhanovskaya, V.V., et al. (eds.) ICCS 2020. LNCS, vol. 12138, pp. 124–140. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-50417-5_10

  5. Kozera, R., Noakes, L., Wiliński, A.: Generic case of Leap-Frog Algorithm for optimal knots selection in fitting reduced data. In: Paszyński, M., Kranzlmüller, D., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M.A. (eds.) ICCS 2021. LNCS, vol. 12745, pp. 337–350. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-77970-2_26

    Chapter  Google Scholar 

  6. Kozera, R., Noakes L., Wilkołazka, M.: Exponential parameterization to fit reduced data. Appl. Math. Comput. 391, 125645 (2021). https://doi.org/10.1016/j.amc.2020.125645

  7. Kozera, R., Wiliński, A.: Fitting dense and sparse reduced data. In: Pejaś, J., El Fray, I., Hyla, T., Kacprzyk, J. (eds.) ACS 2018. AISC, vol. 889, pp. 3–17. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-03314-9_1

    Chapter  Google Scholar 

  8. Kuznetsov, E.B., Yakimovich A.Y.: The best parameterization for parametric interpolation. J. Comput. Appl. Math. 191(2), 239–245 (2006). https://core.ac.uk/download/pdf/81959885.pdf

  9. Kvasov, B.I.: Methods of Shape-Preserving Spline Approximation. World Scientific Pub., Singapore (2000). https://doi.org/10.1142/4172

  10. Matebese, B., Withey, D., Banda M.K.: Modified Newton’s method in the Leapfrog method for mobile robot path planning. In: Dash, S.S., et al. (eds.) ICAIECES 2017, pp. 71–78. Advances in Intelligent Systems and Computing, vol. 668, Springer Nature Singapore (2018). https://doi.org/10.1007/978-981-10-7868-2$_$7

  11. Noakes, L.: A global algorithm for geodesics. J. Aust. Math. Soc. Series A 65(1), 37–50 (1998). https://doi.org/10.1017/S1446788700039380

  12. Noakes, L., Kozera, R.: Nonlinearities and noise reduction in 3-source photometric stereo. J. Math. Imaging Vis. 18(2), 119–127 (2003). https://doi.org/10.1023/A:1022104332058

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ryszard Kozera .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kozera, R., Noakes, L. (2022). Non-Generic Case of Leap-Frog for Optimal Knots Selection in Fitting Reduced Data. In: Groen, D., de Mulatier, C., Paszynski, M., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M.A. (eds) Computational Science – ICCS 2022. ICCS 2022. Lecture Notes in Computer Science, vol 13352. Springer, Cham. https://doi.org/10.1007/978-3-031-08757-8_29

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-08757-8_29

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-08756-1

  • Online ISBN: 978-3-031-08757-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics