Constrained Polynomial Approximation for Inverse Problems in Engineering

  • Paul O’LearyEmail author
  • Roland Ritt
  • Matthew Harker
Conference paper
Part of the Lecture Notes in Mechanical Engineering book series (LNME)


This paper presents the derivation, implementation and testing of a series of algorithms for the least squares approximation of perturbed data by polynomials subject to arbitrary constraints. These approximations are applied to the solution of inverse problems in engineering applications. The generalized nature of the constraints considered enables the generation of vector basis sets which correspond to admissible functions for the solution of inverse initial-, internal- and boundary-value problems. The selection of the degree of the approximation polynomial corresponds to spectral regularization using incomplete sets of basis functions. When applied to the approximation of data, all algorithms yield the vector of polynomial coefficients \(\varvec{\alpha }\), together with the associated covariance matrix \(\mathsf {\Lambda }_{\varvec{\alpha }}\). A matrix algebraic approach is taken to all the derivations. A numerical application example is presented for each of the constraint types presented. Furthermore, a new approach to performing constrained polynomial approximation with constraints on the coefficients is presented.


Constrained polynomial approximation Conditional least squares 



Partial funding for this work was provided by:

1. The Austrian research funding association (FFG) under the scope of the COMET program within the K2 center “IC-MPP” (contract number 859480). This programme is promoted by BMVIT, BMDW and the federal states of Styria, Upper Austria and Tyrol.

2. The Center of Competence for Recycling and Recovery of Waste 4.0 (acronym ReWaste4.0) (contract number 860884) under the scope of the COMET — Competence Centers for Excellent Technologies — is financially supported by BMVIT, BMWFW, and the federal state of Styria, managed by the FFG.


  1. 1.
    Akima, H.: A new method of interpolation and smooth curve fitting based on local procedures. J. ACM 17(4), 589–602 (1970). Scholar
  2. 2.
    Gander, M.J., Wanner, G.: From Euler, Ritz, and Galerkin to modern computing. SIAM Rev. 54(4), 627–666 (2012). Scholar
  3. 3.
    Gelfand, I., Kapranov, M., Zelevinsky, A.: Discriminants, Resultants, and Multidimensional Determinants. Modern Birkhäuser Classics. Birkhäuser, Boston (2008). Scholar
  4. 4.
    Giorgio, C., Michel, B.: Constrained Polynomial Approximation. Wiley-Blackwell (2016)Google Scholar
  5. 5.
    Handscomb, D.: Methods of numerical approximation: lectures delivered at a Summer School held at Oxford University, September 1965. Pergamon Press (2014)Google Scholar
  6. 6.
    Muller, J.M.,: Polynomial approximations with special constraints. In: Elementary Functions. Birkhäuser, Boston (2016)CrossRefGoogle Scholar
  7. 7.
    Klopfenstein, R.W.: Conditional least squares polynomial approximation. Math. Comput. 18(88), 659–662 (1964). Scholar
  8. 8.
    Newbery, A.C.R.: Trigonometric interpolation and curve-fitting. Math. Comput. 24(112), 869–876 (1970)MathSciNetCrossRefGoogle Scholar
  9. 9.
    O’Leary, P., Harker, M.: A framework for the evaluation of inclinometer data in the measurement of structures, pp. 61–1237. IEEE (2012). Scholar
  10. 10.
    O’Leary, P., Harker, M.: Inverse boundary value problems with uncertain boundary values and their relevance to inclinometer measurements. In: Proceedings of the IEEE International Instrumentation and Measurement Technology Conference (I2MTC), pp. 165–169. IEEE (2014).
  11. 11.
    O’Leary, P., Harker, M., Gugg, C.: An inverse problem approach to approximating sensor data in cyber physical systems. In: Proceedings of the International Instrumentation and Measurement Technology Conference (I2MTC), pp. 1717–1722 (2015)Google Scholar
  12. 12.
    Pierce, J.G., Varga, R.S.: Higher order convergence results for the Rayleigh Ritz method aplied to eigenvalue problems. I estimates relating Rayleigh-Ritz and Galerkin approximations to eigenfunctions. SIAM J. Numer. Anal. 9(1), 137–151 (1972)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ritz, W.: Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. Journal fur die Reine und Angewandte Mathematik 1909(135), 1–61 (1909).,,

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Chair of AutomationUniversity of LeobenLeobenAustria

Personalised recommendations