Abstract
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
A full implementation of each method is available at https://www.mathworks.com/matlabcentral/profile/authors/3977359-matthew-harker-paul-o-leary.
- 2.
This simplifies the equations and makes the structure more visible. There is no principle change in the methods for generalized weighting.
- 3.
More stringently we should define all vectors as column vectors and use the transpose operation to obtain the corresponding row vector. However, to maintain consistency here we define the Vandermonde vector as one row of the Vandermonde matrix.
References
Akima, H.: A new method of interpolation and smooth curve fitting based on local procedures. J. ACM 17(4), 589–602 (1970). https://doi.org/10.1145/321607.321609
Gander, M.J., Wanner, G.: From Euler, Ritz, and Galerkin to modern computing. SIAM Rev. 54(4), 627–666 (2012). https://doi.org/10.1137/100804036
Gelfand, I., Kapranov, M., Zelevinsky, A.: Discriminants, Resultants, and Multidimensional Determinants. Modern Birkhäuser Classics. Birkhäuser, Boston (2008). https://books.google.at/books?id=2zgxQVU1hFAC
Giorgio, C., Michel, B.: Constrained Polynomial Approximation. Wiley-Blackwell (2016)
Handscomb, D.: Methods of numerical approximation: lectures delivered at a Summer School held at Oxford University, September 1965. Pergamon Press (2014)
Muller, J.M.,: Polynomial approximations with special constraints. In: Elementary Functions. Birkhäuser, Boston (2016)
Klopfenstein, R.W.: Conditional least squares polynomial approximation. Math. Comput. 18(88), 659–662 (1964). http://www.jstor.org/stable/2002954
Newbery, A.C.R.: Trigonometric interpolation and curve-fitting. Math. Comput. 24(112), 869–876 (1970)
O’Leary, P., Harker, M.: A framework for the evaluation of inclinometer data in the measurement of structures, pp. 61–1237. IEEE (2012). https://doi.org/10.1109/TIM.2011.2180969
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). https://doi.org/10.1109/I2MTC.2014.6860725
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)
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)
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). https://doi.org/10.1515/crll.1909.135.1, http://eudml.org/doc/149295, https://www.degruyter.com/view/j/crll.1909.issue-135/crll.1909.135.1/crll.1909.135.1.xml
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
O’Leary, P., Ritt, R., Harker, M. (2019). Constrained Polynomial Approximation for Inverse Problems in Engineering. In: Abdel Wahab, M. (eds) Proceedings of the 1st International Conference on Numerical Modelling in Engineering . NME 2018. Lecture Notes in Mechanical Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-13-2273-0_19
Download citation
DOI: https://doi.org/10.1007/978-981-13-2273-0_19
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-2272-3
Online ISBN: 978-981-13-2273-0
eBook Packages: EngineeringEngineering (R0)