Polynomial (chaos) approximation of maximum eigenvalue functions

Efficiency and limitations

Abstract

This paper is concerned with polynomial approximations of the spectral abscissa function (defined by the supremum of the real parts of the eigenvalues) of a parameterized eigenvalue problem, which are closely related to polynomial chaos approximations if the parameters correspond to realizations of random variables. Unlike previous work, we highlight the major role of this function smoothness properties. Even if the eigenvalue problem matrices are analytic functions of the parameters, the spectral abscissa function may not be differentiable, and even non-Lipschitz continuous, due to multiple rightmost eigenvalues counted with multiplicity. This analysis demonstrates smoothness properties not only heavily affect the approximation errors of the Galerkin and collocation based polynomial approximations, but also the numerical errors in the evaluation of coefficients in the Galerkin approach with integration methods. A documentation of the experiments, conducted on the benchmark problems through the software Chebfun, is publicly available.

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References

  1. 1.

    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. Dover Publications Inc. (1965). ISBN 0486612724

  2. 2.

    Bos, L., Caliari, M., Marchi, S.D., Vianello, M., Xu, Y.: Bivariate Lagrange interpolation at the Padua points: the generating curve approach. J. Approx. Theory 143(1), 15–25 (2006). https://doi.org/10.1016/j.jat.2006.03.008

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Boussaada, I., Tliba, S., Niculescu, S.I.: A delayed feedback controller for active vibration control: a rightmost-characteristic root assignment based approach. In: 2017 21st International Conference on System Theory, Control and Computing (ICSTCC). IEEE (2017), https://doi.org/10.1109/icstcc.2017.8107073

  4. 4.

    Boussaada, I., Tliba, S., Niculescu, S.I., Ünal, H.U., Vyhlídal, T.: Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems. Application to the control of a mechanical system. Linear Algebra Appl. 542, 589–604 (2018). https://doi.org/10.1016/j.laa.2017.11.022

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Caliari, M., Marchi, S.D., Vianello, M.: Bivariate Lagrange interpolation at the Padua points: Computational aspects. J. Comput. Appl. Math. 221(2), 284–292 (2008). https://doi.org/10.1016/j.cam.2007.10.027

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Crestaux, T., Maître, O. L., Martinez, J.M.: Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Safe. 94(7), 1161–1172 (2009). https://doi.org/10.1016/j.ress.2008.10.008

    Article  Google Scholar 

  7. 7.

    Elman, H.C., Su, T.: Low-rank solution methods for stochastic eigenvalue problems. arXiv:1803.03717 (2018)

  8. 8.

    Fenzi, L., Michiels, W.: Robust stability optimization for linear delay systems in a probabilistic framework. Linear Algebra Appl. 526, 1–26 (2017). https://doi.org/10.1016/j.laa.2017.03.020

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Fenzi, L., Michiels, W.: Experiments on polynomial (chaos) approximation of maximum eigenvalue functions Tutorial. Technical report TW 688. Department of Computer Science, KU Leuven (2018)

  10. 10.

    Fenzi, L., Pilbauer, D., Michiels, W., Vyhlídal, T.: A probabilistic approach towards robust stability optimization, with application to vibration control. In: Stépán, F., Csernák, G. (eds.) Proceedings of the 9th European Nonlinear Dynamics Conference (2017)

  11. 11.

    Ghanem, R., Ghosh, D.: Efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition. Int. J. Numer. Methods Eng. 72(4), 486–504 (2007). https://doi.org/10.1002/nme.2025

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Ghosh, D., Ghanem, R.: Stochastic convergence acceleration through basis enrichment of polynomial chaos expansions. Int. J. Numer. Methods Eng. 73(2), 162–184 (2007). https://doi.org/10.1002/nme.2066

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Hakula, H., Kaarnioja, V., Laaksonen, M.: Approximate methods for stochastic eigenvalue problems. Appl. Math. Comput. 267, 664–681 (2015). https://doi.org/10.1016/j.amc.2014.12.112

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Homma, T., Saltelli, A.: Importance measures in global sensitivity analysis of nonlinear models. Reliab. Eng. Syst. Safe. 52(1), 1–17 (1996). https://doi.org/10.1016/0951-8320(96)00002-6

    Article  Google Scholar 

  15. 15.

    Kangal, F., Meerbergen, K., Mengi, E., Michiels, W.: A subspace method for large-scale eigenvalue optimization. SIAM J. Matrix Anal. Appl. 39(1), 48–82 (2018). https://doi.org/10.1137/16m1070025

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995). ISBN 354058661X

    Google Scholar 

  17. 17.

    Lan, J.C., Dong, X.J., Peng, Z.K., Zhang, W.M., Meng, G.: Uncertain eigenvalue analysis by the sparse grid stochastic collocation method. Acta Mech. Sinica 31(4), 545–557 (2015). https://doi.org/10.1007/s10409-015-0422-9

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Lȯpez, K., Mondiė, S., Garrido, R.: A tuning procedure for the cascade proportional integral retarded controller. IFAC-PapersOnLine 51(14), 61–65 (2018). https://doi.org/10.1016/j.ifacol.2018.07.199

    Article  Google Scholar 

  19. 19.

    Maître, O.P.L., Knio, O.M.: Spectral Methods for Uncertainty Quantification. Springer-Verlag GmbH. ISBN 9048135192 (2010)

  20. 20.

    Marelli, S., Sudret, B: UQLab user manual – Polynomial Chaos Expansions, Report UQLab-V1.1-104, Chair of Risk, Safety & Uncertainty Quantification, ETH Zurich (2018)

  21. 21.

    Michiels, W.: Spectrum-based stability analysis and stabilisation of systems described by delay differential algebraic equations. IET Control Theory Appl. 5 (16), 1829–1842 (2011). https://doi.org/10.1049/iet-cta.2010.0752

    MathSciNet  Article  Google Scholar 

  22. 22.

    Michiels, W., Niculescu, S.I.: Stability, control, and computation for time-delay systems. Cambridge University Press. ISBN 1611973627 (2014)

  23. 23.

    Michiels, W., Engelborghs, K., Vansevenant, P., Roose, D.: Continuous pole placement for delay equations. Automatica 38(5), 747–761 (2002). https://doi.org/10.1016/s0005-1098(01)00257-6

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Michiels, W., Jarlebring, E., Meerbergen, K.: Krylov-based model order reduction of time-delay systems. SIAM J. Matrix Anal. Appl. 32(4), 1399–1421 (2011). https://doi.org/10.1137/100797436

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Michiels, W., Boussaada, I., Niculescu, S.I.: An explicit formula for the splitting of multiple eigenvalues for nonlinear eigenvalue problems and connections with the linearization for the delay eigenvalue problem. SIAM J. Matrix Anal. Appl. 38(2), 599–620 (2017). https://doi.org/10.1137/16m107774x

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Rahman, S., Yadav, V.: Orthogonal polynomial expansion for solving random eigenvalue problems. Int. J. Uncertain. Quantif. 1(2), 163–187 (2011). https://doi.org/10.1615/intjuncertaintyquantification.v1.i2.40

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Ramirez, A., Mondiė, S., Garrido, R., Sipahi, R.: Design of proportional-integral-retarded (PIR,) controllers for second-order LTI systems. IEEE Trans. Autom. Control 61(6), 1688–1693 (2016). https://doi.org/10.1109/tac.2015.2478130

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Ramírez, A., Sipahi, R., Mondiė, S., Garrido, R: An analytical approach to tuning of delay-based controllers for LTI-SISO systems. SIAM J. Control Optim. 55(1), 397–412 (2017). https://doi.org/10.1137/15m1050999

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Saadvandi, M., Meerbergen, K., Desmet, W.: Parametric dominant pole algorithm for parametric model order reduction. J. Comput. Appl. Math. 259, 259–280 (2014). https://doi.org/10.1016/j.cam.2013.09.012

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Sarrouy, E., Dessombz, O., Sinou, J.J.: Stochastic analysis of the eigenvalue problem for mechanical systems using polynomial chaos expansion— application to a finite element rotor. J. Vib. Acoust. 134(5), 1–12 (2012). https://doi.org/10.1115/1.4005842

    Article  MATH  Google Scholar 

  31. 31.

    Sarrouy, E., Dessombz, O., Sinou, J.J.: Piecewise polynomial chaos expansion with an application to brake squeal of a linear brake system. J. Sound Vib. 332(3), 577–594 (2013). https://doi.org/10.1016/j.jsv.2012.09.009

    Article  MATH  Google Scholar 

  32. 32.

    Sobol, I.M.: Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates. Math. Comput. Simul. 55, 271–280 (2001). https://doi.org/10.1016/s0378-4754(00)00270-6

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Sommariva, A., Vianello, M., Zanovello, R.: Nontensorial Clenshaw–Curtis cubature. Numer. Algor. 49, 409–427 (2008). https://doi.org/10.1007/s11075-008-9203-x

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Szudzik, M.P.: The Rosenberg-Strong pairing function. arXiv:1706.04129 (2017)

  35. 35.

    Townsend, A., Webb, M., Olver, S.: Fast polynomial transforms based on Toeplitz and Hankel matrices. Math. Comput. 87, 1913–1934 (2018). https://doi.org/10.1090/mcom/3277

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Trefethen, L.N.: Approximation theory and approximation practice. Cambridge University Press. ISBN 1611972396 (2012)

  37. 37.

    Trefethen, L.N.: Multivariate polynomial approximation in the hypercube. Proc. Amer. Math. Soc. 145(11), 4837–4844 (2017). https://doi.org/10.1090/proc/13623

    MathSciNet  Article  MATH  Google Scholar 

  38. 38.

    Vermiglio, R.: Polynomial chaos expansions for the stability analysis of uncertain delay differential equations. SIAM/ASA J. Uncertain. Quantif. 5(1), 278–303 (2017). https://doi.org/10.1137/15m1029618

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Verriest, E.I., Michiels, W.: Inverse routh table construction and stability of delay equations. Syst. Control Lett. 55(9), 711–718 (2006). https://doi.org/10.1016/j.sysconle.2006.02.002

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Villafuerte, R., Mondiė, S., Garrido, R: Tuning of proportional retarded controllers: theory and experiments. IEEE Trans. Control Syst. Technol. 21(3), 983–990 (2013). https://doi.org/10.1109/tcst.2012.2195664

    Article  Google Scholar 

  41. 41.

    Wang, H., Xiang, S.: On the convergence rates of Legendre approximation. Math. Comput. 81(278), 861–877 (2011). https://doi.org/10.1090/s0025-5718-2011-02549-4

    MathSciNet  Article  MATH  Google Scholar 

  42. 42.

    Williams, M.M.R.: A method for solving stochastic eigenvalue problems II. Appl. Math. Comput. 219(9), 4729–4744 (2013). https://doi.org/10.1016/j.amc.2012.10.089

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Xiang, S.: An improved error bound on Gauss quadrature. Appl. Math. Lett. 58, 42–48 (2016). https://doi.org/10.1016/j.aml.2016.01.015

    MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    Xiang, S., Bornemann, F.: On the convergence rates of Gauss and Clenshaw–Curtis quadrature for functions of limited regularity. SIAM J. Numer. Anal. 50(5), 2581–2587 (2012). https://doi.org/10.1137/120869845

    MathSciNet  Article  MATH  Google Scholar 

  45. 45.

    Xiu, D.: Numerical Methods for Stochastic Computations. Princeton University Press (2010)

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Acknowledgments

The authors would like to thank A. Bultheel for the careful proofreading and his advice, and L. N. Trefethen for pointing to valuable references.

Funding

This work was supported by the project C14/17/072 of the KU Leuven Research Council, by the project G0A5317N of the Research Foundation-Flanders (FWO - Vlaanderen), and by the project UCoCoS, funded by the European Unions Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No 675080.

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Correspondence to Luca Fenzi.

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Appendix: Galerkin approach for the spectral abscissa functions in Example 1

Appendix: Galerkin approach for the spectral abscissa functions in Example 1

The spectral abscissa functions for eigenvalue problems , , and are given by

$$ \begin{array}{lllllll} \alpha^{\text{I}}(\omega)=e^{\omega},\quad \alpha^{\text{II}}(\omega) = \left\{\begin{array}{lllllll} 0, & \text{ if }\omega\in[-1,0),\\ \omega, &\text{ if }\omega\in[0,1]; \end{array}\right.\quad \alpha^{\text{III}}(\omega)=\left\{\begin{array}{llllll} 0, &\text{ if }\omega\in[-1,0),\\ \sqrt{\omega}, &\text{ if }\omega\in[0,1]; \end{array}\right. \end{array} $$
(34)

respectively. In this Appendix, we analytically compute the quantities, which appear in the evaluation of the coefficients ci in (6).

The ρ-norms of the Legendre polynomial are given by \(\|p_{i}\|_{\rho }^{2}=\frac {1}{2i + 1}\) for all \(i\in \mathbb {N}\) (see e.g. 22.2.10 in [1]).

To analytically express the coefficients ci in (6), the evaluation of the ρ-inner product of spectral abscissae (34) and the i th Legendre polynomials is needed. We start analyzing αI(ω) and we furnish the corresponding ρ-inner product iteratively, starting from:

$$\langle \alpha^{\text{I}},p_{0}\rangle_{\rho}=\frac{1}{2}\left( e-e^{-1}\right). $$

Set \(i\in \mathbb {N}\) and i ≥ 1, by using integration by parts, and the following Legendre polynomial properties

$$p_{i}(\pm1) = (\pm1)^{i}, \quad\!\text{and } \frac{\mathrm{d} p_{i + 1}(\omega)}{\mathrm{d}\omega} = \sum\limits_{k = 0}^{\lfloor i/2\rfloor}\frac{p_{i-2k}(\omega)}{\|p_{i-2k}\|_{\rho}^{2}},\quad\! \text{ where } \left\lfloor\frac{i}{2}\right\rfloor = \max_{j\in\mathbb{N}}\left\{j\!\leq\! \frac{i}{2}\right\}, $$

we get

$$\langle\alpha^{\text{I}},p_{i + 1}\rangle_{\rho} = \frac{1}{2}\left( e+\frac{(-1)^{i}}{e} - 2\sum\limits_{k = 0}^{\lfloor i/2\rfloor}\frac{\langle\alpha^{\text{I}},p_{i-2k}\rangle_{\rho}}{\|p_{i-2k}\|_{\rho}^{2}}\right) = \frac{1}{2}\left( e+\frac{(-1)^{i}}{e} - 2\sum\limits_{k = 0}^{\lfloor i/2\rfloor} {c}_{i}^{\text{I}}\right), $$

where \({c}_{i}^{\text {I}}\) indicates the i th coefficients of polynomial approximation (5) of αI(ω).

For all \(i\in \mathbb {N}\), the ρ-inner product of the spectral abscissae αII(ω) and αIII(ω) can be evaluated by using the relation 22.13.8 and 22.13.9 in [1]:

$$\begin{array}{@{}rcl@{}} \langle\alpha^{\text{II}},p_{i}\rangle_{\rho}&=&\left\{\begin{array}{llllll} \frac{(-1)^{j}{\Gamma}\left( j-\frac{1}{2}\right)}{4{\Gamma}\left( -\frac{1}{2}\right){\Gamma}\left( j + 2\right)}, &{\text{if }i = 2j},\\ \frac{1}{6}, &\text{if }i = 1,\\ 0, &\text{if }i = 2j + 1,\ i>1 \end{array}\right.\\ \langle\alpha^{\text{III}},p_{i}\rangle_{\rho}&=&\left\{\begin{array}{lllll} \frac{(-1)^{j}{\Gamma}\left( j-\frac{1}{4}\right){\Gamma}\left( \frac{3}{4}\right)}{4{\Gamma}\left( -\frac{1}{4}\right){\Gamma}\left( j+\frac{7}{4}\right)}, &\text{if }i = 2j,\\ \frac{(-1)^{j}{\Gamma}\left( j+\frac{1}{4}\right){\Gamma}\left( \frac{5}{4}\right)} {4{\Gamma}\left( \frac{1}{4}\right){\Gamma}\left( j+\frac{9}{4}\right)}, &\text{if }i = 2j + 1, \end{array}\right. \end{array} $$

where Γ(⋅) denotes the Gamma function. The two last inner product are, in fact, rational numbers which can be computed without any error via a symbolic software.

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Fenzi, L., Michiels, W. Polynomial (chaos) approximation of maximum eigenvalue functions. Numer Algor 82, 1143–1169 (2019). https://doi.org/10.1007/s11075-018-00648-9

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Keywords

  • Polynomial approximation
  • Polynomial chaos
  • Eigenvalue analysis
  • Interpolation
  • Integration methods