Polynomial (chaos) approximation of maximum eigenvalue functions

Efficiency and limitations
  • Luca FenziEmail author
  • Wim Michiels
Original Paper


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.


Polynomial approximation Polynomial chaos Eigenvalue analysis Interpolation Integration methods 


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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 information

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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Computer Science, NUMA SectionKU LeuvenLeuvenBelgium

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