Parametric Model Reduction via Interpolating Orthonormal Bases

  • Ralf ZimmermannEmail author
  • Kristian Debrabant
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)


In projection-based model reduction (MOR), orthogonal coordinate systems of comparably low dimension are used to produce ansatz subspaces for the efficient emulation of large-scale numerical simulation models. Constructing such coordinate systems is costly as it requires sample solutions at specific operating conditions of the full system that is to be emulated. Moreover, when the operating conditions change, the subspace construction has to be redone from scratch.

Parametric model reduction (pMOR) is concerned with developing methods that allow for parametric adaptations without additional full system evaluations. In this work, we approach the pMOR problem via the quasi-linear interpolation of orthogonal coordinate systems. This corresponds to the geodesic interpolation of data on the Stiefel manifold. As an extension, it enables to interpolate the matrix factors of the (possibly truncated) singular value decomposition. Sample applications to a problem in mathematical finance are presented.


  1. 1.
    P.-A. Absil, R. Mahony, R. Sepulchre, Optimization Algorithms on Matrix Manifolds (Princeton University Press, Princeton, 2008)CrossRefGoogle Scholar
  2. 2.
    P. Benner, S. Gugercin, K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483–531 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    F. Black, M. Scholes, The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973)MathSciNetCrossRefGoogle Scholar
  4. 4.
    T. Bui-Thanh, M. Damodaran, K. Willcox, Proper orthogonal decomposition extensions for parametric applications in transonic aerodynamics. AIAA J. 42(8), 1505–1516 (2004)CrossRefGoogle Scholar
  5. 5.
    K. Carlberg, C. Farhat, J. Cortial, D. Amsallem, The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows. J. Comput. Phys. 242(0), 623–647 (2013)Google Scholar
  6. 6.
    S. Chaturantabut, D. Sorensen, Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Edelman, T.A. Arias, S.T. Smith, The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20(2), 303–353 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    P.A. LeGresley, J.J. Alonso, Investigation of non-linear projection for POD based reduced order models for aerodynamics, in Paper 2001-0926, 39th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 2001Google Scholar
  9. 9.
    R. Pinnau, Model reduction via proper orthogonal decomposition, in Model Order Reduction: Theory, Research Aspects and Applications, ed. by W.H.A. Schilders, H.A. Van der Vorst, J. Rommes. Springer Series Mathematics in Industry, vol. 13 (Springer, Berlin, 2008), pp. 95–109Google Scholar
  10. 10.
    Q. Rentmeesters, Algorithms for data fitting on some common homogeneous spaces, Ph.D. thesis, Université Catholique de Louvain, Louvain, Belgium, July 2013Google Scholar
  11. 11.
    R. Zimmermann, Gradient-enhanced surrogate modeling based on proper orthogonal decomposition. J. Comput. Appl. Math. 237(1), 403–418 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    R. Zimmermann, Local Parametrization of Subspaces on Matrix Manifolds via Derivative Information, pp. 379–387 (Springer, Cham, 2016)Google Scholar
  13. 13.
    R. Zimmermann, A matrix-algebraic algorithm for the Riemannian logarithm on the Stiefel manifold under the canonical metric. SIAM J. Matrix Anal. Appl. 38(2), 322–342 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    R. Zimmermann, A. Vendl, S. Görtz, Reduced-order modeling of steady flows subject to aerodynamic constraints. AIAA J. 52(2), 255–266 (2014)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of Southern DenmarkOdenseDenmark

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