Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T09:37:08.087Z Has data issue: false hasContentIssue false
  • English
  • Français

Error estimates for Galerkin reduced-order models of thesemi-discrete wave equation

Published online by Cambridge University Press:  18 December 2013

D. Amsallem  and
U. Hetmaniuk
D. Amsallem
Affiliation:
Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, USA.. amsallem@stanford.edu
U. Hetmaniuk
Affiliation:
Department of Applied Maths, University of Washington, Box 353925, Seattle, WA 98195-3925, USA.; hetmaniu@uw.edu
Get access

Abstract

Galerkin reduced-order models for the semi-discrete wave equation, that preserve thesecond-order structure, are studied. Error bounds for the full state variables are derivedin the continuous setting (when the whole trajectory is known) and in the discrete settingwhen the Newmark average-acceleration scheme is used on the second-order semi-discreteequation. When the approximating subspace is constructed using the proper orthogonaldecomposition, the error estimates are proportional to the sums of the neglected singularvalues. Numerical experiments illustrate the theoretical results.

Keywords

Model order reductionproper orthogonal decompositionwave equation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 1 , January 2014 , pp. 135 - 163
Copyright
© EDP Sciences, SMAI 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amsallem, D. and Farhat, C., An interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J. 46 (2008) 1803–1813. Google Scholar
Amsallem, D., Cortial, J., Carlberg, K. and Farhat, C., A method for interpolating on manifolds structural dynamics reduced-order models. Int. J. Numer. Methods Eng. 80 (2009) 1241–1258. Google Scholar
Amsallem, D., Cortial, J. and Farhat, C., Toward real-time computational-fluid-dynamics-based aeroelastic computations using a database of reduced-order information. AIAA J. 48 (2010) 2029–2037. Google Scholar
D. Amsallem and J. Roychowdhury, ModSpec: An open, flexible specification framework for multi-domain device modelling. 2011 IEEE/ACM International Conference on Computer-Aided Design (ICCAD) (2011) 367–374.
D. Amsallem and C. Farhat, On the stability of linearized reduced-order models: descriptor vs non-descriptor form. 42nd AIAA Fluid Dynamics Conference and Exhibit (2012) 25–28 New Orleans, LA (2012).
A. Antoulas, Approximation of large-scale dynamical systems. SIAM, Philadelphia (2005).
C. Beattie and S. Gugercin, Krylov-based model reduction of second-order systems with proportional damping, in Proc. 44th CDC/ECC (2005) 2278–2283.
Beattie, C. and Gugercin, S., Interpolatory projection methods for structure-preserving model reduction. Systems Control Lett. 58 (2009) 225–232. Google Scholar
Bui-Thanh, T., Damodoran, M. and Willcox, K., Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition. AIAA J. 42 (2004) 1505–1516. Google Scholar
Chapelle, D., Gariah, A. and Sainte-Marie, J., Galerkin approximation with proper orthogonal decomposition: new error estimates and illustrative examples. ESAIM: M2AN 46 (2012) 731–757. Google Scholar
Chaturantabut, S. and Sorensen, D., A state space error estimate for POD-DEIM nonlinear model reduction. SIAM J. Numer. Anal. 50 (2012) 46–63. Google Scholar
Guyan, R., Reduction of stiffness and mass matrices. AIAA J. 3 (1965) 380–380. Google Scholar
Han, S. and Feeny, B.. Enhanced proper orthogonal decomposition for the modal analysis of homogeneous structures. J. Vibration Control 8 (2002) 19–40. Google Scholar
S. Herkt, M. Hinze and R. Pinnau, Convergence analysis of Galerkin POD for linear second order evolution equations. Hamburger Beiträge zur Angewandten Math. 2011–06 (2011).
Hetmaniuk, U. and Lehoucq, R., Uniform accuracy of eigenpairs from a shift-invert Lanczos method. SIAM J. Matrix Anal. Appl. 28 (2006) 927–948. Google Scholar
Homescu, C., Petzold, L. and Serban, R., Error estimation for reduced-order models of dynamical systems. SIAM Rev. 49 (2007) 277–299. Google Scholar
T. Hughes, The finite element method: linear static and dynamic finite element analysis. Prentice–Hall (1987).
Huynh, D. B., Knezevic, D. and Patera, A., A Laplace transform certified reduced basis method; application to the heat equation and wave equation. C.R. Acad. Sci. Paris, Série I 349 (2011) 401–405. Google Scholar
Karhunen, K., Zur Spektraltheorie Stochastischer Prozesse. Ann. Acad. Sci. Fennicae 34 (1946). Google Scholar
Kerschen, G., Golinval, J.C., Vakakis, A. and Bergman, L., The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlinear Dyn. 41 (2005) 147–169. Google Scholar
Kunisch, K. and Volkwein, S., Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90 (2001) 117–148. Google Scholar
Kunisch, K. and Volkwein, S., Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40 (2002) 492–515. Google Scholar
K. Kunisch and S. Volkwein, Crank−Nicholson Galerkin proper orthogonal decomposition approximations for a general equation in fluid dynamics. 18th GAMM Seminar on Multigrid and Related Methods for Optimization Problems, Leipzig (2002) 97–114.
Kunisch, K. and Volkwein, S., Optimal snapshot location for computing POD basis functions. ESAIM: M2AN 44 (2010) 509–529. Google Scholar
Lass, O. and Volkwein, S.. Adaptive POD basis computation for parameterized nonlinear systems using optimal snapshot location. Konstanzer Schriften Math. 304 (2012) 1–27. Google Scholar
J. Lienemann, D. Billger, E. B. Rudnyi, A. Greiner and J.G. Korvink, MEMS compact modeling meets model order reduction: examples of the application of Arnoldi methods to microsystems devices. Technical Proceedings of the 2004 Nanotechnology conference and trade show, Nanotech 2004, March 1-7, Boston, MA 2 (2004) 303–306.
Lieu, T. and Farhat, C., Adaptation of aeroelastic reduced-order models and application to an F-16 configuration. AIAA J. 45 (2007) 1244–1269. Google Scholar
M. Loeve. Fonctions aléatoires de second ordre. C.R. Acad. Sci. Paris, 220 (1945).
Oberwolfach benchmark collection. (2005). Available at http://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/.
A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics, Number 37 in Texts in Applied Mathematics. Springer (2000).
Rathinam, M. and Petzold, L., A new look at proper orthogonal decomposition. SIAM J. Numer. Anal. 41 (2003) 1893–1925. Google Scholar
E. W. Sachs and M. Schu, A priori error estimates for reduced order models in finance. ESAIM: M2AN. Doi:10.1051/m2na/2012039.
Sirovich, L., Turbulence and the dynamics of coherent structures. Parts I-II. Quarterly of Applied Mathematics XVL (1987) 561–590. Google Scholar
A. Tan, Reduced basis methods for 2nd order wave equation: application to one dimensional seismic problem. Masters thesis, Singapore-MIT Alliance, National University of Singapore (2006).
Thomas, J.P., Dowell, E. and Hall, K., Three-dimensional transonic aeroelasticity using proper orthogonal decomposition-based reduced order models. J. Aircraft 40 (2003) 544–551. Google Scholar
K. Veroy, C. Prud’homme, D. Rovas, and A. Patera, A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. AIAA Pap. 2003-3847 (2003).
Veroy, K. and Patera, A., Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-based a posteriori error bounds. Int. J. Numer. Methods Eng. 47 (2005) 773–788. Google Scholar
S. Volkwein, Model reduction using proper orthogonal decomposition. Lect. Notes (2011) 1–43. Available at http://www.math.uni-konstanz.de/numerik/personen/volkwein/teaching/POD-Vorlesung.pdf.