Galerkin reduced-order models for the semi-discrete wave equation, that preserve the second-order structure, are studied. Error bounds for the full state variables are derived in the continuous setting (when the whole trajectory is known) and in the discrete setting when the Newmark average-acceleration scheme is used on the second-order semi-discrete equation. When the approximating subspace is constructed using the proper orthogonal decomposition, the error estimates are proportional to the sums of the neglected singular values. Numerical experiments illustrate the theoretical results.
Mots clés : model order reduction, proper orthogonal decomposition, wave equation
@article{M2AN_2014__48_1_135_0,
author = {Amsallem, D. and Hetmaniuk, U.},
title = {Error estimates for {Galerkin} reduced-order models of the semi-discrete wave equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {135--163},
publisher = {EDP-Sciences},
volume = {48},
number = {1},
year = {2014},
doi = {10.1051/m2an/2013099},
mrnumber = {3177840},
zbl = {1290.65087},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2013099/}
}
TY - JOUR AU - Amsallem, D. AU - Hetmaniuk, U. TI - Error estimates for Galerkin reduced-order models of the semi-discrete wave equation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 135 EP - 163 VL - 48 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2013099/ DO - 10.1051/m2an/2013099 LA - en ID - M2AN_2014__48_1_135_0 ER -
%0 Journal Article %A Amsallem, D. %A Hetmaniuk, U. %T Error estimates for Galerkin reduced-order models of the semi-discrete wave equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 135-163 %V 48 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2013099/ %R 10.1051/m2an/2013099 %G en %F M2AN_2014__48_1_135_0
Amsallem, D.; Hetmaniuk, U. Error estimates for Galerkin reduced-order models of the semi-discrete wave equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 1, pp. 135-163. doi : 10.1051/m2an/2013099. http://archive.numdam.org/articles/10.1051/m2an/2013099/
[1] and , An interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J. 46 (2008) 1803-1813.
[2] , , and , A method for interpolating on manifolds structural dynamics reduced-order models. Int. J. Numer. Methods Eng. 80 (2009) 1241-1258. | Zbl
[3] , and , Toward real-time computational-fluid-dynamics-based aeroelastic computations using a database of reduced-order information. AIAA J. 48 (2010) 2029-2037.
[4] and , ModSpec: An open, flexible specification framework for multi-domain device modelling. 2011 IEEE/ACM International Conference on Computer-Aided Design (ICCAD) (2011) 367-374.
[5] and , 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).
[6] , Approximation of large-scale dynamical systems. SIAM, Philadelphia (2005). | MR | Zbl
[7] and , Krylov-based model reduction of second-order systems with proportional damping, in Proc. 44th CDC/ECC (2005) 2278-2283.
[8] and , Interpolatory projection methods for structure-preserving model reduction. Systems Control Lett. 58 (2009) 225-232. | MR | Zbl
[9] , and , Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition. AIAA J. 42 (2004) 1505-1516.
[10] , and , Galerkin approximation with proper orthogonal decomposition: new error estimates and illustrative examples. ESAIM: M2AN 46 (2012) 731-757. | Numdam | MR | Zbl
[11] and , A state space error estimate for POD-DEIM nonlinear model reduction. SIAM J. Numer. Anal. 50 (2012) 46-63. | MR | Zbl
[12] , Reduction of stiffness and mass matrices. AIAA J. 3 (1965) 380-380.
[13] and . Enhanced proper orthogonal decomposition for the modal analysis of homogeneous structures. J. Vibration Control 8 (2002) 19-40. | Zbl
[14] , and , Convergence analysis of Galerkin POD for linear second order evolution equations. Hamburger Beiträge zur Angewandten Math. 2011-06 (2011). | Zbl
[15] and , Uniform accuracy of eigenpairs from a shift-invert Lanczos method. SIAM J. Matrix Anal. Appl. 28 (2006) 927-948. | MR | Zbl
[16] , and , Error estimation for reduced-order models of dynamical systems. SIAM Rev. 49 (2007) 277-299. | MR | Zbl
[17] , The finite element method: linear static and dynamic finite element analysis. Prentice-Hall (1987). | MR | Zbl
[18] , and , 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. | Zbl
[19] , Zur Spektraltheorie Stochastischer Prozesse. Ann. Acad. Sci. Fennicae 34 (1946). | MR | Zbl
[20] , , and , The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlinear Dyn. 41 (2005) 147-169. | MR | Zbl
[21] and , Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90 (2001) 117-148. | MR | Zbl
[22] and , Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40 (2002) 492-515. | MR | Zbl
[23] and , 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. | MR | Zbl
[24] and , Optimal snapshot location for computing POD basis functions. ESAIM: M2AN 44 (2010) 509-529. | Numdam | MR | Zbl
[25] and . Adaptive POD basis computation for parameterized nonlinear systems using optimal snapshot location. Konstanzer Schriften Math. 304 (2012) 1-27.
[26] , , , and , 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.
[27] and , Adaptation of aeroelastic reduced-order models and application to an F-16 configuration. AIAA J. 45 (2007) 1244-1269.
[28] . Fonctions aléatoires de second ordre. C.R. Acad. Sci. Paris, 220 (1945). | Zbl
[29] Oberwolfach benchmark collection. (2005). Available at http://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/.
[30] , , and , Numerical Mathematics, Number 37 in Texts in Applied Mathematics. Springer (2000). | MR | Zbl
[31] and , A new look at proper orthogonal decomposition. SIAM J. Numer. Anal. 41 (2003) 1893-1925. | MR | Zbl
[32] and , A priori error estimates for reduced order models in finance. ESAIM: M2AN. Doi:10.1051/m2na/2012039. | Numdam | Zbl
[33] , Turbulence and the dynamics of coherent structures. Parts I-II. Quarterly of Applied Mathematics XVL (1987) 561-590. | MR | Zbl
[34] , Reduced basis methods for 2nd order wave equation: application to one dimensional seismic problem. Masters thesis, Singapore-MIT Alliance, National University of Singapore (2006).
[35] , and , Three-dimensional transonic aeroelasticity using proper orthogonal decomposition-based reduced order models. J. Aircraft 40 (2003) 544-551.
[36] , , , and , A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. AIAA Pap. 2003-3847 (2003).
[37] and , 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. | MR | Zbl
[38] , 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.
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