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.

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.

DOI : 10.1051/m2an/2013099
Classification : 65L20, 65M12, 65M15
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] D. Amsallem and C. Farhat, An interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J. 46 (2008) 1803-1813.

[2] D. Amsallem, J. Cortial, K. Carlberg and C. Farhat, A method for interpolating on manifolds structural dynamics reduced-order models. Int. J. Numer. Methods Eng. 80 (2009) 1241-1258. | Zbl

[3] D. Amsallem, J. Cortial and C. Farhat, Toward real-time computational-fluid-dynamics-based aeroelastic computations using a database of reduced-order information. AIAA J. 48 (2010) 2029-2037.

[4] 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.

[5] 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).

[6] A. Antoulas, Approximation of large-scale dynamical systems. SIAM, Philadelphia (2005). | MR | Zbl

[7] C. Beattie and S. Gugercin, Krylov-based model reduction of second-order systems with proportional damping, in Proc. 44th CDC/ECC (2005) 2278-2283.

[8] C. Beattie and S. Gugercin, Interpolatory projection methods for structure-preserving model reduction. Systems Control Lett. 58 (2009) 225-232. | MR | Zbl

[9] T. Bui-Thanh, M. Damodoran and K. Willcox, Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition. AIAA J. 42 (2004) 1505-1516.

[10] D. Chapelle, A. Gariah and J. Sainte-Marie, Galerkin approximation with proper orthogonal decomposition: new error estimates and illustrative examples. ESAIM: M2AN 46 (2012) 731-757. | Numdam | MR | Zbl

[11] S. Chaturantabut and D. Sorensen, A state space error estimate for POD-DEIM nonlinear model reduction. SIAM J. Numer. Anal. 50 (2012) 46-63. | MR | Zbl

[12] R. Guyan, Reduction of stiffness and mass matrices. AIAA J. 3 (1965) 380-380.

[13] S. Han and B. Feeny. Enhanced proper orthogonal decomposition for the modal analysis of homogeneous structures. J. Vibration Control 8 (2002) 19-40. | Zbl

[14] 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). | Zbl

[15] U. Hetmaniuk and R. Lehoucq, Uniform accuracy of eigenpairs from a shift-invert Lanczos method. SIAM J. Matrix Anal. Appl. 28 (2006) 927-948. | MR | Zbl

[16] C. Homescu, L. Petzold and R. Serban, Error estimation for reduced-order models of dynamical systems. SIAM Rev. 49 (2007) 277-299. | MR | Zbl

[17] T. Hughes, The finite element method: linear static and dynamic finite element analysis. Prentice-Hall (1987). | MR | Zbl

[18] D. B. Huynh, D. Knezevic and A. Patera, 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] K. Karhunen, Zur Spektraltheorie Stochastischer Prozesse. Ann. Acad. Sci. Fennicae 34 (1946). | MR | Zbl

[20] G. Kerschen, J.C. Golinval, A. Vakakis and L. Bergman, 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] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90 (2001) 117-148. | MR | Zbl

[22] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40 (2002) 492-515. | MR | Zbl

[23] 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. | MR | Zbl

[24] K. Kunisch and S. Volkwein, Optimal snapshot location for computing POD basis functions. ESAIM: M2AN 44 (2010) 509-529. | Numdam | MR | Zbl

[25] O. Lass and S. Volkwein. Adaptive POD basis computation for parameterized nonlinear systems using optimal snapshot location. Konstanzer Schriften Math. 304 (2012) 1-27.

[26] 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.

[27] T. Lieu and C. Farhat, Adaptation of aeroelastic reduced-order models and application to an F-16 configuration. AIAA J. 45 (2007) 1244-1269.

[28] M. Loeve. 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] A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics, Number 37 in Texts in Applied Mathematics. Springer (2000). | MR | Zbl

[31] M. Rathinam and L. Petzold, A new look at proper orthogonal decomposition. SIAM J. Numer. Anal. 41 (2003) 1893-1925. | MR | Zbl

[32] E. W. Sachs and M. Schu, A priori error estimates for reduced order models in finance. ESAIM: M2AN. Doi:10.1051/m2na/2012039. | Numdam | Zbl

[33] L. Sirovich, Turbulence and the dynamics of coherent structures. Parts I-II. Quarterly of Applied Mathematics XVL (1987) 561-590. | MR | Zbl

[34] 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).

[35] J.P. Thomas, E. Dowell and K. Hall, Three-dimensional transonic aeroelasticity using proper orthogonal decomposition-based reduced order models. J. Aircraft 40 (2003) 544-551.

[36] 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).

[37] K. Veroy and A. Patera, 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] 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.

Cité par Sources :