In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems involving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral reduced-basis approximation space, and (ii) a stable and inexpensive interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each instance, we discuss the reduced-basis approximation and the associated offline-online computational procedures. Numerical results are presented to assess our approach.
Mots-clés : reduced-basis methods, parametrized PDEs, non-affine parameter dependence, offine-online procedures, elliptic PDEs, parabolic PDEs, nonlinear PDEs
@article{M2AN_2007__41_3_575_0,
author = {Grepl, Martin A. and Maday, Yvon and Nguyen, Ngoc C. and Patera, Anthony T.},
title = {Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {575--605},
publisher = {EDP-Sciences},
volume = {41},
number = {3},
year = {2007},
doi = {10.1051/m2an:2007031},
mrnumber = {2355712},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2007031/}
}
TY - JOUR AU - Grepl, Martin A. AU - Maday, Yvon AU - Nguyen, Ngoc C. AU - Patera, Anthony T. TI - Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2007 SP - 575 EP - 605 VL - 41 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2007031/ DO - 10.1051/m2an:2007031 LA - en ID - M2AN_2007__41_3_575_0 ER -
%0 Journal Article %A Grepl, Martin A. %A Maday, Yvon %A Nguyen, Ngoc C. %A Patera, Anthony T. %T Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations %J ESAIM: Modélisation mathématique et analyse numérique %D 2007 %P 575-605 %V 41 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2007031/ %R 10.1051/m2an:2007031 %G en %F M2AN_2007__41_3_575_0
Grepl, Martin A.; Maday, Yvon; Nguyen, Ngoc C.; Patera, Anthony T. Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 3, pp. 575-605. doi : 10.1051/m2an:2007031. http://www.numdam.org/articles/10.1051/m2an:2007031/
[1] , and, Automatic choice of global shape functions in structural analysis. AIAA Journal 16 (1978) 525-528.
[2] , Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl. Numer. Math. 43 (2002) 9-44. | Zbl
[3] , Parametric families of reduced finite element models: Theory and applications. Mechanical Syst. Signal Process. 10 (1996) 381-394.
[4] ,, and, An “empirical interpolation” method: Application to efficient reduced-basis discretization of partial differential equations. C. R. Acad. Sci. Paris Sér. I Math. 339 (2004) 667-672. | Zbl
[5] and, On the reduced basis method. Z. Angew. Math. Mech. 75 (1995) 543-549. | Zbl
[6] , and, Proper orthogonal decomposition extensions for parametric applications in transonic aerodynamics (AIAA Paper 2003-4213), in Proceedings of the 15th AIAA Computational Fluid Dynamics Conference (2003).
[7] and, Model-order reduction of nonlinear MEMS devices through arclength-based Karhunen-Loéve decomposition, in Proceeding of the IEEE international Symposium on Circuits and Systems 2 (2001) 457-460.
[8] and, A quadratic method for nonlinear model order reduction, in Proceeding of the international Conference on Modeling and Simulation of Microsystems (2000) 477-480.
[9] , and, Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows. SIAM J. Scientific Computing 21 (2000) 1419-1434. | Zbl
[10] , Problems and results on the theory of interpolation, II. Acta Math. Acad. Sci. 12 (1961) 235-244. | Zbl
[11] and, On the error behavior of the reduced basis technique for nonlinear finite element approximations. Z. Angew. Math. Mech. 63 (1983) 21-28. | Zbl
[12] , Reduced-Basis Approximations for Time-Dependent Partial Differential Equations: Application to Optimal Control. Ph.D. thesis, Massachusetts Institute of Technology (2005).
[13] and, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: M2AN 39 (2005) 157-181. | Numdam | Zbl
[14] ,,, and, Certified rapid solution of parametrized partial differential equations for real-time applications2007) pp. 197-212.
[15] and, Solution to the time-harmonic Maxwell's equations in a waveguide: use of higher-order derivatives for solving the discrete problem. SIAM J. Numer. Anal. 34 (1997) 1306-1330. | Zbl
[16] , Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms. Academic Press, Boston (1989). | MR | Zbl
[17] and, A reduced basis method for control problems governed by PDEs, in Control and Estimation of Distributed Parameter Systems, W. Desch, F. Kappel and K. Kunisch Eds., Birkhäuser (1998) 153-168. | Zbl
[18] and, A reduced-order method for simulation and control of fluid flows. J. Comp. Phys. 143 (1998) 403-425. | Zbl
[19] , Quelques Méthodes de Résolution des Problèmes aux Limites Non-linéaires. Dunod (1969). | MR | Zbl
[20] ,,, and, Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris Sér. I Math. 331 (2000) 153-158. | Zbl
[21] , and, Global a priori convergence theory for reduced-basis approximation of single-parameter symmetric coercive elliptic partial differential equations. C. R. Acad. Sci. Paris Sér. I Math. 335 (2002) 289-294. | Zbl
[22] and, Efficient model reduction in non-linear dynamics using the Karhunen-Loève expansion and dual-weighted-residual methods. Comp. Mech. 31 (2003) 179-191. | Zbl
[23] , Reduced-Basis Approximation and A Posteriori Error Bounds for Nonaffine and Nonlinear Partial Differential Equations: Application to Inverse Analysis. Ph.D. thesis, Singapore-MIT Alliance, National University of Singapore (2005).
[24] , and, Certified real-time solution of parametrized partial differential equations, in Handbook of Materials Modeling, S. Yip Ed., Kluwer Academic Publishing, Springer (2005) pp. 1523-1558.
[25] and, Reduced basis technique for nonlinear analysis of structures. AIAA Journal 18 (1980) 455-462.
[26] , The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10 (1989) 777-786. | Zbl
[27] , Projection-based approaches for model reduction of weakly nonlinear systems, time-varying systems, in IEEE Transactions On Computer-Aided Design of Integrated Circuit and Systems 22 (2003) 171-187.
[28] , Estimation of the error in the reduced basis method solution of nonlinear equations. Math. Comp. 45 (1985) 487-496. | Zbl
[29] ,,,, and, Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Eng. 124 (2002) 70-80.
[30] and, Numerical Approximation of Partial Differential Equations. Springer, 2nd edition (1997). | MR | Zbl
[31] , and, Numerical Mathematics, Texts in Applied Mathematics, Vol. 37. Springer, New York (1991). | MR | Zbl
[32] and, A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices, in IEEE Transactions On Computer-Aided Design of Integrated Circuit and Systems 22 (2003) 155-170.
[33] , On the theory and error estimation of the reduced basis method for multi-parameter problems. Nonlinear Anal. Theory Methods Appl. 21 (1993) 849-858. | Zbl
[34] , An introduction to the approximation of functions. Dover Publications Inc., New York (1981). | MR | Zbl
[35] , Balancing for nonlinear systems. Syst. Control Lett. 21 (1993) 143-153. | Zbl
[36] , Turbulence and the dynamics of coherent structures, part 1: Coherent structures. Quart. Appl. Math. 45 (1987) 561-571. | Zbl
[37] , Reduced Basis Approximation and A Posteriori Error Estimation for Many-Parameter Problems. Ph.D. thesis, Massachusetts Institute of Technology (2007) (in preparation).
[38] and, Certified real-time solution of the parametrized steady incompressible Navier-stokes equations; Rigorous reduced-basis a posteriori error bounds. Internat. J. Numer. Meth. Fluids 47 (2005) 773-788. | Zbl
[39] , and, A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations: “Convex inverse” bound conditioners. ESAIM: COCV 8 (2002) 1007-1028. Special Volume: A tribute to J.-L. Lions. | Numdam | Zbl
[40] ,, and, A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations (AIAA Paper 2003-3847), in Proceedings of the 16th AIAA Computational Fluid Dynamics Conference (2003).
[41] , and, Reduced-order modeling of multiscreen frequency-selective surfaces using Krylov-based rational interpolation. IEEE Trans. Antennas Propag. 49 (2001) 801-813.
Cité par Sources :
