Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 3, p. 575-605

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.

DOI : https://doi.org/10.1051/m2an:2007031
Classification:  35J25,  35J60,  35K15,  35K55
Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {41},
number = {3},
year = {2007},
pages = {575-605},
doi = {10.1051/m2an:2007031},
zbl = {pre05289385},
mrnumber = {2355712},
language = {en},
url = {http://www.numdam.org/item/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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 3, pp. 575-605. doi : 10.1051/m2an:2007031. http://www.numdam.org/item/M2AN_2007__41_3_575_0/

 B.O. Almroth, P. Stern and F.A. Brogan, Automatic choice of global shape functions in structural analysis. AIAA Journal 16 (1978) 525-528.

 Z.J. Bai, Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl. Numer. Math. 43 (2002) 9-44. | Zbl 1012.65136

 E. Balmes, Parametric families of reduced finite element models: Theory and applications. Mechanical Syst. Signal Process. 10 (1996) 381-394.

 M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, 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 1061.65118

 A. Barrett and G. Reddien, On the reduced basis method. Z. Angew. Math. Mech. 75 (1995) 543-549. | Zbl 0832.65047

 T.T. Bui, M. Damodaran and K. Willcox, 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).

 J. Chen and S-M. Kang, 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.

 Y. Chen and J. White, A quadratic method for nonlinear model order reduction, in Proceeding of the international Conference on Modeling and Simulation of Microsystems (2000) 477-480.

 E.A. Christensen, M. Brøns and J.N. Sørensen, Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows. SIAM J. Scientific Computing 21 (2000) 1419-1434. | Zbl 0959.35018

 P. Erdös, Problems and results on the theory of interpolation, II. Acta Math. Acad. Sci. 12 (1961) 235-244. | Zbl 0098.04103

 J.P. Fink and W.C. Rheinboldt, On the error behavior of the reduced basis technique for nonlinear finite element approximations. Z. Angew. Math. Mech. 63 (1983) 21-28. | Zbl 0533.73071

 M. Grepl, Reduced-Basis Approximations for Time-Dependent Partial Differential Equations: Application to Optimal Control. Ph.D. thesis, Massachusetts Institute of Technology (2005).

 M.A. Grepl and A.T. Patera, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: M2AN 39 (2005) 157-181. | Numdam | Zbl 1079.65096

 M.A. Grepl, N.C. Nguyen, K. Veroy, A.T. Patera and G.R. Liu, Certified rapid solution of parametrized partial differential equations for real-time applications2007) pp. 197-212.

 P. Guillaume and M. Masmoudi, 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 0885.49029

 M.D. Gunzburger, Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms. Academic Press, Boston (1989). | MR 1017032 | Zbl 0697.76031

 K. Ito and S.S. Ravindran, 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 0908.93025

 K. Ito and S.S. Ravindran, A reduced-order method for simulation and control of fluid flows. J. Comp. Phys. 143 (1998) 403-425. | Zbl 0936.76031

 J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non-linéaires. Dunod (1969). | MR 259693 | Zbl 0189.40603

 L. Machiels, Y. Maday, I.B. Oliveira, A.T. Patera and D.V. Rovas, 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 0960.65063

 Y. Maday, A.T. Patera and G. Turinici, 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 1009.65066

 M. Meyer and H.G. Matthies, Efficient model reduction in non-linear dynamics using the Karhunen-Loève expansion and dual-weighted-residual methods. Comp. Mech. 31 (2003) 179-191. | Zbl 1038.74559

 N.C. Nguyen, 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).

 N.C. Nguyen, K. Veroy and A.T. Patera, 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.

 A.K. Noor and J.M. Peters, Reduced basis technique for nonlinear analysis of structures. AIAA Journal 18 (1980) 455-462.

 J.S. Peterson, The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10 (1989) 777-786. | Zbl 0672.76034

 J.R. Phillips, 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.

 T.A. Porsching, Estimation of the error in the reduced basis method solution of nonlinear equations. Math. Comp. 45 (1985) 487-496. | Zbl 0586.65040

 C. Prud'Homme, D. Rovas, K. Veroy, Y. Maday, A.T. Patera and G. Turinici, Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Eng. 124 (2002) 70-80.

 A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer, 2nd edition (1997). | MR 1299729 | Zbl 0803.65088

 A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, Texts in Applied Mathematics, Vol. 37. Springer, New York (1991). | MR 2265914 | Zbl 0957.65001

 M. Rewienski and J. White, 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.

 W.C. Rheinboldt, 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 0802.65068

 T.J. Rivlin, An introduction to the approximation of functions. Dover Publications Inc., New York (1981). | MR 634509 | Zbl 0489.41001

 J.M.A. Scherpen, Balancing for nonlinear systems. Syst. Control Lett. 21 (1993) 143-153. | Zbl 0785.93042

 L. Sirovich, Turbulence and the dynamics of coherent structures, part 1: Coherent structures. Quart. Appl. Math. 45 (1987) 561-571. | Zbl 0676.76047

 S. Sugata, Reduced Basis Approximation and A Posteriori Error Estimation for Many-Parameter Problems. Ph.D. thesis, Massachusetts Institute of Technology (2007) (in preparation).

 K. Veroy and A.T. Patera, 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 1134.76326

 K. Veroy, D. Rovas and A.T. Patera, 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 1092.35031

 K. Veroy, C. Prud'Homme, D.V. Rovas and A.T. Patera, 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).

 D.S. Weile, E. Michielssen and K. Gallivan, Reduced-order modeling of multiscreen frequency-selective surfaces using Krylov-based rational interpolation. IEEE Trans. Antennas Propag. 49 (2001) 801-813.