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Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations

Published online by Cambridge University Press:  02 August 2007

Martin A. Grepl ,
Yvon Maday ,
Ngoc C. Nguyen  and
Anthony T. Patera
Martin A. Grepl
Affiliation:
Massachusetts Institute of Technology, Room 3-264, Cambridge, MA, USA.
Yvon Maday
Affiliation:
Université Pierre et Marie Curie-Paris6, UMR 7598 Laboratoire Jacques-Louis Lions, B.C. 187, 75005 Paris, France. maday@ann.jussieu.fr Division of Applied Mathematics, Brown University.
Ngoc C. Nguyen
Affiliation:
Massachusetts Institute of Technology, Room 37-435, Cambridge, MA, USA.
Anthony T. Patera
Affiliation:
Massachusetts Institute of Technology, Room 3-266, Cambridge, MA, USA.
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Abstract

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.

Keywords

Reduced-basis methodsparametrized PDEsnon-affine parameter dependenceoffine-online procedureselliptic PDEsparabolic PDEsnonlinear PDEs.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 3 , May 2007 , pp. 575 - 605
Copyright
© EDP Sciences, SMAI, 2007

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References

Almroth, B.O., Stern, P. and Brogan, F.A., Automatic choice of global shape functions in structural analysis. AIAA Journal 16 (1978) 525–528. CrossRef
Bai, Z.J., Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl. Numer. Math. 43 (2002) 9–44. CrossRef
Balmes, E., Parametric families of reduced finite element models: Theory and applications. Mechanical Syst. Signal Process. 10 (1996) 381–394. CrossRef
Barrault, M., Maday, Y., Nguyen, N.C. and Patera, A.T., 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. CrossRef
Barrett, A. and Reddien, G., On the reduced basis method. Z. Angew. Math. Mech. 75 (1995) 543–549. CrossRef
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.
Christensen, E.A., Brøns, M. and Sørensen, J.N., Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows. SIAM J. Scientific Computing 21 (2000) 1419–1434. CrossRef
Erdös, P., Problems and results on the theory of interpolation, II. Acta Math. Acad. Sci. 12 (1961) 235–244. CrossRef
Fink, J.P. and Rheinboldt, W.C., On the error behavior of the reduced basis technique for nonlinear finite element approximations. Z. Angew. Math. Mech. 63 (1983) 21–28. CrossRef
M. Grepl, Reduced-Basis Approximations for Time-Dependent Partial Differential Equations: Application to Optimal Control. Ph.D. thesis, Massachusetts Institute of Technology (2005).
Grepl, M.A. and Patera, A.T., A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: M2AN 39 (2005) 157–181. CrossRef
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 applications, in Proceedings of the 2nd Sandia Workshop of PDE-Constrained Optimization: Towards Real-Time and On-Line PDE-Constrained Optimization, SIAM Computational Science and Engineering Book Series (2007) pp. 197–212.
Guillaume, P. and Masmoudi, M., 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. CrossRef
M.D. Gunzburger, Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms. Academic Press, Boston (1989).
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.
Ito, K. and Ravindran, S.S., A reduced-order method for simulation and control of fluid flows. J. Comp. Phys. 143 (1998) 403–425. CrossRef
J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non-linéaires. Dunod (1969).
Machiels, L., Maday, Y., Oliveira, I.B., Patera, A.T. and Rovas, D.V., Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris Sér. I Math. 331 (2000) 153–158. CrossRef
Maday, Y., Patera, A.T. and Turinici, G., 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. CrossRef
Meyer, M. and Matthies, H.G., Efficient model reduction in non-linear dynamics using the Karhunen-Loève expansion and dual-weighted-residual methods. Comp. Mech. 31 (2003) 179–191. CrossRef
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.
Noor, A.K. and Peters, J.M., Reduced basis technique for nonlinear analysis of structures. AIAA Journal 18 (1980) 455–462.
Peterson, J.S., The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10 (1989) 777–786. CrossRef
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. CrossRef
Porsching, T.A., Estimation of the error in the reduced basis method solution of nonlinear equations. Math. Comp. 45 (1985) 487–496. CrossRef
Prud'homme, C., Rovas, D., Veroy, K., Maday, Y., Patera, A.T. and Turinici, G., Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Eng. 124 (2002) 70–80. CrossRef
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer, 2nd edition (1997).
A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, Texts in Applied Mathematics, Vol. 37. Springer, New York (1991).
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. CrossRef
Rheinboldt, W.C., On the theory and error estimation of the reduced basis method for multi-parameter problems. Nonlinear Anal. Theory Methods Appl. 21 (1993) 849–858. CrossRef
T.J. Rivlin, An introduction to the approximation of functions. Dover Publications Inc., New York (1981).
Scherpen, J.M.A., Balancing for nonlinear systems. Syst. Control Lett. 21 (1993) 143–153. CrossRef
Sirovich, L., Turbulence and the dynamics of coherent structures, part 1: Coherent structures. Quart. Appl. Math. 45 (1987) 561–571. CrossRef
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.
Veroy, K., Rovas, D. and Patera, A.T., 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. CrossRef
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).
Weile, D.S., Michielssen, E. and Gallivan, K., Reduced-order modeling of multiscreen frequency-selective surfaces using Krylov-based rational interpolation. IEEE Trans. Antennas Propag. 49 (2001) 801–813. CrossRef