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

Martin A. Grepl; Yvon Maday; Ngoc C. Nguyen; Anthony T. Patera

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

- Volume: 41, Issue: 3, page 575-605
- ISSN: 0764-583X

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topGrepl, Martin A., et al. "Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations." ESAIM: Mathematical Modelling and Numerical Analysis 41.3 (2007): 575-605. <http://eudml.org/doc/250042>.

@article{Grepl2007,

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.
},

author = {Grepl, Martin A., Maday, Yvon, Nguyen, Ngoc C., Patera, Anthony T.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Reduced-basis methods; parametrized PDEs; non-affine parameter dependence; offine-online procedures; elliptic PDEs; parabolic PDEs; nonlinear PDEs.; Galerkin method; numerical results; reduced-basis methods; offline-online procedures; nonlinear PDEs},

language = {eng},

month = {8},

number = {3},

pages = {575-605},

publisher = {EDP Sciences},

title = {Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations},

url = {http://eudml.org/doc/250042},

volume = {41},

year = {2007},

}

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: Mathematical Modelling and Numerical Analysis

DA - 2007/8//

PB - EDP Sciences

VL - 41

IS - 3

SP - 575

EP - 605

AB -
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.

LA - eng

KW - Reduced-basis methods; parametrized PDEs; non-affine parameter dependence; offine-online procedures; elliptic PDEs; parabolic PDEs; nonlinear PDEs.; Galerkin method; numerical results; reduced-basis methods; offline-online procedures; nonlinear PDEs

UR - http://eudml.org/doc/250042

ER -

## References

top- B.O. Almroth, P. Stern and F.A. Brogan, Automatic choice of global shape functions in structural analysis. AIAA Journal16 (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. Zbl1012.65136
- 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. Zbl1061.65118
- A. Barrett and G. Reddien, On the reduced basis method. Z. Angew. Math. Mech.75 (1995) 543–549. Zbl0832.65047
- 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 Systems2 (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 Computing21 (2000) 1419–1434. Zbl0959.35018
- P. Erdös, Problems and results on the theory of interpolation, II. Acta Math. Acad. Sci.12 (1961) 235–244. Zbl0098.04103
- 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. Zbl0533.73071
- 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: M2AN39 (2005) 157–181. Zbl1079.65096
- 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 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. Zbl1228.65223
- 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. Zbl0885.49029
- 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. Zbl0908.93025
- K. Ito and S.S. Ravindran, A reduced-order method for simulation and control of fluid flows. J. Comp. Phys.143 (1998) 403–425. Zbl0936.76031
- J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non-linéaires. Dunod (1969). Zbl0189.40603
- 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. Zbl0960.65063
- 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. Zbl1009.65066
- 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. Zbl1038.74559
- 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 Journal18 (1980) 455–462.
- J.S. Peterson, The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput.10 (1989) 777–786. Zbl0672.76034
- 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 Systems22 (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. Zbl0586.65040
- 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). Zbl1151.65339
- A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, Texts in Applied Mathematics, Vol. 37. Springer, New York (1991). Zbl1136.65002
- 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 Systems22 (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. Zbl0802.65068
- T.J. Rivlin, An introduction to the approximation of functions. Dover Publications Inc., New York (1981). Zbl0489.41001
- J.M.A. Scherpen, Balancing for nonlinear systems. Syst. Control Lett.21 (1993) 143–153. Zbl0785.93042
- L. Sirovich, Turbulence and the dynamics of coherent structures, part 1: Coherent structures. Quart. Appl. Math.45 (1987) 561–571. Zbl0676.76047
- 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. Fluids47 (2005) 773–788. Zbl1134.76326
- 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: COCV8 (2002) 1007–1028. Special Volume: A tribute to J.-L. Lions. Zbl1092.35031
- 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.

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