- B.O. Almroth, P. Stern and F.A. Brogan, Automatic choice of global shape functions in structural analysis. AIAA Journal 16 (1978) 525–528. [CrossRef]
- Z.J. Bai, Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl. Numer. Math. 43 (2002) 9–44. [CrossRef] [MathSciNet]
- E. Balmes, Parametric families of reduced finite element models: Theory and applications. Mechanical Syst. Signal Process. 10 (1996) 381–394. [CrossRef]
- 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.
- A. Barrett and G. Reddien, On the reduced basis method. Z. Angew. Math. Mech. 75 (1995) 543–549. [MathSciNet]
- 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. [CrossRef]
- P. Erdös, Problems and results on the theory of interpolation, II. Acta Math. Acad. Sci. 12 (1961) 235–244. [CrossRef]
- 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. [CrossRef] [MathSciNet]
- 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. [CrossRef] [EDP Sciences]
- 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.
- 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. [CrossRef] [MathSciNet]
- 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.
- K. Ito and S.S. Ravindran, A reduced-order method for simulation and control of fluid flows. J. Comp. Phys. 143 (1998) 403–425. [CrossRef] [MathSciNet]
- J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non-linéaires. Dunod (1969).
- 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.
- 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.
- 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. [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.
- A.K. Noor and J.M. Peters, Reduced basis technique for nonlinear analysis of structures. AIAA Journal 18 (1980) 455–462. [CrossRef]
- J.S. Peterson, 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.
- T.A. Porsching, Estimation of the error in the reduced basis method solution of nonlinear equations. Math. Comp. 45 (1985) 487–496. [CrossRef] [MathSciNet]
- 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. [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.
- 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. [CrossRef] [MathSciNet]
- T.J. Rivlin, An introduction to the approximation of functions. Dover Publications Inc., New York (1981).
- J.M.A. Scherpen, Balancing for nonlinear systems. Syst. Control Lett. 21 (1993) 143–153. [NASA ADS] [CrossRef] [EDP Sciences] [MathSciNet] [PubMed]
- L. Sirovich, Turbulence and the dynamics of coherent structures, part 1: Coherent structures. Quart. Appl. Math. 45 (1987) 561–571. [MathSciNet]
- 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.
- 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. [CrossRef] [EDP Sciences]
- 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. [CrossRef]
Volume 41, Number 3, May-June 2007
|Page(s)||575 - 605|
|Published online||02 August 2007|