Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-18T06:33:40.699Z Has data issue: false hasContentIssue false
  • English
  • Français

A posteriori error bounds for reduced-basis approximations ofparametrized parabolic partial differential equations

Published online by Cambridge University Press:  15 March 2005

Martin A. Grepl  and
Anthony T. Patera
Martin A. Grepl
Affiliation:
Massachusetts Institute of Technology, Room 3-264, Cambridge, MA, USA.
Anthony T. Patera
Affiliation:
Massachusetts Institute of Technology, Room 3-266, Cambridge, MA, USA. patera@mit.edu (corresponding author).
Get access

Abstract

In this paper, we extend the reduced-basis methods and associated a posteriori error estimators developed earlier for elliptic partial differential equations to parabolic problems with affine parameter dependence. The essential new ingredient is the presence of time in the formulation and solution of the problem – we shall “simply” treat time as an additional, albeit special, parameter. First, we introduce the reduced-basis recipe – Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter-time space – and develop a new greedy adaptive procedure to “optimally” construct the parameter-time sample set. Second, we propose error estimation and adjoint procedures that provide rigorous and sharp bounds for the error in specific outputs of interest: the estimates serve a priori to construct our samples, and a posteriori to confirm fidelity. Third, based on the assumption of affine parameter dependence, we develop offline-online computational procedures: in the offline stage, we generate the reduced-basis space; in the online stage, given a new parameter value, we calculate the reduced-basis output and associated error bound. The operation count for the online stage depends only on N (typically small) and the parametric complexity of the problem; the method is thus ideally suited for repeated, rapid, reliable evaluation of input-output relationships in the many-query or real-time contexts.

Keywords

Parabolic partial differential equationsdiffusion equationparameter-dependent systemsreduced-basis methodsoutput boundsGalerkin approximationa posteriori error estimation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 1 , January 2005 , pp. 157 - 181
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Almroth, B.O., Stern, P. and Brogan, F.A., Automatic choice of global shape functions in structural analysis. AIAA J. 16 (1978) 525–528. CrossRef
Atwell, J.A. and King, B.B., Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math. Comput. Mod. 33 (2001) 1–19. CrossRef
Balmes, E., Parametric families of reduced finite element models: Theory and applications. Mech. Syst. Signal Process. 10 (1996) 381–394. CrossRef
Balsa-Canto, E., Alonso, A.A. and Banga, J.R., Reduced-order models for nonlinear distributed process systems and their application in dynamic optimization. Indust. Engineering Chemistry Res. 43 (2004) 3353–3363. CrossRef
H.T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems. Systems & Control: Foundations & Applications. Birkhäuser (1989).
M. Barrault, N.C. Nguyen, Y. Maday 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. 339 (2004) 667–672.
Barrett, A. and Reddien, G., On the reduced basis method. Z. Angew. Math. Mech. 75 (1995) 543–549. CrossRef
R. Becker and R. Rannacher, Weighted a posteriori error control in finite element methods. In ENUMATH 95 Proc. World Sci. Publ., Singapore (1997).
D. Bertsimas and J.N. Tsitsiklis, Introduction to Linear Optimization. Athena Scientific (1997).
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. Sci. Comput. 21 (2000) 1419–1434. CrossRef
W. Desch, F. Kappel and K. Kunisch, Eds., Control and Estimation of Distributed Parameter Systems, volume 126 of International Series of Numerical Mathematics. Birkhäuser (1998).
El-Farra, N.H. and Christofides, P.D., Coordinating feedback and switching for control of spatially distributed processes. Comput. Chemical Engineering 28 (2004) 111–128. 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) (in progress).
K.-H. Hoffmann, G. Leugering and F. Tröltzsch, Eds., Optimal Control of Partial Differential Equations, volume 133 of International Series of Numerical Mathematics. Birkhäuser (1998).
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. Comput. Phys. 143 (1998) 403–425. CrossRef
Ito, K. and Ravindran, S.S., Reduced basis method for optimal control of unsteady viscous flows. Int. J. Comput. Fluid Dyn. 15 (2001) 97–113. CrossRef
Lall, S., Marsden, J.E. and Glavaski, S., A subspace approach to balanced truncation for model reduction of nonlinear control systems. Int. J. Robust Nonlinear Control 12 (2002) 519–535. CrossRef
Lin Le, M.e, Estimation of the error in the reduced basis method solution of differential algebraic equation systems. SIAM J. Numer. Anal. 28 (1991) 512–528.
J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer (1971).
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 331 (2000) 153–158.
Y. Maday, A.T. Patera and D.V. Rovas, A blackbox reduced-basis output bound method for noncoercive linear problems, in Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar Volume XIV, D. Cioranescu and J.-L. Lions Eds., Elsevier Science B.V. (2002) 533–569.
Mattingly, M., Bailey, E.A., Dutton, A.W., Roemer, R.B. and Devasia, S., Reduced-order modeling for hyperthermia: An extended balanced-realization-based approach. IEEE Transactions on Biomedical Engineering 45 (1998) 1154–1162. CrossRef
Mattingly, M., Roemer, R.B. and Devasia, S., Exact temperature tracking for hyperthermia: A model-based approach. IEEE Trans. Control Systems Technology 8 (2000) 979–992. CrossRef
Moore, B.C., Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Automat. Control 26 (1981) 17–32. CrossRef
Nagy, D.A., Modal representation of geometrically nonlinear behaviour by the finite element method. Comput. Structures 10 (1979) 683–688. CrossRef
Noor, A.K. and Peters, J.M., Reduced basis technique for nonlinear analysis of structures. AIAA J. 18 (1980) 455–462.
I.B. Oliveira and A.T. Patera, Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations. Optim. Engineering (2005) (submitted).
Park, H.M., Yoon, T.Y. and Kim, O.Y., Optimal control of rapid thermal processing systems by empirical reduction of modes. Ind. Eng. Chem. Res. 38 (1999) 3964–3975. CrossRef
Peterson, J.S., The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10 (1989) 777–786. CrossRef
Porsching, T.A., Estimation of the error in the reduced basis method solution of nonlinear equations. Math. Comp. 45 (1985) 487–496. CrossRef
Porsching, T.A. and Lin Le, M.e, The reduced basis method for initial value problems. SIAM J. Numer. Anal. 24 (1987) 1277–1287. 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 Engineering 124 (2002) 70–80. CrossRef
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer, 2nd edition (1997).
Ravindaran, S.S., A reduced-order approach for optimal control of fluids using proper orthogonal decomposition. Int. J. Numer. Meth. Fluids 34 (2000) 425–448. 3.0.CO;2-W>CrossRef
Rheinboldt, W.C., On the theory and error estimation of the reduced basis method for multi-parameter problems. Nonlinear Anal. 21 (1993) 849–858. CrossRef
D.V. Rovas, L. Machiels and Y. Maday, Reduced-basis output bound methods for parabolic problems. IMA J. Appl. Math. (2005) (submitted).
D.V. Rovas, Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA (2002).
Sirovich, L. and Kirby, M., Low-dimensional procedure for the characterization of human faces. J. Opt. Soc. Amer. A 4 (1987) 519–524. CrossRef
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. Methods Fluids (2005) (to appear).
K. Veroy, C. Prud'homme and A.T. Patera, Reduced-basis approximation of the viscous Burgers equation: Rigorous a posteriori error bounds. C. R. Acad. Sci. Paris, Sér. I 337 (2003) 619–624. 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 (June 2003).
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. Willcox and J. Peraire, Balanced model reduction via the proper orthogonal decomposition, in 15th AIAA Computational Fluid Dynamics Conference, AIAA (June 2001).