# A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations

Martin A. Grepl; Anthony T. Patera

- Volume: 39, Issue: 1, page 157-181
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topGrepl, Martin A., and Patera, Anthony T.. "A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 39.1 (2005): 157-181. <http://eudml.org/doc/245117>.

@article{Grepl2005,

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 $W_N$ 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.},

author = {Grepl, Martin A., Patera, Anthony T.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {parabolic partial differential equations; diffusion equation; parameter-dependent systems; reduced-basis methods; output bounds; Galerkin approximation; a posteriori error estimation; parabolic equations; numerical results},

language = {eng},

number = {1},

pages = {157-181},

publisher = {EDP-Sciences},

title = {A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations},

url = {http://eudml.org/doc/245117},

volume = {39},

year = {2005},

}

TY - JOUR

AU - Grepl, Martin A.

AU - Patera, Anthony T.

TI - A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2005

PB - EDP-Sciences

VL - 39

IS - 1

SP - 157

EP - 181

AB - 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 $W_N$ 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.

LA - eng

KW - parabolic partial differential equations; diffusion equation; parameter-dependent systems; reduced-basis methods; output bounds; Galerkin approximation; a posteriori error estimation; parabolic equations; numerical results

UR - http://eudml.org/doc/245117

ER -

## References

top- [1] B.O. Almroth, P. Stern and F.A. Brogan, Automatic choice of global shape functions in structural analysis. AIAA J. 16 (1978) 525–528.
- [2] J.A. Atwell and B.B. King, Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math. Comput. Mod. 33 (2001) 1–19. Zbl0964.93032
- [3] E. Balmes, Parametric families of reduced finite element models: Theory and applications. Mech. Syst. Signal Process. 10 (1996) 381–394.
- [4] E. Balsa-Canto, A.A. Alonso and J.R. Banga, Reduced-order models for nonlinear distributed process systems and their application in dynamic optimization. Indust. Engineering Chemistry Res. 43 (2004) 3353–3363.
- [5] H.T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems. Systems & Control: Foundations & Applications. Birkhäuser (1989). Zbl0695.93020MR1045629
- [6] 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. Zbl1061.65118
- [7] A. Barrett and G. Reddien, On the reduced basis method. Z. Angew. Math. Mech. 75 (1995) 543–549. Zbl0832.65047
- [8] R. Becker and R. Rannacher, Weighted a posteriori error control in finite element methods. In ENUMATH 95 Proc. World Sci. Publ., Singapore (1997). Zbl0968.65083
- [9] D. Bertsimas and J.N. Tsitsiklis, Introduction to Linear Optimization. Athena Scientific (1997). Zbl0997.90505
- [10] 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. Sci. Comput. 21 (2000) 1419–1434. Zbl0959.35018
- [11] 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). Zbl0889.00041MR1627934
- [12] N.H. El-Farra and P.D. Christofides, Coordinating feedback and switching for control of spatially distributed processes. Comput. Chemical Engineering 28 (2004) 111–128.
- [13] 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
- [14] 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).
- [15] 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). Zbl0921.00021MR1725022
- [16] 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
- [17] K. Ito and S.S. Ravindran, A reduced-order method for simulation and control of fluid flows. J. Comput. Phys. 143 (1998) 403–425. Zbl0936.76031
- [18] K. Ito and S.S. Ravindran, Reduced basis method for optimal control of unsteady viscous flows. Int. J. Comput. Fluid Dyn. 15 (2001) 97–113. Zbl1036.76011
- [19] S. Lall, J.E. Marsden and S. Glavaski, A subspace approach to balanced truncation for model reduction of nonlinear control systems. Int. J. Robust Nonlinear Control 12 (2002) 519–535. Zbl1006.93010
- [20] M. Lin Lee, Estimation of the error in the reduced basis method solution of differential algebraic equation systems. SIAM J. Numer. Anal. 28 (1991) 512–528. Zbl0737.65058
- [21] J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer (1971). Zbl0203.09001MR271512
- [22] 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. Zbl0960.65063
- [23] 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. Zbl1006.65124
- [24] M. Mattingly, E.A. Bailey, A.W. Dutton, R.B. Roemer and S. Devasia, Reduced-order modeling for hyperthermia: An extended balanced-realization-based approach. IEEE Transactions on Biomedical Engineering 45 (1998) 1154–1162.
- [25] M. Mattingly, R.B. Roemer and S. Devasia, Exact temperature tracking for hyperthermia: A model-based approach. IEEE Trans. Control Systems Technology 8 (2000) 979–992.
- [26] B.C. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Automat. Control 26 (1981) 17–32. Zbl0464.93022
- [27] D.A. Nagy, Modal representation of geometrically nonlinear behaviour by the finite element method. Comput. Structures 10 (1979) 683–688. Zbl0406.73071
- [28] A.K. Noor and J.M. Peters, Reduced basis technique for nonlinear analysis of structures. AIAA J. 18 (1980) 455–462.
- [29] 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). Zbl1171.65404
- [30] H.M. Park, T.Y. Yoon and O.Y. Kim, Optimal control of rapid thermal processing systems by empirical reduction of modes. Ind. Eng. Chem. Res. 38 (1999) 3964–3975.
- [31] J.S. Peterson, The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10 (1989) 777–786. Zbl0672.76034
- [32] T.A. Porsching, Estimation of the error in the reduced basis method solution of nonlinear equations. Math. Comp. 45 (1985) 487–496. Zbl0586.65040
- [33] T.A. Porsching and M. Lin Lee, The reduced basis method for initial value problems. SIAM J. Numer. Anal. 24 (1987) 1277–1287. Zbl0639.65039
- [34] 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 Engineering 124 (2002) 70–80.
- [35] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer, 2nd edition (1997). Zbl0803.65088MR1299729
- [36] S.S. Ravindaran, A reduced-order approach for optimal control of fluids using proper orthogonal decomposition. Int. J. Numer. Meth. Fluids 34 (2000) 425–448. Zbl1005.76020
- [37] W.C. Rheinboldt, On the theory and error estimation of the reduced basis method for multi-parameter problems. Nonlinear Anal. 21 (1993) 849–858. Zbl0802.65068
- [38] D.V. Rovas, L. Machiels and Y. Maday, Reduced-basis output bound methods for parabolic problems. IMA J. Appl. Math. (2005) (submitted). Zbl1101.65099MR2241309
- [39] D.V. Rovas, Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA (2002).
- [40] L. Sirovich and M. Kirby, Low-dimensional procedure for the characterization of human faces. J. Opt. Soc. Amer. A 4 (1987) 519–524.
- [41] 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). Zbl1134.76326MR2123791
- [42] 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. Zbl1036.65075
- [43] 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).
- [44] 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. Zbl1092.35031
- [45] K. Willcox and J. Peraire, Balanced model reduction via the proper orthogonal decomposition, in 15th AIAA Computational Fluid Dynamics Conference, AIAA (June 2001).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.