A Posteriori Error Estimation for Reduced-Basis Approximation of Parametrized Elliptic Coercive Partial Differential Equations: “Convex Inverse” Bound Conditioners

Karen Veroy; Dimitrios V. Rovas; Anthony T. Patera

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 8, page 1007-1028
  • ISSN: 1292-8119

Abstract

top
We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with affine parameter dependence. The essential components are (i ) (provably) rapidly convergent global reduced-basis approximations – Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii ) a posteriori error estimation – relaxations of the error-residual equation that provide inexpensive bounds for the error in the outputs of interest; and ( iii ) off-line/on-line computational procedures – methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage – in which, given a new parameter value, we calculate the output of interest and associated error bound – depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. In our earlier work we develop a rigorous a posteriori error bound framework for reduced-basis approximations of elliptic coercive equations. The resulting error estimates are, in some cases, quite sharp: the ratio of the estimated error in the output to the true error in the output, or effectivity , is close to (but always greater than) unity. However, in other cases, the necessary “bound conditioners” – in essence, operator preconditioners that (i ) satisfy an additional spectral “bound” requirement, and (ii ) admit the reduced-basis off-line/on-line computational stratagem – either can not be found, or yield unacceptably large effectivities. In this paper we introduce a new class of improved bound conditioners: the critical innovation is the direct approximation of the parametric dependence of the inverse of the operator (rather than the operator itself); we thereby accommodate higher-order (e.g., piecewise linear) effectivity constructions while simultaneously preserving on-line efficiency. Simple convex analysis and elementary approximation theory suffice to prove the necessary bounding and convergence properties.

How to cite

top

Veroy, Karen, Rovas, Dimitrios V., and Patera, Anthony T.. "A Posteriori Error Estimation for Reduced-Basis Approximation of Parametrized Elliptic Coercive Partial Differential Equations: “Convex Inverse” Bound Conditioners." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 1007-1028. <http://eudml.org/doc/90639>.

@article{Veroy2010,
abstract = { We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with affine parameter dependence. The essential components are (i ) (provably) rapidly convergent global reduced-basis approximations – Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii ) a posteriori error estimation – relaxations of the error-residual equation that provide inexpensive bounds for the error in the outputs of interest; and ( iii ) off-line/on-line computational procedures – methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage – in which, given a new parameter value, we calculate the output of interest and associated error bound – depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. In our earlier work we develop a rigorous a posteriori error bound framework for reduced-basis approximations of elliptic coercive equations. The resulting error estimates are, in some cases, quite sharp: the ratio of the estimated error in the output to the true error in the output, or effectivity , is close to (but always greater than) unity. However, in other cases, the necessary “bound conditioners” – in essence, operator preconditioners that (i ) satisfy an additional spectral “bound” requirement, and (ii ) admit the reduced-basis off-line/on-line computational stratagem – either can not be found, or yield unacceptably large effectivities. In this paper we introduce a new class of improved bound conditioners: the critical innovation is the direct approximation of the parametric dependence of the inverse of the operator (rather than the operator itself); we thereby accommodate higher-order (e.g., piecewise linear) effectivity constructions while simultaneously preserving on-line efficiency. Simple convex analysis and elementary approximation theory suffice to prove the necessary bounding and convergence properties. },
author = {Veroy, Karen, Rovas, Dimitrios V., Patera, Anthony T.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Elliptic partial differential equations; reduced-basis methods; output bounds; Galerkin approximation; a posteriori error estimation; convex analysis.; elliptic partial differential equations; output bounds; a posteriori error estimation; convex analysis},
language = {eng},
month = {3},
pages = {1007-1028},
publisher = {EDP Sciences},
title = {A Posteriori Error Estimation for Reduced-Basis Approximation of Parametrized Elliptic Coercive Partial Differential Equations: “Convex Inverse” Bound Conditioners},
url = {http://eudml.org/doc/90639},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Veroy, Karen
AU - Rovas, Dimitrios V.
AU - Patera, Anthony T.
TI - A Posteriori Error Estimation for Reduced-Basis Approximation of Parametrized Elliptic Coercive Partial Differential Equations: “Convex Inverse” Bound Conditioners
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 1007
EP - 1028
AB - We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with affine parameter dependence. The essential components are (i ) (provably) rapidly convergent global reduced-basis approximations – Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii ) a posteriori error estimation – relaxations of the error-residual equation that provide inexpensive bounds for the error in the outputs of interest; and ( iii ) off-line/on-line computational procedures – methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage – in which, given a new parameter value, we calculate the output of interest and associated error bound – depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. In our earlier work we develop a rigorous a posteriori error bound framework for reduced-basis approximations of elliptic coercive equations. The resulting error estimates are, in some cases, quite sharp: the ratio of the estimated error in the output to the true error in the output, or effectivity , is close to (but always greater than) unity. However, in other cases, the necessary “bound conditioners” – in essence, operator preconditioners that (i ) satisfy an additional spectral “bound” requirement, and (ii ) admit the reduced-basis off-line/on-line computational stratagem – either can not be found, or yield unacceptably large effectivities. In this paper we introduce a new class of improved bound conditioners: the critical innovation is the direct approximation of the parametric dependence of the inverse of the operator (rather than the operator itself); we thereby accommodate higher-order (e.g., piecewise linear) effectivity constructions while simultaneously preserving on-line efficiency. Simple convex analysis and elementary approximation theory suffice to prove the necessary bounding and convergence properties.
LA - eng
KW - Elliptic partial differential equations; reduced-basis methods; output bounds; Galerkin approximation; a posteriori error estimation; convex analysis.; elliptic partial differential equations; output bounds; a posteriori error estimation; convex analysis
UR - http://eudml.org/doc/90639
ER -

References

top
  1. M.A. Akgun, J.H. Garcelon and R.T. Haftka, Fast exact linear and non-linear structural reanalysis and the Sherman-Morrison-Woodbury formulas. Int. J. Numer. Meth. Engrg.50 (2001) 1587-1606.  Zbl0971.74076
  2. E. Allgower and K. Georg, Simplicial and continuation methods for approximating fixed-points and solutions to systems of equations. SIAM Rev.22 (1980) 28-85.  Zbl0432.65027
  3. B.O. Almroth, P. Stern and F.A. Brogan, Automatic choice of global shape functions in structural analysis. AIAA J.16 (1978) 525-528.  
  4. M. Avriel, Nonlinear Programming: Analysis and Methods. Prentice-Hall, Inc., Englewood Cliffs, NJ (1976).  
  5. E. Balmes, Parametric families of reduced finite element models. Theory and applications. Mech. Systems and Signal Process.10 (1996) 381-394.  
  6. A. Barrett and G. Reddien, On the Reduced Basis Method. Z. Angew. Math. Mech.75 (1995) 543-549.  Zbl0832.65047
  7. T.F. Chan and W.L. Wan, Analysis of projection methods for solving linear systems with multiple right-hand sides. SIAM J. Sci. Comput.18 (1997) 1698.  Zbl0888.65033
  8. C. Farhat, L. Crivelli and F.X. Roux, Extending substructure based iterative solvers to multiple load and repeated analyses. Comput. Meth. Appl. Mech. Engrg.117 (1994) 195-209.  Zbl0851.73059
  9. 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
  10. 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
  11. Y. Maday, A.T. Patera and G. Turinici, Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations. C. R. Acad. Sci. Paris Sér. I Math. (submitted).  Zbl1009.65066
  12. Y. Maday, A.T. Patera and G. Turinici, A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations. J. Sci. Comput. (accepted).  Zbl1014.65115
  13. A.K. Noor and J.M. Peters, Reduced basis technique for nonlinear analysis of structures. AIAA J.18 (1980) 455-462.  
  14. J.S. Peterson, The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput.10 (1989) 777-786.  Zbl0672.76034
  15. T.A. Porsching, Estimation of the error in the reduced basis method solution of nonlinear equations. Math. Comput.45 (1985) 487-496.  Zbl0586.65040
  16. 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. FluidsEngrg. 124 (2002) 70-80.  
  17. W.C. Rheinboldt, Numerical analysis of continuation methods for nonlinear structural problems. Comput. & Structures13 (1981) 103-113.  Zbl0465.65030
  18. W.C. Rheinboldt, On the theory and error estimation of the reduced basis method for multi-parameter problems. Nonlinear Anal. Theor. Meth. Appl.21 (1993) 849-858.  Zbl0802.65068
  19. E.L. Yip, A note on the stability of solving a rank-p modification of a linear system by the Sherman-Morrison-Woodbury formula. SIAM J. Sci. Stat. Comput.7 (1986) 507-513.  Zbl0628.65020

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.