# 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

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topVeroy, 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- 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
- 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
- B.O. Almroth, P. Stern and F.A. Brogan, Automatic choice of global shape functions in structural analysis. AIAA J.16 (1978) 525-528.
- M. Avriel, Nonlinear Programming: Analysis and Methods. Prentice-Hall, Inc., Englewood Cliffs, NJ (1976).
- E. Balmes, Parametric families of reduced finite element models. Theory and applications. Mech. Systems and Signal Process.10 (1996) 381-394.
- A. Barrett and G. Reddien, On the Reduced Basis Method. Z. Angew. Math. Mech.75 (1995) 543-549. Zbl0832.65047
- 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
- 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
- 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
- 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 approximations of single-parameter symmetric coercive elliptic partial differential equations. C. R. Acad. Sci. Paris Sér. I Math. (submitted). Zbl1009.65066
- 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
- A.K. Noor and J.M. Peters, Reduced basis technique for nonlinear analysis of structures. AIAA J.18 (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
- T.A. Porsching, Estimation of the error in the reduced basis method solution of nonlinear equations. Math. Comput.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. FluidsEngrg. 124 (2002) 70-80.
- W.C. Rheinboldt, Numerical analysis of continuation methods for nonlinear structural problems. Comput. & Structures13 (1981) 103-113. Zbl0465.65030
- 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
- 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

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