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 ${W}_{N}$ 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.

Classification : 35J50, 65N15

Mots clés : elliptic partial differential equations, reduced-basis methods, output bounds, Galerkin approximation, a posteriori error estimation, convex analysis

@article{COCV_2002__8__1007_0, author = {Veroy, Karen and Rovas, Dimitrios V. and Patera, Anthony T.}, title = {A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations : ``convex inverse'' bound conditioners}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1007--1028}, publisher = {EDP-Sciences}, volume = {8}, year = {2002}, doi = {10.1051/cocv:2002041}, zbl = {1092.35031}, language = {en}, url = {www.numdam.org/item/COCV_2002__8__1007_0/} }

Veroy, Karen; Rovas, Dimitrios V.; 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, Tome 8 (2002) , pp. 1007-1028. doi : 10.1051/cocv:2002041. http://www.numdam.org/item/COCV_2002__8__1007_0/

[1] Fast exact linear and non-linear structural reanalysis and the Sherman-Morrison-Woodbury formulas. Int. J. Numer. Meth. Engrg. 50 (2001) 1587-1606. | Zbl 0971.74076

, and ,[2] Simplicial and continuation methods for approximating fixed-points and solutions to systems of equations. SIAM Rev. 22 (1980) 28-85. | MR 554709 | Zbl 0432.65027

and ,[3] Automatic choice of global shape functions in structural analysis. AIAA J. 16 (1978) 525-528.

, and ,[4] Nonlinear Programming: Analysis and Methods. Prentice-Hall, Inc., Englewood Cliffs, NJ (1976). | MR 489892 | Zbl 0361.90035

,[5] Parametric families of reduced finite element models. Theory and applications. Mech. Systems and Signal Process. 10 (1996) 381-394.

,[6] On the Reduced Basis Method. Z. Angew. Math. Mech. 75 (1995) 543-549. | MR 1347913 | Zbl 0832.65047

and ,[7] Analysis of projection methods for solving linear systems with multiple right-hand sides. SIAM J. Sci. Comput. 18 (1997) 1698. | MR 1480631 | Zbl 0888.65033

and ,[8] Extending substructure based iterative solvers to multiple load and repeated analyses. Comput. Meth. Appl. Mech. Engrg. 117 (1994) 195-209. | Zbl 0851.73059

, and ,[9] On the error behavior of the reduced basis technique for nonlinear finite element approximations. Z. Angew. Math. Mech. 63 (1983) 21-28. | MR 701832 | Zbl 0533.73071

and ,[10] Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris Sér. I Math. 331 (2000) 153-158. | MR 1781533 | Zbl 0960.65063

, , , and ,[11] 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). | Zbl 1009.65066

, and ,[12] A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations. J. Sci. Comput. (accepted). | Zbl 1014.65115

, and ,[13] Reduced basis technique for nonlinear analysis of structures. AIAA J. 18 (1980) 455-462.

and ,[14] The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10 (1989) 777-786. | MR 1000745 | Zbl 0672.76034

,[15] Estimation of the error in the reduced basis method solution of nonlinear equations. Math. Comput. 45 (1985) 487-496. | MR 804937 | Zbl 0586.65040

,[16] Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Engrg. 124 (2002) 70-80.

, , , , and ,[17] Numerical analysis of continuation methods for nonlinear structural problems. Comput. & Structures 13 (1981) 103-113. | MR 616722 | Zbl 0465.65030

,[18] On the theory and error estimation of the reduced basis method for multi-parameter problems. Nonlinear Anal. Theor. Meth. Appl. 21 (1993) 849-858. | MR 1249664 | Zbl 0802.65068

,[19] 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. | Zbl 0628.65020

,