A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations : “convex inverse” bound conditioners

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

ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 1007-1028.

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.

[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. | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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). | Zbl

[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). | Zbl

[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. | MR | Zbl

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

[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. Fluids Engrg. 124 (2002) 70-80.

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

[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. | MR | Zbl

[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. | Zbl