We present in this article two components: these components can in fact serve various goals independently, though we consider them here as an ensemble. The first component is a technique for the rapid and reliable evaluation prediction of linear functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential features are (i) (provably) rapidly convergent global reduced-basis approximations - Galerkin projection onto a space spanned by solutions of the governing partial differential equation at selected points in parameter space; (ii) a posteriori error estimation - relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous 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. This component is ideally suited - considering the operation count of the online stage - for the repeated and rapid evaluation required in the context of parameter estimation, design, optimization, and real-time control. The second component is a framework for distributed simulations. This framework comprises a library providing the necessary abstractions/concepts for distributed simulations and a small set of tools - namely SimTeXand SimLaB- allowing an easy manipulation of those simulations. While the library is the backbone of the framework and is therefore general, the various interfaces answer specific needs. We shall describe both components and present how they interact.
Mots-clés : mathematical framework, reduced-basis methods, error bounds, computational framework, simulations repository, distributed and parallel computing, CORBA, C++
@article{M2AN_2002__36_5_747_0,
author = {Prud'homme, Christophe and Rovas, Dimitrios V. and Veroy, Karen and Patera, Anthony T.},
title = {A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {747--771},
publisher = {EDP-Sciences},
volume = {36},
number = {5},
year = {2002},
doi = {10.1051/m2an:2002035},
zbl = {1024.65104},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2002035/}
}
TY - JOUR AU - Prud'homme, Christophe AU - Rovas, Dimitrios V. AU - Veroy, Karen AU - Patera, Anthony T. TI - A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2002 SP - 747 EP - 771 VL - 36 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2002035/ DO - 10.1051/m2an:2002035 LA - en ID - M2AN_2002__36_5_747_0 ER -
%0 Journal Article %A Prud'homme, Christophe %A Rovas, Dimitrios V. %A Veroy, Karen %A Patera, Anthony T. %T A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations %J ESAIM: Modélisation mathématique et analyse numérique %D 2002 %P 747-771 %V 36 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2002035/ %R 10.1051/m2an:2002035 %G en %F M2AN_2002__36_5_747_0
Prud'homme, Christophe; Rovas, Dimitrios V.; Veroy, Karen; Patera, Anthony T. A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations. ESAIM: Modélisation mathématique et analyse numérique, Special issue on Programming, Tome 36 (2002) no. 5, pp. 747-771. doi : 10.1051/m2an:2002035. http://www.numdam.org/articles/10.1051/m2an:2002035/
[1] , and, Fast exact linear and non-linear structural reanalysis and the Sherman-Morrison-Woodbury formulas. Int. J. Numer. Methods Engrg. 50 (2001) 1587-1606. | Zbl
[2] and, Simplicial and continuation methods for approximating fixed-points and solutions to systems of equations. SIAM Rev. 22 (1980) 28-85. | Zbl
[3] , and, Automatic choice of global shape functions in structural analysis. AIAA Journal 16 (1978) 525-528.
[4] and, On the reduced basis method. Z. Angew. Math. Mech. 75 (1995) 543-549. | Zbl
[5] and, Analysis of projection methods for solving linear systems with multiple right-hand sides. SIAM J. Sci. Comput. 18 (1997) 1698-1721. | Zbl
[6] ,,, and, The topological design of multifunctional cellular metals. Prog. Mater. Sci. 46 (2001) 309-327.
[7] , and, Extending substructure based iterative solvers to multiple load and repeated analyses. Comput. Methods Appl. Mech. Engrg. 117 (1994) 195-209. | Zbl
[8] and, On the error behavior of the reduced basis technique for nonlinear finite element approximations. Z. Angew. Math. Mech. 63 (1983) 21-28. | Zbl
[9] , and, A posteriori finite element output bounds for the incompressible Navier-Stokes equations; Application to a natural convection problem. J. Comput. Phys. 172 (2001) 401-425. | Zbl
[10] ,, and, Blackbox reduced-basis output bound methods for shape optimization2000) 429-436.
[11] , and, A general formulation for a posteriori bounds for output functionals of partial differential equations; Application to the eigenvalue problem. C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 823-828. | Zbl
[12] , and, Global a priori convergence theory for reduced-basis approximation of single-parameter symmetric coercive elliptic partial differential equations. C. R. Acad. Sci. Paris Sér. I Math. 335 (2002) 1-6. | Zbl
[13] and, Reduced basis technique for nonlinear analysis of structures. AIAA Journal 18 (1980) 455-462.
[14] and, A general output bound result: Application to discretization and iteration error estimation and control. Math. Models Methods Appl. Sci. 11 (2001) 685-712. | Zbl
[15] and, A general output bound result: Application to discretization and iteration error estimation and control. Math. Models Methods Appl. Sci. (2000). MIT FML Report 98-12-1. | Zbl
[16] , The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10 (1989) 777-786. | Zbl
[17] , Estimation of the error in the reduced basis method solution of nonlinear equations. Math. Comp. 45 (1985) 487-496. | Zbl
[18] , A Framework for Reliable Real-Time Web-Based Distributed Simulations. MIT (to appear).
[19] ,,,, and, Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bounds methods. J. Fluids Engrg. 124 (2002) 70-80.
[20] , Numerical analysis of continuation methods for nonlinear structural problems. Comput. Structures 13 (1981) 103-113. | Zbl
[21] , On the theory and error estimation of the reduced basis method for multi-parameter problems. Nonlinear Anal. 21 (1993) 849-858. | Zbl
[22] , Reduced-Basis Output Bound Methods for Partial Differential Equations. Ph.D. thesis, MIT (in progress).
[23] , Reduced Basis Methods Applied to Problems in Elasticity: Analysis and Applications. Ph.D. thesis, MIT (in progress).
[24] and, Optimal truss plates. Internat. J. Solids Structures 38 (2001) 5165-5183. | Zbl
[25] , A note on the stability of solving a rank- modification of a linear system by the Sherman-Morrison-Woodbury formula. SIAM J. Sci. Stat. Comput. 7 (1986) 507-513. | Zbl
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