Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-19T13:37:15.944Z Has data issue: false hasContentIssue false
  • English
  • Français

A Mathematical and Computational Framework for Reliable Real-Time Solution of Parametrized Partial Differential Equations

Published online by Cambridge University Press:  15 October 2002

Christophe Prud'homme ,
Dimitrios V. Rovas ,
Karen Veroy  and
Anthony T. Patera
Christophe Prud'homme
Affiliation:
Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-266, 77 Massachusetts Ave., Cambridge, MA 02139, USA. prudhomm@MIT.EDU.
Dimitrios V. Rovas
Affiliation:
Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-266, 77 Massachusetts Ave., Cambridge, MA 02139, USA. prudhomm@MIT.EDU.
Karen Veroy
Affiliation:
Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-266, 77 Massachusetts Ave., Cambridge, MA 02139, USA. prudhomm@MIT.EDU.
Anthony T. Patera
Affiliation:
Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-266, 77 Massachusetts Ave., Cambridge, MA 02139, USA. prudhomm@MIT.EDU.
Get access

Abstract

We present in this article two components: these components can in fact serve various goalsindependently, though we consider them here as an ensemble. The first component is a technique forthe rapid and reliable evaluation prediction of linear functional outputs of elliptic (andparabolic) partial differential equations with affine parameter dependence. The essential features are (i) (provably) rapidly convergent globalreduced–basis approximations — Galerkin projection onto a spaceW N spanned by solutions of the governing partial differentialequation at N selected points in parameter space; (ii) a posteriori error estimation — relaxations of the error–residualequation that provide inexpensive yet sharp and rigorous bounds forthe error in the outputs of interest; and (iii) off–line/on–linecomputational procedures — methods which decouple the generationand projection stages of the approximation process. This component is ideally suited — consideringthe operation count of the online stage — for the repeated and rapid evaluation required in thecontext of parameter estimation, design, optimization, andreal–time control. The second component is a framework for distributed simulations. This frameworkcomprises a library providing the necessary abstractions/concepts for distributed simulations and asmall set of tools — namely SimTeXand SimLaB— allowing an easy manipulation of thosesimulations. While the library is the backbone of the framework and is therefore general, thevarious interfaces answer specific needs. We shall describe both components and present how theyinteract.

Keywords

Mathematical frameworkreduced-basis methodserror boundscomputational frameworksimulations repositorydistributed and parallel computingCORBAC++.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 5: Special issue on Programming , September 2002 , pp. 747 - 771
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Akgun, M.A., Garcelon, J.H. and Haftka, R.T., Fast exact linear and non-linear structural reanalysis and the Sherman-Morrison-Woodbury formulas. Int. J. Numer. Methods Engrg. 50 (2001) 1587-1606. CrossRef
Allgower, E. and Georg, K., Simplicial and continuation methods for approximating fixed-points and solutions to systems of equations. SIAM Rev. 22 (1980) 28-85. CrossRef
Almroth, B.O., Stern, P. and Brogan, F.A., Automatic choice of global shape functions in structural analysis. AIAA Journal 16 (1978) 525-528. CrossRef
Barrett, A. and Reddien, G., On the reduced basis method. Z. Angew. Math. Mech. 75 (1995) 543-549. CrossRef
Chan, T.F. and Wan, W.L., Analysis of projection methods for solving linear systems with multiple right-hand sides. SIAM J. Sci. Comput. 18 (1997) 1698-1721. CrossRef
Evans, A.G., Hutchinson, J.W., Fleck, N.A., Ashby, M.F. and Wadley, H.N.G., The topological design of multifunctional cellular metals. Prog. Mater. Sci. 46 (2001) 309-327. CrossRef
Farhat, C., Crivelli, L. and Roux, F.X., Extending substructure based iterative solvers to multiple load and repeated analyses. Comput. Methods Appl. Mech. Engrg. 117 (1994) 195-209. CrossRef
Fink, J.P. and Rheinboldt, W.C., On the error behavior of the reduced basis technique for nonlinear finite element approximations. Z. Angew. Math. Mech. 63 (1983) 21-28. CrossRef
L. Machiels, J. Peraire and A.T. Patera, 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.
Y. Maday, L. Machiels, A.T. Patera and D.V. Rovas, Blackbox reduced-basis output bound methods for shape optimization, in Proceedings 12 th International Domain Decomposition Conference, Chiba, Japan (2000) 429-436.
Y. Maday, A.T. Patera and J. Peraire, 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.
Maday, Y., Patera, A.T. and Turinici, G., 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.
Noor, A.K. and Peters, J.M., Reduced basis technique for nonlinear analysis of structures. AIAA Journal 18 (1980) 455-462.
Patera, A.T. and Rønquist, E.M., A general output bound result: Application to discretization and iteration error estimation and control. Math. Models Methods Appl. Sci. 11 (2001) 685-712. CrossRef
A.T. Patera and E.M. Rønquist, 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.
Peterson, J.S., The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10 (1989) 777-786. CrossRef
Porsching, T.A., Estimation of the error in the reduced basis method solution of nonlinear equations. Math. Comp. 45 (1985) 487-496. CrossRef
C. Prud'homme, A Framework for Reliable Real-Time Web-Based Distributed Simulations. MIT (to appear).
Prud'homme, C., Rovas, D., Veroy, K., Maday, Y., Patera, A.T. and Turinici, G., Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bounds methods. J. Fluids Engrg. 124 (2002) 70-80. CrossRef
Rheinboldt, W.C., Numerical analysis of continuation methods for nonlinear structural problems. Comput. Structures 13 (1981) 103-113. CrossRef
Rheinboldt, W.C., On the theory and error estimation of the reduced basis method for multi-parameter problems. Nonlinear Anal. 21 (1993) 849-858. CrossRef
D. Rovas, Reduced-Basis Output Bound Methods for Partial Differential Equations. Ph.D. thesis, MIT (in progress).
K. Veroy, Reduced Basis Methods Applied to Problems in Elasticity: Analysis and Applications. Ph.D. thesis, MIT (in progress).
Wicks, N. and Hutchinson, J. W., Optimal truss plates. Internat. J. Solids Structures 38 (2001) 5165-5183. CrossRef
Yip, E.L., 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. CrossRef