Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods

Journal of Fluids Engineering, Transactions of the ASME

Volumn 124, Issue 1, 2002, Pages 70-80

  Prud'homme, C ^a   Rovas, D V ^a   Veroy, K ^a   Machiels, L ^a   Maday, Y ^b   Patera, A T ^a   Turinici, G ^c  

^a Massachusetts Institute of Technology Cambridge   (United States)

^b SORBONNE UNIVERSITÉ   (France)

^c INRIA ROCQUENCOURT   (France)

Author keywords

[No Author keywords available]

Indexed keywords

FINITE ELEMENT APPROXIMATION; NONLINEAR STRUCTURAL ANALYSIS; REAL-TIME CONTROL;

MATHEMATICAL MODELS; NONLINEAR SYSTEMS; OPTIMIZATION; PARAMETER ESTIMATION; PARTIAL DIFFERENTIAL EQUATIONS; PERTURBATION TECHNIQUES;

REAL TIME SYSTEMS;

EID: 0003321083     PISSN: 00982202     EISSN: None     Source Type: Journal    
DOI: 10.1115/1.1448332     Document Type: Article

References (28)

1. Chan, T. F , and Wan, W. L., 1997, "Analysis of projection methods for solving linear systems with multiple right-hand sides," SIAM J Sci. Comput. (USA), 18, No. 6, p. 1698.
2. Farhat, C., Crivelli, L., and Roux, F , 1994, "Extending substructure based iterative solvers to multiple load and repeated analyses," Comput. Methods Appl. Mech. Eng., 117, No. 1-2, pp 195-209
3. Akgun, M. A , Garcelon, J. H., and Haftka, R. T., 2001, "Fast exact linear and non-linear structural reanalysis and the Sherman-Morrison-Woodbury formulas," Int. J. Numer, Methods Eng., 50, No 7, pp. 1587-1606.
4. Yip, E. L., 1986, "A note on the stability of solving a rank-p modification of a linear system by the Sherman-Morrison-Woodbury formula," SIAM (Soc. Ind Appl Math) J. Sci. Stat. Comput., 7, No 2, pp 507-513.
5. Allgower, E , and Georg, K., 1980, "Simplicial and continuation methods for approximating fixed-points and solutions to systems of equations," SIAM Rev. 22, No 1, pp. 28-85.
6. Rheinboldt, W., 1981, "Numerical analysis of continuation methods for nonlinear structural problems," Comput. Struct., 13, No 1-3, pp. 103-113.
7. Almroth, B. O , Stern, P., and Brogan, F A , 1978, "Automatic choice of global shape functions in structural analysis," AIAA J. 16, pp. 525-528.
8. Noor, A. K., and Peters, J M , 1980, "Reduced basis technique for nonlinear analysis of structures," AIAA J , 18, No. 4, pp. 455-462.
9. Balmes, E., 1996, "Parametric families of reduced finite element models theory and applications," Mech Syst. Signal Process , 10, No. 4, pp. 381-394.
10. Barrett, A , and Reddien, G , 1995, "On the reduced basis method," Z. Angew Math Mech , 75, No. 7, pp. 543-549.
11. Fink, J. P., and Rheinboldt, W. C., 1983, "On the error behaviour of the reduced basis technique for nonlinear finite element approximations," Z. Angew. Math. Mech , 63, pp. 21-28.
12. Peterson, J. S., 1989, "The reduced basis method for incompressible viscous flow calculations," SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput., 10, No 4, pp 777-786.
13. Porsching, T. A , 1985, "Estimation of the error in the reduced basis method solution of nonlinear equations," Math. Comput., 45, No. 172, pp. 487-496.
14. Rheinboldt, W , 1993, "On the theory and error estimation of the reduced basis method for multi-parameter problems," Nonlinear Analysis, Theory, Methods and Applications, 21, No 11, pp. 849-858.
15. Veroy, K., Leurent, T , Prud'homme, C., Rovas, D., and Patera, A., 2002, "Reliable real-time solution of parametrized elliptic partial differential equations: Application to elasticity," Proceedings SMA Symposium 2002.
16. Maday, Y., Machiels, L , Patera, A. T., and Rovas, D. V. 2000, "Blackbox reduced-basis output bound methods for shape optimization," Proceedings 12th International Domain Decomposition Conference, eds. T Chan, et al., ddm org, pp. 429-436.
17. Evans, A. G., Hutchinson, J. W., Fleck, N. A., Ashby, M. F., and Wadley, H. N. G., 2001, "The topological design of multifunctional cellular metals," Prog. Mater. Sci., 46, No. 3-4, pp. 309-327
18. Wicks, N., and Hutchinson, J W., 2001, "Optimal truss plates," Int. J Solids Struct., 38, No. 30-31, pp 5165-5183.
19. Maday, Y., Patera, A., 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 Série I. Submitted.
20. Maday, Y., Patera, A. T., and Peraire, J., 1999, "A general formulation for a posteriori bounds for output functional of partial differential equations; Application to the eigenvalue problem," C. R. Acad. Sci. Paris, Série I, 328, pp. 823-828
21. Machiels, L., Peraire, J., and Patera, A. T., 2001, "A posteriori finite element output bounds for the incompressible Navier-Stokes equations: Application to a natural convection problem," J. Comput. Phys., 172, pp. 401-425.
22. Patera, A T., and Ronquist, E. M., 2001, "A general output bound result Application to discretization and iteration error estimation and control," Math. Models Methods Appl. Sci , 11, No 4, pp. 685-712.
23. Maday, Y. Patera, A. T , and Rovas, D. V , 2001, "A blackbox reduced-basis output bound method for noncoercive linear problems," Seminaive du College de France J. L. Lions, Series in Applied Mathematics, eds. P. G. Ciarlet and P. L Lions, Elsevier-Gauthier-Villars, Vol 7, accepted, July 2002.
24. Machiels, L., Maday, Y , Oliveira, I. B., Patera, A., and Rovas, D., 2000, "Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems," C. R. Acad Sci Paris. Série I, 331, No. 2, pp. 153-158.
25. Antoulas, A., and Sorensen, D., 2001 "Approximation of large-scale dynamical systems: An overview," Technical report, Rice University.
26. Sirovich, L., and Kirby, M , 1987, "Low-dimensional procedure for the characterization of human faces," J Opt. Soc. Am. A. 4, No. 3, pp 519-524
27. Machiels, L , Patera, A., and Rovas, D. 2001, "Reduced basis output bound methods for parabolic problems," Comput. Methods Appl. Mech. Eng , Submitted.
28. Machiels, L , Maday, Y , and Patera, A T., 2000, "A flux-free nodal Neumann subproblem approach to output bounds for partial differential equations," C. R. Acad. Sci. Paris, Série I, 330, No. 3, pp. 249-254.