A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations

Computer Methods in Applied Mechanics and Engineering

Volumn 199, Issue 23-24, 2010, Pages 1603-1626

  Nouy, Anthony ^a  

^a UNIVERSITÉ DE NANTES   (France)

Author keywords

Evolution problems; Generalized Spectral Decomposition; Model reduction; Proper Generalized Decomposition (PGD); Proper Orthogonal Decomposition (POD); Separation of variables

Indexed keywords

EVOLUTION PROBLEM; GENERALIZED SPECTRAL DECOMPOSITION; PROPER ORTHOGONAL DECOMPOSITIONS; SEPARATION OF VARIABLES; SPECTRAL DECOMPOSITION;

COMPUTER SIMULATION; EIGENVALUES AND EIGENFUNCTIONS; GALERKIN METHODS; PARTIAL DIFFERENTIAL EQUATIONS; TENSORS;

COMPUTATIONAL FLUID DYNAMICS;

EID: 77951238456     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2010.01.009     Document Type: Article

References (38)

1. Ammar A., Mokdad B., Chinesta F., Keunings R. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids. J. Non-Newton. Fluid Mech. 2006, 139(3):153-176.
2. Ammar A., Mokdad B., Chinesta F., Keunings R. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids. Part II: Transient simulation using space-time separated representations. J. Non-Newton. Fluid Mech. 2007, 144(2-3):98-121.
3. Amsallem D., Farhat C. Interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J. 2008, 46(7):1803-1813.
4. Atwell J.A., King B.B. Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math. Comput. Modell. 2001, 33(1-3):1-19.
5. Berkooz G., Holmes P., Lumley J.L. The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 1993, 25:539-575.
6. Beylkin G., Mohlenkamp M.J. Algorithms for numerical analysis in high dimensions. SIAM J. Sci. Comput. 2005, 26(6):2133-2159.
7. Boyaval S., Le Bris C., Maday Y., Nguyen N.C., Patera A.T. A reduced basis approach for variational problems with stochastic parameters: application to heat conduction with variable robin coefficient. Comput. Methods Appl. Mech. Engrg. 2009, 198(41-44):3187-3206.
8. Le Bris C., Lelievre T., Maday Y. Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations. Construct. Approx. 2009, 30(3):621-651.
9. Chinesta F., Ammar A., Lemarchand F., Beauchene P., Boust F. Alleviating mesh constraints: model reduction, parallel time integration and high resolution homogenization. Comput. Methods Appl. Mech. Engrg. 2008, 197(5):400-413.
10. Christensen E.A., Brons M., Sorensen J.M. Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows. SIAM J. Sci. Comput. 2000, 21(4):1419-1434.
11. Doostan A., Iaccarino G. A least-squares approximation of partial differential equations with high-dimensional random inputs. J. Comput. Phys. 2009, 228(12):4332-4345.
12. Dureisseix D., Ladevèze P., Schrefler B.A. A computational strategy for multiphysics problems - application to poroelasticity. Int. J. Numer. Methods Engrg. 2003, 56(10):1489-1510.
13. Ekeland I., Teman R. Convex Analysis and Variational Problems. Classics in Applied Mathematics 1999, SIAM.
14. Eriksson K., Johnson C., Thomée V. Time discretization of parabolic problems by the discontinuous Galerkin method. RAIRO Model. Math. Anal. 1985, 19:611-643.
15. Gunzburger M.D., Peterson J.S., Shadid J.N. Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data. Comput. Methods Appl. Mech. Engrg. 2007, 196(4-6):1030-1047.
16. Henri T., Yvon J.-P. Convergence estimates of pod-Galerkin methods for parabolic problems. Proceedings of the 21st IFIP TC7 Conference on System Modeling and Optimization, July 21-25, 2003, Sophia Antipolis, France 2003, 295-306. Springer-Verlag.
17. A. Iollo, S. Lanteri, J.-A. Désidéri, Stability properties of POD-Galerkin approximations for the compressible Navier-Stokes equations, Technical Report 3589, INRIA, 1998.
18. Karhunen K. Zur spektraltheorie stochastischer prozesse. Ann. Acad. Sci. Fenn. 1946, 34.
19. Kunisch K., Volkwein S. Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 2001, 90(1):117-148.
20. Kunisch K., Volkwein S. Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 2002, 40(2):492-515.
21. Ladevèze P. Nonlinear Computational Structural Mechanics - New Approaches and Non-Incremental Methods of Calculation 1999, Springer-Verlag.
22. Ladevèze P., Nouy A. On a multiscale computational strategy with time and space homogenization for structural mechanics. Comput. Methods Appl. Mech. Engrg. 2003, 192:3061-3087.
23. Ladevèze P., Passieux J.C., Néron D. The LATIN multiscale computational method and the Proper Generalized Decomposition. Comput. Methods Appl. Mech. Engrg. 2010, 199(21-22):1287-1296.
24. T. Lieu, M. Lesoinne, Parameter adaptation of reduced order models for three-dimensional flutter analysis, AIAA Paper 888, 2004.
25. J.L. Lions, E. Magenes, Non-homogenous boundary value problems and applications, 1972.
26. Loève M. Fonctions aléatoires du second ordre. C.R. Acad. Sci. Paris 1945, 220.
27. Nouy A. A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg. 2007, 196(45-48):4521-4537.
28. Nouy A. Generalized spectral decomposition method for solving stochastic finite element equations: invariant subspace problem and dedicated algorithms. Comput. Methods Appl. Mech. Engrg. 2008, 197:4718-4736.
29. Nouy A. Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations. Arch. Comput. Methods Engrg. 2009, 16(3):251-285.
30. A. Nouy, Proper generalized decompositions and separated representations for the numerical solution of high dimensional stochastic problems, Arch. Comput. Methods Engrg., in press.
31. Nouy A., Ladevèze P. Multiscale computational strategy with time and space homogenization: a radial-type approximation technique for solving micro problems. Int. J. Multiscale Comput. Engrg. 2004, 170(2):557-574.
32. Nouy A., Le Maître O.P. Generalized spectral decomposition method for stochastic non-linear problems. J. Comput. Phys. 2009, 228(1):202-235.
33. Pares N., Diez P., Huerta A. Bounds of functional outputs for parabolic problems. Part 1: Exact bounds of the discontinuous Galerkin time discretization. Comput. Methods Appl. Mech. Engrg. 2008, 197(19-20):1641-1660.
34. Riesz F., Sz.-Nagy B. Functional Analysis 1990, Dover Publications, New York, NY.
35. Rovas D.V., Machiels L., Maday Y. Reduced-basis output bound methods for parabolic problems. IMA J. Numer. Anal. 2006, 26(3):423-445.
36. R. Schmidt, M. Glauser, Improvements in low dimensional tools for flow-structure interaction problems: using global pod, AIAA Paper 2004-0889, 2004.
37. Schötzau D., Schwab C., Werder T., Gerdes K. hp-Discontinuous Galerkin time stepping for parabolic problems. Comput. Methods Appl. Mech. Engrg. 2001, 49-50(190):6685-6708.
38. V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, Chapter: The Discontinuous Galerkin Time Stepping Method, Springer Series in Computational Mathematics, Springer, 1997, pp. 181-208.