Abstract
This paper revisits a powerful discretization technique, the Proper Generalized Decomposition—PGD, illustrating its ability for solving highly multidimensional models. This technique operates by constructing a separated representation of the solution, such that the solution complexity scales linearly with the dimension of the space in which the model is defined, instead the exponentially-growing complexity characteristic of mesh based discretization strategies. The PGD makes possible the efficient solution of models defined in multidimensional spaces, as the ones encountered in quantum chemistry, kinetic theory description of complex fluids, genetics (chemical master equation), financial mathematics, … but also those, classically defined in the standard space and time, to which we can add new extra-coordinates (parametric models, …) opening numerous possibilities (optimization, inverse identification, real time simulations, …).
Similar content being viewed by others
References
Achdou Y, Pironneau O (2005) Computational methods for option pricing. SIAM frontiers in applied mathematics. SIAM, Philadelphia
Ammar A, Ryckelynck D, Chinesta F, Keunings R (2006) On the reduction of kinetic theory models related to finitely extensible dumbbells. J Non-Newton Fluid Mech 134:136–147
Ammar A, Mokdad B, Chinesta F, Keunings R (2006) A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. J Non-Newton Fluid Mech 139:153–176
Ammar A, Mokdad B, Chinesta F, Keunings R (2007) A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Part II: transient simulation using space-time separated representations. J Non-Newton Fluid Mech 144:98–121
Ammar A, Chinesta F (2008) Circumventing curse of dimensionality in the solution of highly multidimensional models encountered in quantum mechanics using meshfree finite sums decomposition. In: Lecture notes on computational science and engineering. Springer, Berlin, pp 1–17
Ammar A, Normandin M, Daim F, Gonzalez D, Cueto E, Chinesta F (2010) Non-incremental strategies based on separated representations: Applications in computational rheology. Commun Math Sci 8(3):671–695
Chinesta F, Ammar A, Cueto E (2010) On the use of Proper Generalized Decompositions for solving the multidimensional chemical master equation. Eur J Comput Mech 19:53–64
Ammar A, Chinesta F, Diez P, Huerta A (2010) An error estimator for separated representations of highly multidimensional models. Comput Methods Appl Mech Eng 199:1872–1880
Ammar A, Chinesta F, Cueto E (2010) Coupling finite elements and separated representations. Int J Multiscale Comput Eng (in press)
Beylkin G, Mohlenkamp M (2005) Algorithms for numerical analysis in high dimensions. SIAM J Sci Comput 26(6):2133–2159
Bird BB, Curtiss CF, Armstrong RC, Hassager O (1987) Dynamics of polymeric liquids. In: Kinetic theory, vol 2. Wiley, New York
Bungartz HJ, Griebel M (2004) Sparse grids. Acta Numer 13:1–123
Cancès E, Defranceschi M, Kutzelnigg W, Le Bris C, Maday Y (2003) Computational quantum chemistry: a primer. Handbook of numerical analysis, vol X. Elsevier, Amsterdam, pp 3–270
Chaidron G, Chinesta F (2002) On the steady solution of non-linear advection problems in steady recirculating flows. Comput Methods Appl Mech Eng 191:1159–1172
Chinesta F, Chaidron G (2001) On the steady solution of linear advection problems in steady recirculating flows. J Non Newton Fluid Mechanics 98:65–80
Chinesta F, Chaidron G (2003) A characteristics strategy for solving advection equations in 2D steady flows containing recirculating areas. Comput Methods Appl Mech Eng 192:37–38, 4217–4235
Chinesta F, Ammar A, Falco A, Laso M (2007) On the reduction of stochastic kinetic theory models of complex fluids. Model Simul Mater Sci Eng 15:639–652
Chinesta F, Ammar A, Lemarchand F, Beauchene P, Boust F (2008) Alleviating mesh constraints: Model reduction, parallel time integration and high resolution homogenization. Comput Methods Appl Mech Eng 197(5):400–413
Chinesta F, Ammar A, Joyot P (2008) The nanometric and micrometric scales of the structure and mechanics of materials revisited: An introduction to the challenges of fully deterministic numerical descriptions. Int J Multiscale Comput Eng 6(3):191–213
Chinesta F, Ammar A, Cueto E (2010) Proper generalized decomposition of multiscale models. Int J Numer Methods Eng 83(8–9):1114–1132
Ammar A, Chinesta F, Falco A (2010) On the convergence of a greedy rank-one update algorithm for a class of linear systems. Arch Comput Methods Eng (submitted). doi:10.1007/s11831-010-9048-z
Donea J, Huerta A (2002) Finite element methods for flow problems. Wiley, New York
Fang J, Kroger M, Ottinger HC (2000) A thermodynamically admissible reptation model for fast flows of entangled polymers. II. Model predictions for shear and extensional flows. J Rheol 40:1293–1318
Fish J, Guttal R (1996) The s-version of the finite element method for laminated composites. Int J Numer Methods Eng 39:3641–3662
Gonzalez D, Ammar A, Chinesta F, Cueto E (2010) Recent advances in the use of separated representations. Int J Numer Methods Eng (in press)
Heinrich JC, Zienkiewicz OC (1979) The finite element method and upwinding techniques in the numerical solution of convection dominated flow problems. In: AMD, vol 34. Am Soc Mech Eng, New York, pp 105–136
Hughes TJR, Brooks AN (1979) A multidimensional upwind scheme with no crosswind diffusion. In: Hughes TJR (ed) Finite element methods for convection dominated flows. AMD, vol 34. American Society of Mechanical Engineering, New York
Joyot P, Bonithon G, Chinesta F, Villon P (2010) Non-incremental boundary element discretization of parabolic models based on the use of proper generalized decompositions. Eng Anal Bound Elements (submitted). doi:10.1016/j.enganabound.2010.07.007
Ladeveze P (1999) Nonlinear computational structural mechanics. Springer, New York
Ladeveze P, Passieux JCh, Neron D (2010) The LATIN multiscale computational method and the proper orthogonal decomposition. Comput Methods Appl Mech Eng 199(21–22):1287–1296
Levy A, Le Corre S, Poitou A, Soccard E (2010) Ultrasonic welding of thermoplastic composites, modeling of the process using time homogenization. Int J Multiscale Comput Eng (in press)
Nouy A, Ladeveze P (2004) Multiscale computational strategy with time and space homogenization: A radial-type approximation technique for solving microproblems. Int J Multiscale Comput Eng 170(2)
Maday Y, Ronquist EM (2004) The reduced basis element method: application to a thermal fin problem. SIAM J Sci Comput 26(1):240–258
Mokdad B, Pruliere E, Ammar A, Chinesta F (2007) On the simulation of kinetic theory models of complex fluids using the Fokker-Planck approach. Appl Rheol 17(2):1–14
Mokdad B, Ammar A, Normandin M, Chinesta F, Clermont JR (2010) A fully deterministic micro-macro simulation of complex flows involving reversible network fluid models. Math Comput Simul 80:1936–1961
Niroomandi S, Alfaro I, Cueto E, Chinesta F (2008) Real-time deformable models of non-linear tissues by model reduction techniques. Comput Methods Programs Biomed 91:223–231
Niroomandi S, Alfaro I, Cueto E, Chinesta F (2010) Model order reduction for hyperelastic materials. Int J Numer Methods Eng 81(9):1180–1206
Niroomandi S, Alfaro I, Cueto E, Chinesta F (2010) Accounting for large deformations in real-time simulations of soft tissues based on reduced order models. Comput Methods Programs Biomed (submitted)
Park HM, Cho DH (1996) The use of the Karhunen-Loève decomposition for the modelling of distributed parameter systems. Chem Eng Sci 51:81–98
Pruliere E, Ammar A, El Kissi N, Chinesta F (2009) Recirculating flows involving short fiber suspensions: Numerical difficulties and efficient advanced micro-macro solvers. Arch Comput Methods Eng State Art Rev 16:1–30
Pruliere E, Ferec J, Chinesta F, Ammar A (2010) An efficient reduced simulation of residual stresses in composites forming processes. Int J Mater Form 3(2):1341–1352
Pruliere E, Chinesta F, Ammar A (2010) On the deterministic solution of parametric models by using the proper generalized decomposition. Math Comput Simul (submitted). doi:10.1016/j.matcom.2010.07.015
Rank E, Krause R (1987) A multiscale finite element method. Comput Struct 64(1–4):139–144
Rassias TM, Simsa J (1995) Finite sums decompositions in mathematical analysis. Wiley, New York
Rvachev VL, Sheiko TI (1995) R-functions in boundary value problems in mechanics. Appl Mech Rev 48:151–188
Ryckelynck D, Hermanns L, Chinesta F, Alarcon E (2005) An efficient “a priori” model reduction for boundary element models. Eng Anal Bound Elem 29:796–801
Ryckelynck D (2005) A priori hyperreduction method: an adaptive approach. J Comput Phys 202:346–366
Ryckelynck D, Chinesta F, Cueto E, Ammar A (2006) On the a priori model reduction: overview and recent developments. Arch Comput Methods Eng State Art Rev 13(1):91–128
Sukumar N, Chopp D, Moes N, Belytschko T (2001) Modeling holes and inclusions by level sets in the extended finite element method. Comput Methods Appl Mech Eng 190:6183–6200
Author information
Authors and Affiliations
Corresponding author
Additional information
Research partially supported by the Spanish Ministry of Science ad Innovation through grant CICYT-DPI2008-00918.
Rights and permissions
About this article
Cite this article
Chinesta, F., Ammar, A. & Cueto, E. Recent Advances and New Challenges in the Use of the Proper Generalized Decomposition for Solving Multidimensional Models. Arch Computat Methods Eng 17, 327–350 (2010). https://doi.org/10.1007/s11831-010-9049-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11831-010-9049-y