Preliminary list of abstracts

Note: We are using the fantastic MathJax JavaScript library to typeset the mathematics on this web page. You can right-click on a formula to zoom in or to select the rendering engine you like (all under 'Settings'). Under Firefox the default renderer turns out to be MathML (which is quick), but you might prefer the HTML-CSS renderer for more faithful LaTeX rendering. If you encounter any problems with this setup: then please email us!

Click here to go back to the list of abstracts.

Efficient solution of highly multidimensional models by using the proper generalized decomposition: Towards a change of paradigm in computational mechanics?
Keywords: Multidimensional models; Curse of dimensionality; Proper Generalized Decomposition
Tue, 14:25--14:50
  • Chinesta, Francisco (EADS Chair - GEM Institute CNRS - Ecole Centrale Nantes, Ecole Centrale de Nantes, France)
  • Leygue, Adrien (EADS Chair at GEM Institute, Ecole Centrale de Nantes, France)
  • Cueto, Elias (I3A, University of Zaragoza, Spain)
  • Keunings, Roland (ICTEAM, Université Catholique de Louvain, Belgium)

Numerous models encountered in science and engineering remain nowadays, despite the impressive progresses attained recently in computational simulation techniques, intractable when the usual and well experienced discretization techniques are applied for their numerical simulation. A first challenging issue concerns the treatment of highly multidimensional models arising from quantum mechanics or kinetic theory models of solids or fluids, including micro and nano-structured complex fluids, stochastic problems and parametric models. The main challenge in the treatment of this kind of models is related to the curse of dimensionality because when one applies standard mesh based discretization the number of degrees of freedom involved scales exponentially with the dimension of the space concerned. Moreover, in the context of problems optimization of inverse identification many direct problems must be solved. Again, alternative advanced computational techniques are urgently needed. By introducing all the model or process parameters, initial or boundary conditions, parameters defining the shape, applied loads, etc. as extra-coordinates in the model, optimization or inverse analysis only need the solution only once of the resulting multidimensional model, that some time involves hundreds of dimensions. The curse of dimensionality related to the solution of such kind of models can be efficiently circumvented by applying the proper generalized decomposition, a separated representation constructor that allows solving efficiently models involving hundreds of dimensions. In this works we present numerous applications in computer science and engineering, where highly multidimensional models never until now treated were successfully solved [1-4].

[1] A. Ammar, B. Mokdad, F. Chinesta, R. Keunings. A New Family of Solvers for Some Classes of Multidimensional Partial Differential Equations Encountered in Kinetic Theory Modeling of Complex Fluids. Journal of Non-Newtonian Fluid Mechanics, 139, 153-176, (2006).
[2] A. Ammar, B. Mokdad, F. Chinesta, R. Keunings. 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 Representation. Journal of Non-Newtonian Fluid Mechanicas, 144, 98-121, (2007).
[3] F. Chinesta, A. Ammar, E. Cueto Recent Advances and New Challenges in the Use of the Proper Generalized Decomposition for Solving Multidimensional Models. Archives of Computational Methods in Engineering – State of the Art Reviews, 17/4, 327-350 (2010).
[4] F. Chinesta, A. Ammar, A. Leygue, R. Keunings An Overview of the Proper Generalized Decomposition with Applications in Computational Rheology. Journal of Non Newtonian Fluid Mechanics. 10.1016/j.jnnfm.2010.12.012