3 Proper generalized decomposition

Abstract

: The so-called “reduced” models have always been very popular and often essential in engineering to analyze the behavior of structures and materials, especially in dynamics. They highlight the relevant information and lead, moreover, to less expensive and more robust calculations. In addition to conventional reduction methods, a generation of reduction strategies is now being developed, such as proper generalized decomposition (PGD), which is the subject of this chapter. The primary feature of these strategies is to be very general and to offer enormous potential for solving problems beyond the reach of industrial computing codes. It is typically the case when trying to take into account the uncertainties or the variations of parameters or nonlinear problems with very large number of degrees of freedom, in the presence of several scales or interactions between several physics. These methods, along with the notions of “offline” and “online” calculations, also open the way to new approaches where simulation and analysis can be carried out almost in real-time. What distin-guishes PGD from proper orthogonal decomposition (POD) and reduced basis is the calculation procedure that does not differentiate between the different variables pa-rameters/time/space. In other terms, we can say that we minimize or make stationary a residual defined over the parameters-time-space domain. PGD with time/space separation and the classical greedy computation technique were introduced in the 1980s as part of the LATIN solver [66, 67] for solving nonlinear time-dependent problems with the terminology “time/space radial approximation.” The corpus of literature de-votedtothismethod is vast[68, 77]but remainedinthe form oftime/spaceseparations for many years. A more general separated representation was more recently employed in [5, 6] for approximating the solution of multidimensional partial differential equa-tions.In[93],suchseparatedrepresentationsarealsoconsideredforsolvingstochastic equations. PGD is the common name coined in 2010 by the authors of this chapter for these techniques because it can be viewed as an extension of the classical POD. To-day,manyworksuseanddevelopthePGDinextremelyvariedfields.Inthischapterwe In this chapter we use different notations to be consistent with the referred publica-tions. In any case, notation will be appropriately defined before being used to avoid any possible confusion.