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 distinguishes PGD from proper orthogonal decomposition (POD) and reduced basis is the calculation procedure that does not differentiate between the different variables parameters/ 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 devoted to this method is vast [68, 77] but remained in the form of time/space separations for many years. A more general separated representation was more recently employed in [5, 6] for approximating the solution of multidimensional partial differential equations. In [93], such separated representations are also considered for solving stochastic 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. Today, many works use and develop the PGD in extremely varied fields. In this chapter we revisit the fundamentals, variants, and applications of PGD, covering different kinds of separated representations of the involved unknown fields as well as different constructors able to address a variety of linear and nonlinear models, elliptic, parabolic, and hyperbolic.
