Keywords: model order reduction, proper generalised decomposition, parametric models, high dimensional models, curse of dimensionality.

This chapter focuses on the development of a new simulation paradigm enabling the solving of models until now never solved and on introducing spectacular CPU time savings (of the order of millions) that combined with supercomputing could revolutionise future ICT (information and communication technologies) at the heart of science and technology. A new paradigm is proposed for simulation-based engineering sciences called proper generalized decomposition (PGD) which has proved its tremendous possibilities in the context of some basic demonstrators. Its transfer for treating large-scale models will require further developments that are in progress.

Many problems in science and engineering today remain intractable, in spite of the impressive progress attained in mechanical modelling, numerical analysis, discretisation techniques and computer science during the last decade, because their numerical complexity, or the restrictions imposed by different requirements make them unaffordable for today’s technologies.

The following different challenging scenarios for efficient numerical simulations are enumerated:

1. The first one concerns models that are defined in high dimensional spaces,
2. A second category of problems involves multiscale problems not necessarily defined in high-dimensional spaces, but whose spectrum of characteristic times or lengths is so wide that standard incremental discretisation techniques cannot be applied.
3. Other challenging problems are defined in degenerated geometrical domains. By this we mean that at least one of the characteristic dimensions of the domain is smaller by several orders of magnitude than the others.
4. Many problems in process control, parametric modelling, inverse identification, and process or shape optimization, usually require, when approached with standard techniques, the direct computation of a very large number of solutions of the model concerned for particular values of the problem parameters.
5. Traditionally, simulation-based engineering sciences (SBES) relied on the use of static data inputs to perform the simulations. This data could be parameters of the model(s) or boundary conditions. Dynamic data-driven application systems (DDDAS) entail the ability to dynamically incorporate additional data into an executing application, and in reverse, the ability of an application to dynamically steer the measurement process.
6. Augmented reality is another area in which efficient (fast and accurate) simulation is urgently needed. Light computing platforms are appealing alternatives to heavy computing platforms that in general are expensive and whose use requires technical knowledge.

The proper generalized decomposition based model enables efficient alternatives to all the above listed numerical challenges to be addressed.

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