Model Order Reduction. Towards a New Generation of Computational Vademecums

Despite recent advances in physical modelling, numerical analysis and computational capacity, many problems in science and engineering remain unsolvable today, because their numerical complexity is sometimes simply unimaginable. In this article, we take a closer look at a number of them, in particular:

• models involving several coordinates or those defined in domains where at least one of the representative dimensions is much smaller than the other characteristic dimensions, both of which lead to prohibitive discretizations;

• transient problems in which the spectrum of characteristic times is too wide;

• real-time simulations ;

• and finally, problems that have to be solved several times, such as control problems, parametric analysis, inverse analysis, quantification and propagation of uncertainty, or optimization, requiring the resolution of a large number of direct problems.

It's in these types of scenarios that the usual simulation techniques prove ineffective, unless combined with adequate computing power that is often out of reach for small and medium-sized companies.

In order to democratize numerical simulation, freeing it from the aforementioned difficulties and the need to resort to "large-scale means", beyond the reach of most potential users, new simulation techniques have emerged and developed rapidly from both fundamental and application standpoints: these are the so-called model reduction techniques which are reviewed in this article.

We'll start with a brief description of some of the current bottlenecks in numerical simulation. Then we'll tackle an issue of vital importance: the difference between data and information. We will show that a colossal amount of data resulting from the discretization of a model may conceal only a tiny amount of information. Based on this observation, we will then see that it is possible to define approximation bases that are reduced in size but contain almost all the information needed to reconstruct the desired solution. This is the fundamental idea behind model reduction techniques based on POD (Proper Orthogonal Decomposition – decomposition orthogonal to eigenvalues).

We will show that these techniques, based on POD and its variants, can accelerate computations without requiring too many resources, but that certain problems, such as multidimensional models and degenerate domains, remain out of reach. What's more, the quality control of solutions and the enrichment of reduced bases remain, in most cases, still a very active subject of research.

We'll then see that the use of separate representations...