Current simulation challenges
Model Order Reduction. Towards a New Generation of Computational Vademecums

As already mentioned in the introduction, many problems in the sciences cannot be solved because the models involved are too numerically complex.

We can distinguish various scenarios that still represent real challenges for numerical simulation:

• the first concerns models defined in high-dimensional spaces. These models can be found in quantum chemistry or in the microstructural description of complex fluids. These models suffer from the dreaded "curse of dimensionality". If in a space of dimension N we consider M nodes per dimension, the resulting hypothetical mesh will involve M^N nodes. If M ∼ 10³ (a rather crude description in practice) and N ∼ 30 (a relatively simple model as we'll see later), the numerical complexity reaches the value of 10⁹⁰. Recall that 10⁸⁰ represents the presumed number of elementary particles in the universe....