In a seminal article in 1929, P.A.M. Dirac wrote: "The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation."
This quote is more relevant than ever now that modern model order reduction and numerical methods (e.g. reduced bases combined with a posteriori error estimators, geometric optimization methods, hierarchical tensors, randomized linear algebra, quantum embedding methods...) combined with current or expected computing power (exascale computers, quantum computers) are opening new perspectives. This field of research is generating intense activity in computational physics, chemistry and materials science, and it is vital for our community not to miss out on this essential field of research for addressing the scientific challenges of the 21st century (energy production and storage, drug design, atomic and molecular-scale engineering...).
In this lecture, I will illustrate the richness of this scientific field in terms of mathematical and numerical methods, as well as applications, and discuss some current challenges.
