Final Report Summary - BREAD (Breaking the curse of dimensionality: numerical challenges in high dimensional analysis and simulation)
The first pillar of the project was of theoretical nature: identify the fundamental mathematical principles which allow to circumvent the curse of dimensionality and how they come into play in the afore mentioned applications. This was achieved for a mild class of linear and nonlinear parametrised PDEs for which sparse high-dimensional polynomial approximations can be rigorously proved to converge with rates that are immune to the curse of dimensionality.
The second pillar of the project was of numerical nature: develop computational strategy exploiting and benefiting from these mathematical principle. Our main achievement was to introduce a class of algorithms that are non-intrusive in the sense that they only require the evaluation of a finite number M of particular evaluations corresponding to well-chosen parameter instances, and which provably converge as M grows with a similar rate as predicted by the theoretical approximation results.