ncts

The approximation of the law of stochastic dynamics in finite time (weak approximation) and in long time for ergodic dynamics has recently become an important topic of stochastic numerics, with applications in molecular dynamics, optimisation, finance, and machine learning. As the dimension of the popular problems is typically high, the question of finding new cheap, robust, and accurate discretisations is central. In this talk, we present the methodology for the creation of high order methods in the weak sense and for the invariant measure, the algebraic formalism of exotic trees and series for the calculations, but also for its fundamental algebraic and geometric properties, and we apply this new approach to the creation of new high order methods. To deal with dynamics under constraints, we present an extension to projection methods, and a uniformly accurate integrator for dynamics in the neighbourhood of manifolds. We shall conclude with exciting ongoing works on intrinsic methods for the integration of stochastic dynamics on manifolds.