ANR MaStoC - Manifolds and Stochastic Computations
Abstract:
Physical phenomena are often subject to geometric constraints, such as fixed distances, angles between atoms in a molecule, or fixed energy for Hamiltonian systems.
Such phenomena are modelled with deterministic and stochastic differential equations on manifolds, and are now heavily used in a variety of applications including chemistry, stochastic optimisation, quantum physics, and machine learning.
For the simulation of such systems, it is of utmost importance that the numerical method used to approximate the differential equation preserves the geometric constraints.
In other words, it is necessary that the method lies on the manifold, especially for sampling measures on manifolds.
In the deterministic context, Runge-Kutta methods or Lie-group methods are famous examples of numerical methods used for the integration of differential equations with high order of accuracy.
There exist high order stochastic integrators in a Euclidean setting, but the only alternatives on manifolds are variants of the Euler scheme, have only order one of accuracy, and are almost all extrinsic, that is, they depend on an embedding of the manifold in a Euclidean space of much higher dimension.
The aim of the project MaStoC (Manifolds and Stochastic Computations) is the design of new versatile numerical methods of high accuracy for solving stochastic differential equations on manifolds.
The approach relies on the development of stochastic Lie-group methods, together with algebraic formalisms for the calculation of order conditions, the development of a rigorous error analysis on general Riemannian manifolds, and the design of new robust methods for tackling stochastic multiscale dynamics subject to constraints.
The project MaStoC will build the foundations of the emerging field of intrinsic stochastic numerics on manifolds and will provide a collection of new efficient methods that will in particular allow for future interdisciplinary collaborations.