Reinhold Schneider
Hierarchical Tensor Representation for Langevin Dynamics and Fokker Planck Equation
Hierarchical Tucker tensor format, introduced by Hackbusch and coworkers, and a particular case of Tensor Trains (TT) (Tyrtyshnikov) have been introduced for high dimensional problems. The parametrization has been known in quantum physics as tensor network states.
There are several ways to cast an approximate numerical solution into a variational framework. The Ritz Galerkin ansatz leads to an optimization problem on Riemannian manifolds. This techniques, in simplified form as an alternating directional search (ALS) or single site DMRG can be used to approximate the eigenstates of Fokker Planck equations for Langevin dynamics arising in molecular dynamics in a thermostat.
This approach can be carried out to approximate the meta-stable eigen-functions of the corresponding Backward Kolmogorov operator by numerical approximation of the transfer operator (also know as Koopman operator), and vice versa the Fokker Planck operator. The algorithm is applies on Monte Carlo integration for setting up the Galerkin matrices, similarly than for Markov state models (Schütte et al.), where the samples are given by sufficiently long (discrete) trajectories of the stochastic ODE (Langevin equation). Tensor product approximation helps to avoid the curse of dimensionality.
This is joint work with F. Nüske and F. Noe from FU Berlin.