Coarse-graining Langevin dynamics using reduced-order techniques

TL;DR

This paper introduces a reduced-order modeling approach for Langevin equations in bio-molecular systems, using Galerkin projection onto Krylov subspaces, ensuring fluctuation-dissipation consistency for low-order reductions.

Contribution

It develops a novel reduction technique for Langevin dynamics based on Krylov subspaces, with proofs of moment-matching and fluctuation-dissipation theorem compliance.

Findings

Reduced models satisfy fluctuation-dissipation theorem for orders less than six.

The approach simplifies Langevin dynamics simulation in bio-molecular models.

Implementation details include bi-orthogonalization and efficient matrix operations.

Abstract

This paper considers the reduction of the Langevin equation arising from bio-molecular models. To facilitate the construction and implementation of the reduced models, the problem is formulated as a reduced-order modeling problem. The reduced models can then be directly obtained from a Galerkin projection to appropriately defined Krylov subspaces. The equivalence to a moment-matching procedure, previously implemented in , 2), is proved. A particular emphasis is placed on the reduction of the stochastic noise, which is absent in many order-reduction problems. In particular, for order less than six we can show the reduced model obtained from the subspace projection automatically satisfies the fluctuation-dissipation theorem. Details for the implementations, including a bi-orthogonalization procedure and the minimization of the number of matrix multiplications, will be discussed as well.