An iterative aggregation and disaggregation approach to the calculation of steady-state distributions of continuous processes

TL;DR

This paper introduces an iterative aggregation and disaggregation method to efficiently compute steady-state distributions of continuous processes by mapping them onto discrete spaces and refining solutions through smaller matrices, suitable for complex systems.

Contribution

The paper presents a novel IAD approach that accelerates steady-state computation for continuous processes by combining symbolic mapping with iterative refinement, applicable to high-resolution and complex systems.

Findings

Method efficiently computes steady states for complex atomistic systems.

Applicable to distributed and parallel computing environments.

Demonstrated effectiveness on two numerical examples.

Abstract

A mapping of the process on a continuous configuration space to the symbolic representation of the motion on a discrete state space will be combined with an iterative aggregation and disaggregation (IAD) procedure to obtain steady state distributions of the process. The IAD speeds up the convergence to the unit eigenvector, which is the steady state distribution, by forming smaller aggregated matrices whose unit eigenvector solutions are used to refine approximations of the steady state vector until convergence is reached. This method works very efficiently and can be used together with distributed or parallel computing methods to obtain high resolution images of the steady state distribution of complex atomistic or energy landscape type problems. The method is illustrated in two numerical examples. In the first example the transition matrix is assumed to be known. The second example represents an overdamped Brownian motion process subject to a dichotomously changing external potential.