Nearly reducible finite Markov chains: Theory and algorithms

Area of Science:

• Physics
• Computational Science
• Applied Mathematics

Background:

• Finite Markov chains model stochastic dynamics across diverse scientific fields.
• Analyzing nearly reducible Markov chains is challenging due to ill-conditioning and computational inefficiencies.

Purpose of the Study:

• To review exact, numerically stable methods for analyzing discrete- and continuous-time Markovian networks.
• To address limitations of traditional methods like dense linear algebra and iterative eigendecomposition for ill-conditioned systems.

Main Methods:

• Focus on state reduction procedures as an alternative to traditional numerical techniques.
• Exploration of robust algorithms for computing macroscopic and microscopic dynamical quantities.
• Introduction of the kinetic path sampling algorithm for efficient trajectory simulation.

Main Results:

• State reduction algorithms robustly compute stationary, committor, and visitation probabilities.
• Macroscopic quantities like first passage time distribution moments are reliably calculated.
• Efficient sampling of rare events and identification of key dynamical states are enabled.

Conclusions:

• State reduction and kinetic path sampling offer reliable solutions for analyzing challenging Markovian networks.
• These methods are crucial for understanding kinetically relevant transition mechanisms and dominant system states.
• The reviewed techniques are valuable for practical applications involving rare events in dynamical systems.