In this paper we analyze decompositions of reversible nearly uncoupled
Markov chains into rapidly mixing subchains. We state upper
bounds on the 2nd eigenvalue for restriction and stochastic complementation
chains of reversible Markov chains, as well as a relation between
them. We illustrate the obtained bounds analytically for bunkbed
graphs, and furthermore apply them to restricted Markov chains that
arise when analyzing conformation dynamics of a small biomolecule.
We shortly review the uncoupling-coupling method, a Markov chain
Monte Carlo based approach to compute statistical properties of systems like
medium-sized biomolecules. This technique has recently been proposed for the efficient computation of biomolecular conformations. One crucial step of UC is the
decomposition of reversible nearly uncoupled Markov chains into rapidly mixing
subchains. We show how the underlying scheme of uncoupling-coupling can also be
applied to stochastic differential equations where it can be translated into a domain
decomposition technique for partial differential equations.