an:05602065
Zbl 1171.15008
Fritzsche, David; Mehrmann, Volker; Szyld, Daniel B.; Virnik, Elena
An SVD approach to identifying metastable states of Markov chains
EN
ETNA, Electron. Trans. Numer. Anal. 29(2007-2008), 46-69 (2008).
00253105
2008
j
15A18 15B51 60J10 60J20 65F15
Markov chain; stochastic matrix; conformation dynamics; metastable; eigenvalue cluster; singular value decomposition; block diagonal dominance; eigenvectors; algorithm; numerical examples
Summary: Being one of the key tools in conformation dynamics, the identification of metastable states of Markov chains has been subject to extensive research in recent years, especially when the Markov chains represent energy states of biomolecules. Some previous work on this topic involved the computation of the eigenvalue cluster close to one, as well as the corresponding eigenvectors and the stationary probability distribution of the associated stochastic matrix. More recently, since the eigenvalue cluster algorithm may be nonrobust, an optimization approach was developed.
As a possible less costly alternative, we present a singular value decomposition (SVD) approach of identifying metastable states of a stochastic matrix, where we only need the singular vector associated with the second largest singular value. We also introduce a concept of block diagonal dominance on which our algorithm is based. We outline some theoretical background and discuss the advantages of this strategy. Some simulated and real numerical examples illustrate the effectiveness of the proposed algorithm.