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Revisiting the matrix‐free solution of Markov regenerative processes
Author(s) -
Amparore Elvio Gilberto,
Donatelli Susanna
Publication year - 2011
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.819
Subject(s) - preconditioner , markov chain , krylov subspace , mathematics , ergodic theory , matrix (chemical analysis) , computation , mathematical optimization , sparse matrix , computer science , algorithm , iterative method , pure mathematics , statistics , materials science , physics , quantum mechanics , composite material , gaussian
SUMMARY In this paper, we revisit the steady‐state solution method for Markov Regenerative Processes (MRP) proposed in the work by German. This method solves the embedded Markov chain P of the MRP without storing the matrix P explicitly. We address three issues left open in German's Work: 1) the solution method is restricted to Power method; 2) it has been defined only for ergodic MRPs; and 3) no preconditioning is available to speed‐up the computation. This paper discusses how to lift these limitations by extending the algorithm to preconditioned Krylov‐subspace methods and by generalizing it to the non‐ergodic case. An MRP‐specific preconditioner is also proposed, which is built from a sparse approximation of the MRP matrix, computed via simulation. An experimental assessment of the proposed preconditioner is then provided. Copyright © 2011 John Wiley & Sons, Ltd.