Compact Representation of Solution Vectors in Kronecker-Based Markovian Analysis
It is well known that the infinitesimal generator underlying a multi-dimensional Markov chain with a relatively large reachable state space can be represented compactly on a computer in the form of a block matrix in which each nonzero block is expressed as a sum of Kronecker products of smaller matrices. Nevertheless, solution vectors used in the analysis of such Kronecker-based Markovian representations still require memory proportional to the size of the reachable state space, and this becomes a bigger problem as the number of dimensions increases. The current paper shows that it is possible to use the hierarchical Tucker decomposition (HTD) to store the solution vectors during Kronecker-based Markovian analysis relatively compactly and still carry out the basic operation of vector-matrix multiplication in Kronecker form relatively efficiently. Numerical experiments on two different problems of varying sizes indicate that larger memory savings are obtained with the HTD approach as the number of dimensions increases.
KeywordsMarkov chains Kronecker products Hierarchical Tucker decomposition Reachable state space Compact vectors
This work is supported by the Alexander von Humboldt Foundation through the Research Group Linkage Programme. The research of the last author is supported by The Scientific and Technological Research Council of Turkey.
- 1.APNN-Toolbox. Abstract Petri Net Notation Toolbox. http://www4.cs.uni-dortmund.de/APNN-TOOLBOX
- 5.Buchholz, P., Kemper, P.: Compact representations of probability distributions in the analysis of superposed GSPNs. In: Proceedings of the 9th International Workshop on Petri Nets and Performance Models, Aachen, Germany, pp. 81–90. IEEE Press, New York, September 2001Google Scholar
- 13.Kressner, D., Macedo, F.: Low-rank tensor methods for communicating Markov processes. In: Norman, G., Sanders, W. (eds.) QEST 2014. LNCS, vol. 8657, pp. 25–40. Springer, Heidelberg (2014)Google Scholar
- 14.Kressner, D., Tobler, C.: htucker — A Matlab toolbox for tensors in hierarchical Tucker format. Technical report 2012-02, Mathematics Institute of Computational Science and Engineering, Lausanne, Switzerland, August 2012. http://anchp.epfl.ch/htucker
- 17.Netlib, A.: Collection of Mathematical Software, Papers, and Databases. http://www.netlib.org