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      Compact representation of solution vectors in Kronecker-based Markovian analysis

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      Author(s)
      Buchholz, P.
      Dayar, Tuğrul
      Kriege, J.
      Orhan, M. Can
      Date
      2016-08
      Source Title
      QEST: International Conference on Quantitative Evaluation of Systems -13th International Conference, QEST 2016
      Print ISSN
      0302-9743
      Publisher
      Springer
      Pages
      260 - 276
      Language
      English
      Type
      Conference Paper
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      Abstract
      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 Kroneckerbased 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. © Springer International Publishing Switzerland 2016.
      Keywords
      Compact vectors
      Hierarchical Tucker decomposition
      Kronecker products
      Markov chains
      Reachable state space
      Chains
      Markov processes
      Matrix algebra
      Vectors
      Compact representation
      Infinitesimal generator
      Kronecker product
      Markovian representation
      Multi-dimensional Markov chains
      Numerical experiments
      Tucker decompositions
      Vector-matrix multiplications
      Vector spaces
      Permalink
      http://hdl.handle.net/11693/37502
      Published Version (Please cite this version)
      http://dx.doi.org/10.1007/978-3-319-43425-4_18
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