Compact Representation of Solution Vectors in Kronecker-Based Markovian Analysis

Abstract

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.

On compact vector formats in the solution of the chemical master equation with backward differentiation

“…Storing the infinitesimal generator matrix in Kronecker form requires the truncated state space to be represented as a union of Cartesian products of subsets of subsystem state spaces. 49,56 Two algorithms for partitioning an arbitrary multidimensional state space into Cartesian products have been proposed. 65 However, these two algorithms are relatively time consuming and do not seem to be suitable for our purposes in this context.…”

“…[45][46][47][48] HTD may be considered as a generalization of the TT decomposition in which basis and core matrices are related through a tree structure with logarithmic depth in the number of dimensions that reduces the complexity of the Tucker decomposition by hierarchically splitting the core tensor into core matrices. An advantage of the HTD representation for our purposes is that the multiplication of a compact solution vector in HTD format with a sum of Kronecker products is a well-defined operation 49 and has recently been used in the steady-state analysis of Kronecker-based CTMCs. 50 In the CME setting, such a Kronecker-based Markovian analysis with vectors in HTD format is possible under the separability assumption of transition rates.…”

On compact solution vectors in Kronecker-based Markovian analysis

“…The tensor train decomposition has been applied in [13] to approximate the solution vector for models where the product space is reachable using an alternating least squares approach or the Power method. To the best of our knowledge, HTD was first applied to hierarchically structured CTMCs in [14]. Therein, it is shown that a compact solution vector in HTD format can be multiplied with a sum of Kronecker products to yield another compact solution vector in HTD format.…”

“…This necessitates some kind of truncation, hence, approximation, to be introduced to the addition operation. To investigate the merit of the approach, the following analysis was performed in [14]. Starting from an initial solution, the compact vector in HTD format was iteratively multiplied with the uniformized generator matrix of a given CTMC in Kronecker form 1000 times.…”

“…We remark that compact representations for solution vectors in Markovian analysis have also been considered from the perspective of binary decision diagrams [16,17]. The proposed compact data structures therein have not been timewise competitive and do not allow the computation of truncation error bounds, whereas the approach investigated in [14] seems to be a step forward.…”