Cartesian Product Partitioning of Multi-Dimensional Reachable State Spaces

Dayar, Tuǧrul; Orhan, M. Can

doi:10.1017/s0269964816000085

Cited by 8 publications

(11 citation statements)

References 17 publications

(39 reference statements)

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“…The infinitesimal generator Q underlying the MC can be viewed as a (J × J) block matrix induced by the Cartesian product partitioning of S as in [7,9] …”

Section: Compact Vectors In Kronecker Settingmentioning

confidence: 99%

“…This is the form of the Kronecker representation in hierarchical Markovian models [3], where rectangularity of the smaller matrices is possible due to the product state space of the modelled system being larger than its reachable state space [9]. When the product state space is equal to the reachable state space, the smaller matrices turn out to be square as in stochastic automata networks [19,20].…”

Section: Introductionmentioning

confidence: 99%

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Compact Representation of Solution Vectors in Kronecker-Based Markovian Analysis

Buchholz

Dayar

Kriege

et al. 2016

Quantitative Evaluation of Systems

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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.

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“…The infinitesimal generator Q underlying the MC can be viewed as a (J × J) block matrix induced by the Cartesian product partitioning of S as in [7,9] …”

Section: Compact Vectors In Kronecker Settingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Compact Representation of Solution Vectors in Kronecker-Based Markovian Analysis

Buchholz

Dayar

Kriege

et al. 2016

Quantitative Evaluation of Systems

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show abstract

“…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 . Two algorithms for partitioning an arbitrary multidimensional state space into Cartesian products have been proposed . However, these two algorithms are relatively time consuming and do not seem to be suitable for our purposes in this context.…”

Section: Implementation Issuesmentioning

confidence: 99%

“…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. Efficiently partitioning an arbitrary truncated state space at each time step requires further research and is out of the scope of this paper.…”

Section: Choosing the Truncated State Spacementioning

confidence: 99%

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

Dayar

Orhan

2018

Numerical Linear Algebra App

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Summary A stochastic chemical system with multiple types of molecules interacting through reaction channels can be modeled as a continuous‐time Markov chain with a countably infinite multidimensional state space. Starting from an initial probability distribution, the time evolution of the probability distribution associated with this continuous‐time Markov chain is described by a system of ordinary differential equations, known as the chemical master equation (CME). This paper shows how one can solve the CME using backward differentiation. In doing this, a novel approach to truncate the state space at each time step using a prediction vector is proposed. The infinitesimal generator matrix associated with the truncated state space is represented compactly, and exactly, using a sum of Kronecker products of matrices associated with molecules. This exact representation is already compact and does not require a low‐rank approximation in the hierarchical Tucker decomposition (HTD) format. During transient analysis, compact solution vectors in HTD format are employed with the exact, compact, and truncated generated matrices in Kronecker form, and the linear systems are solved with the Jacobi method using fixed or adaptive rank control strategies on the compact vectors. Results of simulation on benchmark models are compared with those of the proposed solver and another version, which works with compact vectors and highly accurate low‐rank approximations of the truncated generator matrices in quantized tensor train format and solves the linear systems with the density matrix renormalization group method. Results indicate that there is a reason to solve the CME numerically, and adaptive rank control strategies on compact vectors in HTD format improve time and memory requirements significantly.

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“…Therefore, multidimensional and one-dimensional representations of states will be used interchangeably. Having said this, we define the Cartesian product partitioning of S as in [17].…”

Section: S529mentioning

confidence: 99%

On Vector-Kronecker Product Multiplication with Rectangular Factors

Dayar¹,

Orhan²

2015

SIAM J. Sci. Comput.

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Abstract. The infinitesimal generator matrix underlying a multidimensional Markov chain can be represented compactly by using sums of Kronecker products of small rectangular matrices. For such compact representations, analysis methods based on vector-Kronecker product multiplication need to be employed. When the factors in the Kronecker product terms are relatively dense, vectorKronecker product multiplication can be performed efficiently by the shuffle algorithm. When the factors are relatively sparse, it may be more efficient to obtain nonzero elements of the generator matrix in Kronecker form on the fly and multiply them with corresponding elements of the vector. This work proposes a modification to the shuffle algorithm that multiplies relevant elements of the vector with submatrices of factors in which zero rows and columns are omitted. This approach avoids unnecessary floating-point operations that evaluate to zero during the course of the multiplication and possibly reduces the amount of memory used. Numerical experiments on a large number of models indicate that in many cases the modified shuffle algorithm performs a smaller number of floating-point operations than the shuffle algorithm and the algorithm that generates nonzeros on the fly, sometimes with a minimum number of floating-point operations and as little of memory possible.

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Cartesian Product Partitioning of Multi-Dimensional Reachable State Spaces

Cited by 8 publications

References 17 publications

Compact Representation of Solution Vectors in Kronecker-Based Markovian Analysis

Compact Representation of Solution Vectors in Kronecker-Based Markovian Analysis

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

On Vector-Kronecker Product Multiplication with Rectangular Factors

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