On Vector-Kronecker Product Multiplication with Rectangular Factors

Abstract

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.

Compact Representation of Solution Vectors in Kronecker-Based Markovian Analysis

“…When the factors are relatively sparse, it may be more efficient to obtain nonzeros of the generator in Kronecker form on the fly and multiply them with corresponding elements of the vector [6]. Recently, the shuffle algorithm has been modified so that relevant elements of the vector are multiplied with submatrices of factors in which zero rows and columns are omitted [8]. This approach is shown to avoid unnecessary floating-point operations (flops) that evaluate to zero during the course of the multiplication and possibly reduces the amount of memory used.…”

Performance Analysis and Optimization of the Vector-Kronecker Product Multiplication

“…This comparison is justified by a considerable gain in both memory efficiency and lower CPU usage when using GTA formalism. Dayar and Orhan work [8] is primarily based on optimizing the execution of the shuffle algorithm in order to improve data locality. The optimization also reduces the number of FLOPS, this is accomplished by focusing the computation on the nonzero values of the matrices and thus trying to avoid FLOPS that use zero rows and columns.…”

“…The kronecker product matrix is defined as a block matrix formed with a special multiplication between two matrices [8]. The problem is that given N square matrices A (i) of order n i and a vector x ∈ R 1×L where L = N n=1 n i , the complexity of building this matrix is ( N i=1 n 2 i ).…”

On compact solution vectors in Kronecker-based Markovian analysis

“…In practice, the matrices Q k,h are sparse [3] and held in sparse row format since the nonzeros in each of its rows indicate the possible transitions from the state with that row index. The advantage of partitioning the reachable state space is the elimination of unreachable states from the set of rows and columns of the generator to avoid unnecessary computational effort (see, for instance, [2,15]) due to unreachable states and to use vectors not larger than |R| in the analysis. The Kronecker form of the blocks Q (i,j) in Q has been studied before for a number of models [21][22][23].…”

“…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. The same numerical experiment was performed with a solution vector the same size as the reachable state space size using an improved version of the shuffle algorithm [15]. For a fixed truncation error tolerance strategy in the HTD format, the two approaches were compared for memory, time, and accuracy, leading to the preliminary conclusion that compact vectors in HTD format become more memory efficient as the number of dimensions increases.…”