Comparison of multilevel methods for kronecker-based Markovian representations

The paper presents a class of numerical methods to compute the stationary distribution of Markov chains (MCs) with large and structured state spaces. A popular way of dealing with large state spaces in Markovian modeling and analysis is to employ Kronecker-based representations for the generator matrix and to exploit this matrix structure in numerical analysis methods. This paper presents various multilevel (ML) methods for a broad class of MCs with a hierarchcial Kronecker structure of the generator matrix. The particular ML methods are inspired by multigrid and aggregation-disaggregation techniques, and differ among each other by the type of multigrid cycle, the type of smoother, and the order of component aggregation they use. Numerical experiments demonstrate that so far ML methods with successive over-relaxation as smoother provide the most effective solvers for considerably large Markov chains modeled as HMMs with multiple macrostates

Towards faster numerical solution of Continuous Time Markov Chains stored by symbolic data structures

This work considers different aspects of model-based performance- and dependability analysis. This research area analyses systems (e.g. computer-, telecommunication- or production-systems) in order to quantify their performance and reliability. Such an analysis can be carried out already in the planning phase, without a physically existing system. All aspects treated in this work are based on finite state spaces (i.e. the models only have finitely many states) and a representation of the state graphs by Multi-Terminal Binary Decision Diagrams (MTBDDs). Currently, there are many tools that transform high-level model specifications (e.g. process algebra or Petri-Net) to low-level models (e.g. Markov chains). Markov chains can be represented by sparse matrices. For complex models very large state spaces may occur (this phenomenon is called state space explosion in the literature) and accordingly very large matrices representing the state graphs. The problem of building the model from the specification and storing the state graph can be regarded as solved: There are heuristics for compactly storing the state graph by MTBDD or Kronecker data structure and there are efficient algorithms for the model generation and functional analysis. For the quantitative analysis there are still problems due to the size of the underlying state space. This work provides some methods to alleviate the problems in case of MTBDD-based storage of the state graph. It is threefold: 1. For the generation of smaller state graphs in the model generation phase (which usually are easier to solve) a symbolic elimination algorithm is developed. 2. For the calculation of steady-state probabilities of Markov chains a multilevel algorithm is developed which allows for faster solutions. 3. For calculating the most probable paths in a state graph, the mean time to the first failure of a system and related measures, a path-based solver is developed

Decompositional analysis of Kronecker structured Markov chains

This contribution proposes a decompositional iterative method with low memory requirements for the steadystate analysis ofKronecker structured Markov chains. The Markovian system is formed by a composition of subsystems using the Kronecker sum operator for local transitions and the Kronecker product operator for synchronized transitions. Even though the interactions among subsystems, which are captured by synchronized transitions, need not be weak, numerical experiments indicate that the solver benefits considerably from weak interactions among subsystems, and is to be recommended specifically in this case. © 2008, Kent State University

A High-Throughput Solver for Marginalized Graph Kernels on GPU

We present the design and optimization of a linear solver on General Purpose GPUs for the efficient and high-throughput evaluation of the marginalized graph kernel between pairs of labeled graphs. The solver implements a preconditioned conjugate gradient (PCG) method to compute the solution to a generalized Laplacian equation associated with the tensor product of two graphs. To cope with the gap between the instruction throughput and the memory bandwidth of current generation GPUs, our solver forms the tensor product linear system on-the-fly without storing it in memory when performing matrix-vector dot product operations in PCG. Such on-the-fly computation is accomplished by using threads in a warp to cooperatively stream the adjacency and edge label matrices of individual graphs by small square matrix blocks called tiles, which are then staged in registers and the shared memory for later reuse. Warps across a thread block can further share tiles via the shared memory to increase data reuse. We exploit the sparsity of the graphs hierarchically by storing only non-empty tiles using a coordinate format and nonzero elements within each tile using bitmaps. Besides, we propose a new partition-based reordering algorithm for aggregating nonzero elements of the graphs into fewer but denser tiles to improve the efficiency of the sparse format.We carry out extensive theoretical analyses on the graph tensor product primitives for tiles of various density and evaluate their performance on synthetic and real-world datasets. Our solver delivers three to four orders of magnitude speedup over existing CPU-based solvers such as GraKeL and GraphKernels. The capability of the solver enables kernel-based learning tasks at unprecedented scales

On the convergence of a class of multilevel methods for large sparse Markov chains

This paper investigates the theory behind the steady state analysis of large sparse Markov chains with a recently proposed class of multilevel methods using concepts from algebraic multigrid and iterative aggregation- disaggregation. The motivation is to better understand the convergence characteristics of the class of multilevel methods and to have a clearer formulation that will aid their implementation. In doing this, restriction (or aggregation) and prolongation (or disaggregation) operators of multigrid are used, and the Kronecker-based approach for hierarchical Markovian models is employed, since it suggests a natural and compact definition of grids (or levels). However, the formalism used to describe the class of multilevel methods for large sparse Markov chains has no influence on the theoretical results derived. © 2007 Society for Industrial and Applied Mathematics

On vector-kronecker product multiplication with rectangular factors

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, vector- Kronecker 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. © 2015 Society for Industrial and Applied Mathematics

On compact solution vectors in Kronecker-based Markovian analysis

State based analysis of stochastic models for performance and dependability often requires the computation of the stationary distribution of a multidimensional continuous-time Markov chain (CTMC). The infinitesimal generator underlying a multidimensional CTMC with a large reachable state space can be represented compactly in the form of a block matrix in which each nonzero block is expressed as a sum of Kronecker products of smaller matrices. However, solution vectors used in the analysis of such Kronecker-based Markovian representations require memory proportional to the size of the reachable state space. This implies that memory allocated to solution vectors becomes a bottleneck as the size of the reachable state space increases. Here, it is shown that the hierarchical Tucker decomposition (HTD) can be used with adaptive truncation strategies to store the solution vectors during Kronecker-based Markovian analysis compactly and still carry out the basic operations including vector–matrix multiplication in Kronecker form within Power, Jacobi, and Generalized Minimal Residual methods. Numerical experiments on multidimensional problems of varying sizes indicate that larger memory savings are obtained with the HTD approach as the number of dimensions increases. © 2017 Elsevier B.V

Lumpable continuous-time stochastic automata networks

Cataloged from PDF version of article.The generator matrix of a continuous-time stochastic automata network (SAN) is a sum of tensor products of smaller matrices, which may have entries that are functions of the global state space. This paper specifies easy to check conditions for a class of ordinarily lumpable partitionings of the generator of a continuous-time SAN in which aggregation is performed automaton by automaton. When there exists a lumpable partitioning induced by the tensor representation of the generator, it is shown that an efficient aggregation-iterative disaggregation algorithm may be employed to compute the steady-state distribution. The results of experiments with two SAN models show that the proposed algorithm performs better than the highly competitive block Gauss-Seidel in terms of both the number of iterations and the time to converge to the solution. © 2002 Elsevier Science B.V. All rights reserved

Methodologies synthesis

This deliverable deals with the modelling and analysis of interdependencies between critical infrastructures, focussing attention on two interdependent infrastructures studied in the context of CRUTIAL: the electric power infrastructure and the information infrastructures supporting management, control and maintenance functionality. The main objectives are: 1) investigate the main challenges to be addressed for the analysis and modelling of interdependencies, 2) review the modelling methodologies and tools that can be used to address these challenges and support the evaluation of the impact of interdependencies on the dependability and resilience of the service delivered to the users, and 3) present the preliminary directions investigated so far by the CRUTIAL consortium for describing and modelling interdependencies

Kronecker representation and decompositional analysis of closed queueing networks with phase-type service distributions and arbitrary buffer sizes

Two approximative fixed-point iterative methods based on decomposition for closed queueing networks with Coxian service distributions and arbitrary buffer sizes are extended to include phase-type service distributions. The irreducible Markov chain associated with each subnetwork in the respective decompositions is represented hierarchically using Kronecker products. The two methods are implemented in a software tool capable of computing the steady-state probability vector of each subnetwork by a multilevel method at each fixed-point iteration and are compared with other methods for accuracy and efficiency. Numerical results indicate that there is a niche filled by the two approximative methods