Parallel iterative solution method for large sparse linear

Computations with Markov Chains, 1995

We consider a variant of the well-known Gauss-Seidel method for the solution of Markov chains in steady state. Whereas the standard algorithm visits each state exactly once per iteration in a predetermined order, the alternative approach uses a dynamic strategy. A set of states to be visited is maintained which can grow and shrink as the computation progresses. In this manner, we hope to concentrate the computational work in those areas of the chain in which maximum improvement in the solution can be achieved. We consider the adaptive approach both as a solver in its own right and as a relaxation method within the multi-level algorithm. Experimental results show significant computational savings in both cases.

Journal of Scientific Computing, 2022

Discrete-time discrete-state finite Markov chains are versatile mathematical models for a wide range of real-life stochastic processes. One of most common tasks in studies of Markov chains is computation of the stationary distribution. Without loss of generality, and drawing our motivation from applications to large networks, we interpret this problem as one of computing the stationary distribution of a random walk on a graph. We propose a new controlled, easily distributed algorithm for this task, briefly summarized as follows: at the beginning, each node receives a fixed amount of cash (positive or negative), and at each iteration, some nodes receive 'green light' to distribute their wealth or debt proportionally to the transition probabilities of the Markov chain; the stationary probability of a node is computed as a ratio of the cash distributed by this node to the total cash distributed by all nodes together. Our method includes as special cases a wide range of known, very different, and previously disconnected methods including power iterations, Gauss-Southwell, and online distributed algorithms. We prove exponential convergence of our method, demonstrate its high efficiency, and derive scheduling strategies for the green-light, that achieve convergence rate faster than state-of-the-art algorithms.

Performance modelling of complex and heterogeneous systems based on analytical models are often solved by the analysis of underlying Markovian models. We consider performance models based on Continuous Time Markov Chains (CTMCs) and their solution, that is the analysis of the steady-state distribution, to efficiently derive a set of performance indices. This paper presents a tool that is able to decide whether a set of cooperating CTMCs yields a product-form stationary distribution. In this case, the tool computes the unnormalised steady-state distribution. The algorithm underlying the tool has been presented in [10] by exploiting the recent advances in the theory of product-form models such as the Reversed Compound Agent Theorem (RCAT) . In this paper, we focus on the peculiarities of the formalism adopted to describe the interacting CTMCs and on the software design that may have interesting consequences for the performance community.

Journal of Information Science and Engineering, 2002

The objective of this work is the analysis and prediction of the performance of irregular codes, mainly in their parallel implementations. In particular, this paper focuses on parallel iterative solvers for sparse matrices as a relevant case of study of this kind of codes. An efficient library of solvers and preconditioners was developed using HPF and MPI as parallel platforms. For this library, models to characterize and predict the behavior of the execution of the methods, preconditioners and kernels were introduced. To show the results of these models, a visualization tool with an easy to use GUI interface was implemented. Finally, results of the prediction models for the codes of the parallel library are presented using the visualization tool.

Studies in classification, data analysis, and knowledge organization, 2022

Markov Decision Processes (MDPs) are useful to solve real-world probabilistic planning problems. However, finding an optimal solution in an MDP can take an unreasonable amount of time when the number of states in the MDP is large. In this paper, we present a way to decompose an MDP into Strongly Connected Components (SCCs) and to find dependency chains for these SCCs. We then propose a variant of the Topological Value Iteration (TVI) algorithm, called parallel chained TVI (pcTVI), which is able to solve independent chains of SCCs in parallel leveraging modern multicore computer architectures. The performance of our algorithm was measured by comparing it to the baseline TVI algorithm on a new probabilistic planning domain introduced in this study. Our pcTVI algorithm led to a speedup factor of 20, compared to traditional TVI (on a computer having 32 cores).