Split: a flexible and efficient algorithm to vector-descriptor product

In this paper, we address the problem of calculating basis siphons in Petri nets (PNs) by resorting to the semi-tensor product (STP) of matrices. The proposed appraoch is based on our previous results on the calculation of siphons and minimal siphons in PNs, whose key notion is that of the siphon equation (SE) of PNs. First, we discuss the properties of the SE of PNs in the framework of STP. Second, an efficient recursion algorithm is proposed, which can be applied to computing all basis siphons for any PNs. Some results on the computational complexity of the proposed algorithm, in this paper, are also provided, as well as comparison with the existing approach. Last, an example is presented to illustrate the theoretical results. The proposed approach proved highly effective in calculating all basis siphons of PNs and only requires matrix manipulations.

In this paper, the 8 /spl times/ 8 discrete Hartley transform will be implemented by using quantum elementary gates such as controlled NOT gates and Hadamard gates. Two steps involved are as follows: first, the discrete Hartley transform matrix is decomposed into the product of several sparse unitary matrices using the structure of fast Hartley transform algorithm. Then, each sparse matrix is implemented by elementary quantum gates and cascade them to obtain the final circuit. The design results show that the proposed design method has less circuit complexity than the conventional Cybenko's method.

SUMMARYWe propose a theoretical testing framework and a test generation algorithm for concurrent systems specified with true‐concurrency models, such as Petri nets or networks of automata. The semantic model of computation of such formalisms is labelled event structures, which allow to represent concurrency explicitly. We introduce the notions of strong and weak concurrency: strongly concurrent events must be concurrent in the implementation, while weakly concurrent ones may eventually be ordered. The ioco type conformance relations for sequential systems rely on the observation of sequences of actions and blockings; thus, they are not capable of capturing and exploiting concurrency of non‐sequential behaviours. We propose an extension of ioco for labelled event structures, named co‐ioco, allowing to deal with strong and weak concurrency. We extend the notions of test cases and test execution to labelled event structures and give a test generation algorithm building a complete test suite for co‐ioco. Copyright © 2014 John Wiley & Sons, Ltd.

This paper proposes an improved algorithm for enumerating siphons and minimal siphons of Petri nets (PNs) in the framework of [18]. First, the logical equation of each transition is converted into a matrix equation by using the semi-tensor product (STP) of matrices, and then the matrix equation group of the PNs is obtained. Second, an improved algorithm is proposed to calculate the siphons and minimal siphons, respectively. Finally, two illustrative examples are presented to show the efficiency and application of the improved algorithm.

In this paper, we present a comprehensive modeling technique for bounded Petri net systems (BPNSs) in the framework of the semi‐tensor product (STP) of matrices. The two dynamic properties of BPNSs, namely, reachability and controllability, are investigated systematacially. First, the dynamics of a bounded Petri net system (BPNS), by resorting to the STP of matrices, are expressed in the form of a discrete‐time bilinear equation, which is called the marking evolution equation (MEE) of BPNSs. Second, controllability and transition‐marking adjacency matrix (TMAM) of BPNSs are defined, respectively. Further, several necessary and sufficient conditions for reachability and controllability of BPNSs are given in terms of the MEE and TMAM. Third, an efficient algorithm to verify reachability property of BPNSs, in this paper, is provided, as well as its computational complexity. Finally, an example is presented to illustrate the theoretical results in this paper. The main contribution of this paper is the presentation of a precise mathematical model for BPNSs. The main advantage of the proposed approach is that not only it can be applied to verify whether or not any given marking is reachable from the other in state space, but also it is very convenient to find all firing sequences between any two reachable markings.

Decentralized supervisory structure has drawn much attention in recent years to address the computational complexity in designing supervisory structures for large Petri net model. Many studies are reported in the paradigm of automata while few can be found in the Petri net paradigm. The decentralized supervisory structure can address the computational complexity, but it adds the structural complexity of supervisory structure. This paper proposed a new method of designing a global controller for decentralized systems of a large Petri net model for flexible manufacturing systems. The proposed method can both reduce the computational complexity by decomposition of large Petri net models into several subnets and structural complexity by designing a global supervisory structure that can greatly reduce the cost at the implementation stage. Two efficient algorithms are developed in the proposed method. Algorithm 1 is used to compute decentralized working zones from the given Petri net model for flexible manufacturing systems. Algorithm 2 is used to compute the global controller that enforces the liveness to the decentralized working zones. The ring assembling method is used to reconnect and controlled the working zones via a global controller. The proposed method can be applied to large Petri nets size and, in general, it has less computational and structural complexity. Experimental examples are presented to explore the applicability of the proposed method.

The ratio between processor and main memory performance has been increasing since quite some time, and can safely be expected to do so throughout the oncoming years. In the era of single-core processors, this was mainly observable by increased latency, for example when measured in number of (possibly stalled) CPU clock cycles. Nowadays, with multicore chips, multiple cores share the same connection to off-chip main memory, which effectively reduces available bandwidth, as well. Caches help in both cases: they provide both a much lower latency and a much higher bandwidth by being located on-chip. By holding copies of least recently used memory blocks, caches exploit the fact that programs on the average access memory in ways that often access the same memory cell (temporal locality), or nearby memory cells (spatial locality). However, this natural locality is not enough for scientific computing in HPC. Further improving any existing access locality of given algorithms is very much wanted. In this talk, we present strategies to improve the locality of memory accesses for linear algebra problems occurring in different kinds of applications: (1) an algorithmic approach based on Peano spacefilling curves that leads to inherently cache efficient (cache oblivious) matrix algorithms, such as matrix multiplication or LU decomposition for dense and sparse matrices - on single-core CPUs, as well as in the context of shared-memory multicore platforms. (2) cache optimization strategies for matrix-vector multiplications with very large, sparse matrices, as they occur in the iterative MLEM algorithm, which is used for image reconstruction in nuclear medicine. Here, different cache-aware optimization strategies are combined in order to better exploit large caches, small caches, and single cache lines.

An efficient method for special least squares solution of the complex matrix equation [formula omitted

This work presents the theoretical formulation of a new class of fast Fourier transform algorithms and the experimental results obtained from the implementation of these algorithms on transputers computational structures. The theoretical formulation is based on a new methodology for the tensor or Kronecker product decomposition of the discrete Fourier transform matrix into a product of sparse matrices, with not all matrices having the same order. Special attention is given to what is termed a general twiddle or phase factor operator. Conforming this generalized operator, and the other factors present in a Kronecker product formulation of an algorithm, to the specific topology of a given transputer computational structure results in more efficient software implementations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We define a new restricted type of algebraic circuit (AC) problems, called linear algebraic circuits (LACs), and consider the problem of obtaining efficient solutions for their parallel evaluation. The term efficiency indicates that for some fixed number of processors, the algorithm can compete with the sequential evaluation of ACs. While parallel evaluation of general ACs is P-complete, there are restricted types of ACs such as prefix-sums circuits (ACs in the form of chains) and arithmetic expressions (ACs in the form of trees) for which efficient parallel evaluation algorithms exist. In this work, we consider a restricted type, other than chains or trees, of ACs where at least one input of each multiplication must be a constant. We show that LACs can be evaluated in n/18p1/3 log2 p steps, where n is the size of the circuit and p < n is the number of processors. Thus, for suitable values of p, this algorithm can compete with the sequential evaluation of LACs and achieve speedups of about p1/3.It follows that LACs can be represented by a product of matrices. Thus, computing this product in parallel yields a possible evaluation algorithm for LACs. Parallel computation of a product of matrices is also the main engine used in existing parallel evaluation algorithms for ACs. We show, via a lower bound, that such a naive solution based on matrix multiplication alone, is inherently inefficient regardless of the order in which the product of the matrices is computed and hence a tool complex solution must be devised. Finally, useful applications for parallelizing sequential code are given.

PARALLEL AND LARGE SCALE MATRIX COMPUTATIONS IN CONTROL: SOME IDEAS

We consider complex synchronizations in a generalized net- work of queues with signals (Gnetwork) or Stochastic Au- tomata Networks without functions. Both models allow to describe their continuous time Markov chain as a summation of tensor (or Kronecker) products and sums of local descrip- tion of

We describe the implementation and performance of an efficient parallel Gauss-Seidel algorithm that has been developed for irregular, sparse matrices from electrical power systems applications. Although, Gauss-Seidel algorithms are inherently sequential, by performing specialized orderings on sparse matrices, it is possible to eliminate much of the data dependencies caused by precedence in the calculations. A two-part matrix ordering technique has been developed -- first to partition the matrix into block-diagonal-bordered form using diakoptic techniques and then to multi-color the data in the last diagonal block using graph coloring techniques. The ordered matrices often have extensive parallelism, while maintaining the strict precedence relationships in the Gauss-Seidel algorithm. We present timing results for a parallel Gauss-Seidel solver implemented on the Thinking Machines CM-5 distributed memory multi-processor. The algorithm presented here requires active message remote procedure calls in order to minimize communications overhead and obtain good relative speedup.

For accelerating the convergence of numerical computation of sparse matrices, the classical approach is to partition sparse matrices into block-diagonal structures by graph partitioning algorithms, and then apply distinct preconditioning strategies for different blocks, which enable parallel processing of individual blocks and significantly enhances the efficiency. This paper comprehensively analyzes typical graph partitioning algorithms, including METIS, ParMETIS, hMETIS and Scotch in terms of efficiency and resource consumption. By experiments on standardized test cases, we systematically compare these algorithms and summarizes their respective characteristics and optimal application scenarios for future work.

In this paper, enlightened by the effectiveness of Anderson acceleration (AA), a well‐established technique for accelerating fixed‐point solvers, we first present the Anderson accelerating modulus‐based matrix splitting (AAMS) algorithms for a class of horizontal nonlinear complementarity problems. Then, by introducing the strong semi‐smoothness of the absolute value function, we establish the local convergence theory of the AAMS algorithms. More importantly, we provide the optimal parameter for the AAMS algorithms, which is independent of iteration and applicable to other reduced MS algorithms. Finally, the numerical experiments are conducted to clarify the efficiency and practicability of the AAMS algorithms and its optimal parameter, and the effects of two parameters (both derived from AA) of the AAMS algorithms are analyzed.