Anthony T. Chronopoulos

s -step iterative methods for symmetric linear systems

Abstract In this paper we introduce s -step Conjugate Gradient Method for Symmetric and Positive Definite (SPD) linear systems of equations and discuss its convergence. In the s -step Conjugate Gradient Method iteration s new directions are formed simultaneously from ≎ r i , Ar i ,…, A s −1 r i ≎ and the preceding s directions. All s directions are chosen to be A-orthogonal to the preceding s directions. The approximation to the solution is then advanced by minimizing an error functional simultaneously in all s directions. This intuitively means that the progress towards the solution in one iteration of the s -step method equals the progress made over s consecutive steps of the one-step method. This is proven to be true.

Noncooperative load balancing in distributed systems

In this paper, we present a game theoretic framework for obtaining a user-optimal load balancing scheme in heterogeneous distributed systems. We formulate the static load balancing problem in heterogeneous distributed systems as a noncooperative game among users. For the proposed noncooperative load balancing game, we present the structure of the Nash equilibrium. Based on this structure we derive a new distributed load balancing algorithm. Finally, the performance of our noncooperative load balancing scheme is compared with that of other existing schemes. The main advantages of our load balancing scheme are the distributed structure, low complexity and optimality of allocation for each user.

Load balancing in distributed systems: an approach using cooperative games

In this paper we formulate the static load balancing problem in single class job distributed systems as a cooperative game among computers. It is shown that the Nash Bargaining Solution (NBS) provides a Pareto optimal allocation which is also fair to all jobs. We propose a cooperative load balancing game and present the structure of the NBS For this game an algorithm for computing NBS is derived. We show that the fairness index is, always 1 using NBS which means that the allocation is fair to all jobs. Finally, the performance of our cooperative load balancing scheme is compared with that of other existing schemes.

Game-theoretic static load balancing for distributed systems

In this paper, we present a game theoretic approach to solve the static load balancing problem for single-class and multi-class (multi-user) jobs in a distributed system where the computers are connected by a communication network. The objective of our approach is to provide fairness to all the jobs (in a single-class system) and the users of the jobs (in a multi-user system). To provide fairness to all the jobs in the system, we use a cooperative game to model the load balancing problem. Our solution is based on the Nash Bargaining Solution (NBS) which provides a Pareto optimal solution for the distributed system and is also a fair solution. An algorithm for computing the NBS is derived for the proposed cooperative load balancing game. To provide fairness to all the users in the system, the load balancing problem is formulated as a non-cooperative game among the users who try to minimize the expected response time of their own jobs. We use the concept of Nash equilibrium as the solution of our non-cooperative game and derive a distributed algorithm for computing it. Our schemes are compared with other existing schemes using simulations with various system loads and configurations. We show that our schemes perform near the system optimal schemes and are superior to the other schemes in terms of fairness.

A class of loop self-scheduling for heterogeneous clusters

Distributed Computing Systems are a viable and less expensive alternative to parallel computers. However, a serious difficulty in concurrent programming of a distributed system is how to deal with scheduling and load balancing of such a system which may consist of heterogeneous computers. Distributed scheduling schemes suitable for parallel loops with independent iterations on heterogeneous computer clusters have been designed in the past. In this work we consider a class of Self-Scheduling schemes for parallel loops with independent iterations which have been applied to multiprocessor systems. We extend this type of schemes to heterogeneous distributed systems. We present tests that the distributed versions of these schemes maintain load balanced execution on heterogeneous systems.

Parallel iterative S-step methods for unsymmetric linear systems

Abstract GCR (Generalized Conjugate Residual) and Omin (Orthomin) are iterative methods for approximating the solution of unsymmetric linear systems. The S-step generalization of these methods has been derived and studied in past work. The S-step methods exhibit improved convergence properties. Also, their data locality and parallel properties are enhanced by forming blocks of s search direction vectors. However, s is limited (to s ≤ 5) by numerical stability considerations. The following new contributions are described in this article. The Modified Gram-Schmidt method is used to AT A-orthogonalize the s direction vectors within each S-step block. It is empirically shown that use of values of s, up to s = 16, preserves the numerical stability of the new iterative methods. Finally, the new S-step Omin, implemented on the CRAY C90, attained an execution rate greater than 10 Gflops (Billion Floating Point Operations per sec).

A class of Lanczos-like algorithms implemented on parallel computers

The Lanczos algorithm is most commonly used in approximating a small number of extreme eigenvalues and eigenvectors for symmetric large sparse matrices. Main memory accesses for shared memory systems or global communications (synchronizations) in message passing systems decrease the computation speed. In this paper, the standard Lanczos algorithm is restructured so that only one synchronization point is required; that is, one global communication in a message passing distributed-memory machine or one global communication in a message passing distributed-memory machine or one global memory sweep in a shared-memory machine per each iteration is required. We also introduce the s-step Lanczos method for finding a few eigenvalues of symmetric large sparse matrices in a similar way to the s-step Conjugate Gradient method [2], and we prove that the s-step method generates reduction matrices which are similar to reduction matrices generated by the standard method. One iteration of the s-step Lanczos algorithm corresponds to s iterations of the standard Lanczos algorithm. The s-step method has improved data locality, minimized global communication and has superior parallel properties to the standard method. These algorithms are implemented on a 64-node NCUBE/seven hypercube and a CRAY-2, and performance results are presented.

A game-theoretic approach to joint rate and power control for uplink CDMA communications

Next generation wireless systems will be required to support heterogeneous services with different transmission rates that include real time multimedia transmissions, as well as non-real time data transmissions. In order to provide such flexible transmission rates, efficient use of system resources in next generation systems will require control of both data transmission rate and power for mobile terminals. In this paper we formulate the problem of joint transmission rate and power control for the uplink of a single cell CDMA system as a non-cooperative game. We assume that the utility function depends on both transmission rates and powers and show the existence of Nash equilibrium in the non-cooperative joint transmission rate and power control game (NRPG). We include numerical results obtained from simulations that compare the proposed algorithm with a similar one which is also based on game theory and it also updates the transmission rates and powers simultaneously in a single step.

s -step iterative methods for (non)symmetric (in)definite linear systems

In this paper a class of s-step methods for nonsymmetric linear systems of equations is introduced. These methods are obtained from nonsymmetric generalizations of the conjugate residual method, which apply to nonsymmetric definite systems [S. C. Eisenstat, H. C. Elmans and M. H. Schultz, SIAM J. Numer. Anal., 20 (1983), pp. 345–357]. The s-step methods are derived then in a way similar to obtaining the s-step conjugate gradient [G. E. Forsythe, Numer. Math., 11 (1968), pp. 57–76], [A. T. Chronopoulos and C. W. Gear, J. Comput. Math., 25 (1989), pp. 153–168], [A. T. Chronopoulos, Ph.D. thesis, Dept. of Computer Science, University of Illinois, Urbana, IL, 1986). It is proven that the s-step methods (with

Iterative methods for nonlinear operator equations

s \geqq 2