Sparse matrix computation for air quality forecast data assimilation

Numerical Algorithms - In this paper, we study the ensemble Kalman filter (EnKF) method for chemical species simulation in air quality forecast data assimilation. The main contribution of this...     

Nonsmooth optimization methods for parallel decomposition of multicommodity flow problems

We develop an iterative algorithm based on right-hand side decomposition for the solution of multicommodity network flow problems. At each step of the proposed iterative procedure the coupling constraints are eliminated by subdividing the shared capacity resource among the different commodities and a master problem is constructed which attempts to improve sharing of the resources at each iteration.As the objective function of the master problem is nonsmooth, we apply to it a new optimization technique which does not require the exact solutions of the single commodity flow subproblems. This technique is based on the notion of - subgradients instead of subgradients and is suitable for parallel implementation. Extensions to the nonlinear, convex separable case are also discussed.The work of this author has been supported by the Air Force Office of Scientific Research Grant AFOSR-89-0410.     

Zero data and interpolation problems for rectangular matrix polynomials

Zero data of rectangular matrix polynomials are described in various forms. The basic interpolation problem of constructing rectangular matrix polynomials from their zero data is solved. Certain rectangular factorizations are analyzed in terms of spectral data.     

Alternatives for parallel Krylov subspace basis computation

Numerical methods related to Krylov subspaces are widely used in large sparse numerical linear algebra. Vectors in these subspaces are manipulated via their representation onto orthonormal bases. Nowadays, on serial computers, the method of Arnoldi is considered as a reliable technique for constructing such bases. However, although easily parallelizable, this technique is not as scalable as expected for communications. In this work we examine alternative methods aimed at overcoming this drawback. Since they retrieve upon completion the same information as Arnoldi's algorithm does, they enable us to design a wide family of stable and scalable Krylov approximation methods for various parallel environments. We present timing results obtained from their implementation on two distributed-memory multiprocessor supercomputers: the Intel Paragon and the IBM Scalable POWERparallel SP2. © 1997 John Wiley & Sons, Ltd.     

Combination method for parallel computation in ODE

In this paper, a 4th order parallel computation method with four processes for solving ODEs is discussed. This method is the Runge-Kutta method combined with a linear multistep method, which overcomes the difficulties of the 4th order parallel Runge-Kutta method discussed in [1]. The concept of critical speedup for parallel methods is also defined, and speedups of some methods are analyzed by using this concept.     

Optimal equi-partition of rectangular domains for parallel computation

We present an efficient method for the partitioning of rectangular domains into equi-area sub-domains of minimum total perimeter. For a variety of applications in parallel computation, this corresponds to a load-balanced distribution of tasks that minimize interprocessor communication. Our method is based on utilizing, to the maximum extent possible, a set of optimal shapes for sub-domains. We prove that for a large class of these problems, we can construct solutions whose relative distance from a computable lower bound converges to zero as the problem size tends to infinity. PERIX-GA, a genetic algorithm employing this approach, has successfully solved to optimality million-variable instances of the perimeter-minimization problem and for a one-billion-variable problem has generated a solution within 0.32% of the lower bound. We report on the results of an implementation on a CM-5 supercomputer and make comparisons with other existing codes.This research was partially funded by Air Force Office of Scientific Research grant F496-20-94-1-0036 and National Science Foundation grants CDA-9024618 and CCR-9306807.     

Sufficient conditions for the determination and use of data in the same granular parallel computation process

Parallel algorithms for distributed memory computers should be granular, in which case the set of algorithmic operations is split into sets known as computation grains, or tiles. Conditions are proposed and proved under which data is used in the same granular computation process where it was determined. These conditions can be used to estimate the number of communication operations in alternative versions of parallel algorithms.     

Parallel computation structure for magnetohydrodynamic problems

Translated from Vychislitel'nye Sistemy i Voprosy Prinyatiya Reshenii, pp. 18–27, 1991.     

Generalized Hessian matrix and second-order optimality conditions for problems withC 1,1 data

In this paper, we present a generalization of the Hessian matrix toC ^1,1 functions, i.e., to functions whose gradient mapping is locally Lipschitz. This type of function arises quite naturally in nonlinear analysis and optimization. First the properties of the generalized Hessian matrix are investigated and then some calculus rules are given. In particular, a second-order Taylor expansion of aC ^1,1 function is derived. This allows us to get second-order optimality conditions for nonlinearly constrained mathematical programming problems withC ^1,1 data.     

Practical computation of matrix functions

Several new representations for an analytic function f(A) of a complex matrix A, and in particular for e^At and A^t, are derived, which also are numerically useful in that they avoid the computation of eigenvalues of A.     

Adaptive computation for convection dominated diffusion problems

We derive sharp L∞(L ¹ ) a posteriori error estimate for the convection dominated diffusion equations of the form

$$\frac{{\partial u}}{{\partial t}} + div(vu) - \varepsilon \Delta u = g.$$

The derived estimate is insensitive to the diffusion parameter ε → 0. The problem is discretized implicitly in time via the method of characteristics and in space via continuous piecewise linear finite elements. Numerical experiments are reported to show the competitive behavior of the proposed adaptive method.

     

Adaptive computation for convection dominated diffusion problems

We derive sharp L~∞(L~1) a posteriori error estimate for the convection dominated diffusion equations of the formThe derived estimate is insensitive to the diffusion parameter ε→0. The problem is discretized implicitly in time via the method of characteristics and in space via continuous     

Generalized speculative computation of parallel simulated annealing

Simulated annealing is known to be highly sequential due to dependences between iterations. While the conventional speculative computation with a binary tree has been found effective for parallel simulated annealing, its performance is limited to (logp)-fold speedup due to parallel execution of logp iterations onp processors. This report presents a new approach to parallel simulated annealing, calledgeneralized speculative computation (GSC). The GSC is synchronous, maintaining the same decision sequence as sequential simulated annealing. The use of two loop indices encoded in a single integer eliminates broadcasting of central data structure to all processors. The master-slave parallel programming paradigm simplifies controlling the activities ofp iterations which are executed in parallel onp processors. To verify the performance of GSC, we implemented 100-city to 500-city Traveling Salesman Problems on the AP1000 massively parallel multiprocessor. Execution results on the AP1000 demonstrate that the GSC approach can indeed be an effective method for parallel simulated annealing as it gave over 20-fold speedup on 100 processors.     

Decomposition of linear programs using parallel computation

This paper describes DECOMPAR: an implementation of the Dantzig-Wolfe decomposition algorithm for block-angular linear programs using parallel processing of the subproblems. The software is based on a robust experimental code for LP decomposition and runs on the CRYSTAL multicomputer at the University of Wisconsin-Madison. Initial computational experience is reported. Promising directions in future development of this approach are discussed.Research supported in part by the Office of Naval Research under grant N00014-87-K-0163.     

Massively parallel computation of many-variable polynomial interpolation

In this paper, we present homogeneous polynomials in many variables. We show how the hypercube representation of these polynomials (introduced by Beauzamy et al. in [1], and derived from Bombieri's work in Beauzamy et al. [2]) allows us to build interpolation polynomials, that is, polynomials taking prescribed values at prescribed points in . We then show that the construction is robust and give quantitative estimates on how the constructed polynomial is perturbed if either the data, the points, or both are perturbed. The theorems, constructions, and algorithms answer questions asked by Dr. Ken Clark, U.S. Army Research Office.

In the final part of the paper, we present the explicit algorithms, implemented on the Connection Machines CM200 and CM5 at the Etablissement Technique Central de l'Armement, Arcueil. This algorithm is efficient, especially when the number of variables is high, and it takes all advantage of the massively parallel architecture.     

Vector and parallel computing for matrix balancing

Estimating the entries of a large matrix to satisfy a set of internal consistency relations is a problem with several applications in economics, urban and regional planning, transportation, statistics and other areas. It is known as theMatrix Balancing Problem. Matrix balancing applications arising from the estimation of telecommunication or transportation traffic and from multi-regional trade flows give rise to huge optimization problems. In this report, we show that the RAS algorithm can be specialized for vector and parallel computing and used for the solution of very large problems. The algorithm is specialized for vector computations on a CRAY X-MP and is parallelized on an Alliant FX/8. A variant of the algorithm — developed here for its potential parallelism — turns out to be more efficient than the original algorithm even when implemented serially. We use the algorithms to estimate disaggregated input/output tables and a multi-regional trade flow table of the U.S. The larger problem solved has approximately 12 000 constraints and over 370 000 nonlinear variables. This is the first of two papers that aim at the solution of very large matrix balancing problems. Zenios [20] is using the same algorithm for the same models on a massively parallel Connection Machine CM-2.Research partially supported by NSF grants ECS-8718971 and CCR-8811135, and AFOSR grant 89-0145. Computing resources were made available through the ACRF at Argonne National Laboratory and CRAY Research, Inc.     

Binary tree algebraic computation and parallel algorithms for simple graphs

In this paper we define the binary tree algebraic computation (BTAC) problem and develop an efficient parallel algorithm for solving this problem. A variety of graph problems (minimum covering set, minimum r-dominating set, maximum matching set, etc.) for trees and two terminal series parallel (TTSP) graphs can be converted to instances of the BTAC problem. Thus efficient parallel algorithms for these problems are obtained systematically by using the BTAC algorithm. The parallel computation model is an exclusive read exclusive write PRAM. The algorithms for tree problems run in O(log n) time with O(n) processors. The algorithms for TTSP graph problems run in O(log m) time with O(m) processors where n (m) is the number of vertices (edges) in the input graph. These algorithms are within an O(log n) factor of optimal.     

On the computation of all supported efficient solutions in multi-objective integer network flow problems

This paper presents a new algorithm for identifying all supported non-dominated vectors (or outcomes) in the objective space, as well as the corresponding efficient solutions in the decision space, for multi-objective integer network flow problems. Identifying the set of supported non-dominated vectors is of the utmost importance for obtaining a first approximation of the whole set of non-dominated vectors. This approximation is crucial, for example, in two-phase methods that first compute the supported non-dominated vectors and then the unsupported non-dominated ones. Our approach is based on a negative-cycle algorithm used in single objective minimum cost flow problems, applied to a sequence of parametric problems. The proposed approach uses the connectedness property of the set of supported non-dominated vectors/efficient solutions to find all integer solutions in maximal non-dominated/efficient facets.