In this paper we provide a robust reordering scheme for sparse matrices. The scheme relies on the notion of multisection, a generalization of bisection. The reordering strategy is demonstrated to have consistently good performance in terms of fill reduction when compared with multiple minimum degree and generalized nested dissection. Experimental results show that by using multisection, we obtain an ordering which is consistently as good as or better than both for a wide spectrum of sparse problems. 1 Introduction It is well recognized that finding a fill-reducing ordering is crucial in the success of the numerical solution of sparse linear systems. For symmetric positive-definite systems, the minimum degree [38] and the nested dissection [11] orderings are perhaps the most popular ordering schemes. They represent two opposite approaches to the ordering problem. However, they share a common undesirable characteristic. Both schemes produce generally good orderings, but the ordering qua...

In this paper we present the Generic Graph Component Library (GGCL), a generic programming framework for graph data structures and graph algorithms. Following the theme of the Standard Template Library (STL), the graph algorithms in GGCL do not depend on the particular data structures upon which they operate, meaning a single algorithm can operate on arbitrary concrete representations of graphs. To attain this type of flexibility for graph data structures, which are more complicated than the containers in STL, we introduce several concepts to form the generic interface between the algorithms and the data structures, namely, Ve r ex, Edge, Visitor, andDecorator. We describe the principal abstractions comprising the GGCL, the algorithms and data structures that it provides, and provide examples that demonstrate the use of GGCL to implement some common graph algorithms. Performance results are presented which demonstrate that the use of novel lightweight implementation techniques and stat...

Most of the current techniques for the direct solution of linear equations are based on supernodal or multifrontal approaches. An important feature of these methods is that arithmetic is performed on dense submatrices and Level 2 and Level 3 BLAS (matrixvector and matrix-matrix kernels) can be used. Both sparse LU and QR factorizations can be implemented within this framework. Partitioning and ordering techniques have seen major activity in recent years. We discuss bisection and multisection techniques, extensions to orderings to block triangular form, and recent improvements and modifications to standard orderings such as minimum degree. We also study advances in the solution of indefinite systems and sparse least-squares problems. The desire to exploit parallelism has been responsible for many of the developments in direct methods for sparse matrices over the last ten years. We examine this aspect in some detail, illustrating how current techniques have been developed or ...

In this paper we investigate the additional storage overhead needed for a parallel implementation of finite element applications. In particular, we compare the storage requirements for the factorization of the sparse matrices that would occur on parallel processor versus a uniprocessor. This variation in storage results from the factorization fill-in. We address the question of whether the storage overhead is so large for parallel implementations that it imposes severe limitations on the problem size in contrast to the problems executed sequentially on a uniprocessor. The storage requirements for the parallel implementation is based upon a new ordering scheme, the Combination Mesh-Based scheme. This scheme uses a domain decomposition method which attempts to balance the processors' loads and decrease the interprocessor communication. The storage requirements for the sequential implementation is based upon the Minimum Degree algorithm. The difference between the two storage requirements...

In this review paper, we consider some important developments and trends in algorithm design for the solution of linear systems concentrating on aspects that involve the exploitation of parallelism. We briefly discuss the solution of dense linear systems, before studying the solution of sparse equations by direct and iterative methods. We consider preconditioning techniques for iterative solvers and discuss some of the present research issues in this field. Keywords: linear systems, dense matrices, sparse matrices, tridiagonal systems, parallelism, direct methods, iterative methods, Krylov methods, preconditioning. AMS(MOS) subject classifications: 65F05, 65F50. 1 Introduction Solution methods for systems of linear equations Ax = b; (1) where A is a coefficient matrix of order n and x and b are n-vectors, are usually grouped into two distinct classes: direct methods and iterative methods. However, CCLRC - Rutherford Appleton Laboratory, Oxfordshire, England and CERFACS, Toulouse,...

We review current methods for the direct solution of sparse linear equations. We discuss basic concepts such as fill-in, sparsity orderings, indirect addressing and compare general sparse codes with codes for dense systems. We examine methods for greatly increasing the efficiency when the matrix is symmetric positive definite. We consider frontal and multifrontal methods emphasizing how they can take advantage of vectorization, RISC architectures, and parallelism. Some comparisons are made with other techniques and the availability of software for the direct solution of sparse equations is discussed.

We review the influence of the advent of high performance computing on the solution of linear equations. We will concentrate on direct methods of solution and consider both the case when the coefficient matrix is dense and when it is sparse. We will examine the current performance of software in this area and speculate on what advances we might expect in the early years of the next century. Keywords: sparse matrices, direct methods, parallelism, matrix factorization, multifrontal methods. AMS(MOS) subject classifications: 65F05, 65F50. 1 Current reports available at http://www.cerfacs.fr/algor/algo reports.html. Also appeared as Technical Report RAL-TR-1999-072 from Rutherford Appleton Laboratory, Oxfordshire. 2 duff@cerfacs.fr. Also at Atlas Centre, RAL, Oxon OX11 0QX, England. Rutherford Appleton Laboratory. Contents 1 Introduction 1 2 Building blocks 1 3 Factorization of dense matrices 2 4 Factorization of sparse matrices 4 5 Parallel computation 8 6 Current situation 12 7 F...

this paper is the application of a multivariate optimization technique called "simulated annealing" that can in principle be applied to any minimizant or maximizant in NP-hard problems. In particular, we'll investigate whether it can be practically applied to the minimizants of profile and fill to produce better orderings of A. Previous work in [1] looked at the minimizants of profile, wavefront, and bandwidth. We will compare our results with this work where they overlap. 2 Simulated Annealing (Most of this section follows the presentation in [11], although [8] presented the original idea.) The statistical behaviour of physical systems with large numbers of degrees of freedom inspired the simulated annealing technique. To "anneal" is "to heat (glass, metals, etc.) and then cool slowly to prevent brittleness". This is just one instance of a fundamental observation about nature: given sufficient time and a mechanism to do so, a system will always tend to adjust itself to a minimal energy state. For instance ffl The surface of a lake is flat. ffl (Slowly) cooled liquids form highly-regular crystals. ffl Air molecules spread evenly in a room. All of these represent systems with large numbers of degrees of freedom achieving global minima. Most iterative techniques that attempt to solve global optimization problems are, in a sense, "greedy": as soon as they find a better solution, they adopt it. For this reason, these techniques don't always behave well 5 in the presence of local minima. How, then, are molecules able to "solve" the minimal energy problem globally? Because nature gives them sufficient time and energy to rearrange themselves in a way that ultimately satisfies the global minimum. The Boltzmann distribution permits a system to exist in a state that is energy E...

The PETSc (Portable Extensible Tools for Scientific computing) library arose from research into domain decomposition methods which require combining many different solutions in a single application. The initial efforts tried to use existing numerical software but had limited success. The problems include everything from faulty assumptions about the computing environment (e.g., how many processes there are) to implicit (yet deadly) global state. More recently, PETSc and PVODE have found a way to cooperate, and new techniques that exploit dynamically linked libraries offer a more general approach to interoperable components. The paper highlights some of the issues in building sharable component software and discussing mistakes still often made in designing, maintaining, documenting, and testing components.

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