Vector Processing Computer Research Articles

The direct CI method. A detailed analysis

A thorough analysis of the direct CI method as applied to the case of a general set of reference configurations coupled to all single and double substitutions is presented. It is pointed out that there is no single strategy which proves optimal under all circumstances. A variety of procedures are therefore presented together with rules to enable the selection of the most favourable under a given circumstance. Much emphasis has been placed on organizing the calculations via a series of matrix multiplications, which enables a vector or array processing computer to be used to best effect. Some consideration is given to using an atomic integral (rather than molecular integral) driven scheme for some interactions, thus removing the necessity for a complete transformation of the molecular integrals to a molecular orbital basis, and the advantages and disadvantages of so doing are discussed. Improved procedures for carrying out both full and partial transformations of the molecular integrals are described. A number of test case calculations involving configuration lists of the order of 104 to 105 have been analysed in detail, to give a clear picture of the cost of the various interaction types which arise, and indicating that integrals carrying two external molecular orbital indices account for approximately 60 per cent of the cost in typical cases. Typically, a calculation involving 105 configurations requires approximately two minutes of CRAY-1 computer time, allowing for ten iterations of the diagonalization procedure. The cost of the calculations are found, somewhat surprisingly, to be approximately linear in the dimension of the configuration space, indicating that calculations involving 106 configurations are now quite feasible.

An implementation of a tree code on a SIMD, parallel computer

We describe a fast tree algorithm for gravitational N-body simulation on SIMD parallel computers. The tree construction uses fast, parallel sorts. The sorted lists are recursively divided along their x, y and z coordinates. This data structure is a completely balanced tree (i.e., each particle is paired with exactly one other particle) and maintains good spatial locality. An implementation of this tree-building algorithm on a 16k processor Maspar MP-1 performs well and constitutes only a small fraction (approximately 15%) of the entire cycle of finding the accelerations. Each node in the tree is treated as a monopole. The tree search and the summation of accelerations also perform well. During the tree search, node data that is needed from another processor is simply fetched. Roughly 55% of the tree search time is spent in communications between processors. We apply the code to two problems of astrophysical interest. The first is a simulation of the close passage of two gravitationally, interacting, disk galaxies using 65,636 particles. We also simulate the formation of structure in an expanding, model universe using 1,048,576 particles. Our code attains speeds comparable to one head of a Cray Y-MP, so single instruction, multiple data (SIMD) type computers can be used for these simulations. The cost/performance ratio for SIMD machines like the Maspar MP-1 make them an extremely attractive alternative to either vector processors or large multiple instruction, multiple data (MIMD) type parallel computers. With further optimizations (e.g., more careful load balancing), speeds in excess of today's vector processing computers should be possible.

A comparison of fast chemical kinetic solvers for air quality modeling

Abstract Two methods for solving stiff systems of ordinary differential equations describing nonlinear chemical kinetics in air quality models, namely the hybrid and the quasi-steady state approximation (QSSA) schemes, are compared with respect to their accuracy and computational speed. Their implementation for parallel and vector processing computers is discussed. Tests are conducted using two different photochemical mechanisms. Also, a new test problem is developed to represent various degrees of stiffness encountered in urban and regional air quality simulations. It is concluded that the hybrid scheme is more accurate than the QSSA scheme. Different techniques are considered to eliminate conservation errors and make the solutions more accurate. As for the computational efficiency, the QSSA scheme is approximately two to four times faster than the hybrid scheme.

Power system steady state security analysis using vector processing computers

The main difficulties encountered in applying a vector processor to the solution of a power flow is the sparsity of the power flow matrices which results in very short vectors when the nonzero terms are gathered into a packed vector format. The authors describe a steady-state security analysis code which overcomes the problem of very short vectors. The length of the list of outages is the major problem in security analysis since it usually includes at least each individual piece of equipment taken one at a time, and a large number of cases with multiple outages. The use of contingency selection and screening has opened the way to reducing this burden, but a very large number of individual cases must be run to satisfy the need to be sure that no cases have gone unchecked. The efforts described have extended the computational speed with which this can be accomplished, and the implementation of very high performance security analysis codes has been made possible.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A vector processing algorithm of auxiliary integral evaluation for two‐electron Gaussian integrals

AbstractA six‐term auxiliary integral expression for the two‐electron Gaussian integral is derived on the basis of the Chebyshev polynomial approximation instead of the seven‐term Taylor expansion. This expression and the related recurrence formula enable us to perform a high‐speed calculation on a vector processing computer.

MDL: A system for fast simulation of large layered neural networks

The Model Development Language (or MDL) is a simulation language and system which uses an assignment statement notation tailored for 2-dimensional layered neural networks. The key features are: (1) Operations occur on 2-dimensional arrays which correspond to the layout of many brain structures. (2) A spatial convolution operation models spatial summation occurring in dendritic trees. (3) Arbitrary time invariant expressions and differential equations may be specified by simple assignment statements. (4) Numerical methods are automatically taken care of in the MDL environment rather than in the user's specification of the model. These features allow high level modeling using leaky integrator model neurons and modeling of detailed biophysical properties, such as the Hodgkin-Huxley equations and voltage clamp experiments. Conciseness of the language allows specification of a moderately complex retina model in only one or two pages of code. The parallel structure of the language allows highly optimized implementation on parallel or vector processing computers. Use of the language is illustrated through two ex amples.

Time-dependent wavepacket algorithm for inelastic molecule-molecule scattering

We present a time-dependent wavepacket propagation method for diatom-diatom inelastic scattering. This method is efficient for systems involving large numbers of states, and the algorithm used is suitable for and tested on a vector-processing computer. The numerical test is performed for a model HF-HF system, and the results obtained agree well with those from another calculation performed by the time-independent R-matrix propagation method.

Diagrammatic many-body perturbation expansion for atoms and molecules: VII experiments in vector and parallel processing for fourth-order energy terms involving triply excited intermediate states

The efficient evaluation of the fourth-order energy terms corresponding to linked diagrams involving triply excited intermediate states which arise in the many-body perturbation expansion for the electron correlation energy of closed-shell atoms and molecules on vector processing and parallel processing computers is discussed. The linked-diagram theorem decouples the many-electron problem allowing efficient implementation on parallel processing machines. Furthermore, the computation associated with each of the resulting subproblems is very well suited to vector processing machines. A “dynamic load balancing” technique is employed in both a dedicated and a multi-user environment. Timing tests are reported for the CRAY X-MP/416 computer. Execution rates in excess of 828 Mflops are observed.

Light scattering calculations by the Fredholm integral equation method

The Fredholm integral equation method (FIM), originally introduced by Holtet al. to solve the light scattering problem for ellipsoidal particles, is reinvestigated by taking into account a recent great progress in numerical computers. A numerical code optimized for vector-processing computers is developed, and is applied to the light scattering by spherical and spheroidal particles. The results for these particles are compared with those by the Mie theory and by Asano and Yamamoto, respectively, and it is confirmed that the agreement with both of them is satisfactory. Sample calculations are also performed for the ‘oblique’ incidence, in which the direction of incidence is not parallel nor perpendicular to the symmetry axis of the particle. No difficulties in the computation are found compared with the calculations for the parallel or perpendicular incidence. We study the efficiency factor for polarization (Qpol) in general direction of incidence for spheroidal particles, and discuss the deviation from the Rayleigh approximation.

Simulation of three‐dimensional flow of immiscible fluids within and below the unsaturated zone

This paper presents a two‐phase flow model based on a three‐dimensional, finite‐difference formulation. As three‐dimensional simulations can require substantial computer effort, a numerical technique that takes advantage of vector and parallel processing computer architecture is developed. The model is posed in terms of water saturation and nonwetting fluid pressure. It uses three‐phase capillary pressure and relative permeability relationships to permit simulation within or below the unsaturated zone. A modified formulation of slice successive overtaxation (an iterative matrix solution technique) is introduced. This technique is designed to use parallel processing capabilities of new computers. The model is applied to immiscible fluid flow at two chemical waste landfills near Niagara Falls, New York. At both sites, denser than water, nonaqueous liquids (NAPLs) are present in the groundwater regimes in relatively large quantities. The model applications address several technical concerns at the two sites, including the effectiveness of clay as a geologic barrier to NAPL migration owing to capillary pressure forces, the three‐dimensional aspects of dense NAPL flow, and the sensitivity of NAPL recovery in pumping wells due to various hydrogeologic and fluid properties. The results of the applications show that (1) even under a downward hydraulic gradient, natural differences in capillary pressure relationships for different lithologies can prevent downward migration of NAPL, (2) without any lithologic‐capillary barrier, an upward hydraulic gradient induced by a de watering system can prevent downward migration of NAPL, (3) NAPL recovery at wells is sensitive to relative permeability, a relationship that requires field calibration in many settings, and (4) the three‐dimensional aspects of two‐phase flow and hydrogeologic heterogeneity require explicit treatment in many settings.

Diagrammatic many-body perturbation expansion for atoms and molecules: VI: Experiments in vector processing and parallel processing for second-order energy calculations

The efficient evaluation of the second-order expression in the many-body perturbation theory expansion for the correlation energy on vector processing and parallel processing computers is discussed. It is argued that the linked diagram theorem not only leads to the well known theoretical advantages of the many-body perturbation theory approach which allows the calculation of correlation energies for large (i.e. extended molecules or species containing heavy atoms) systems but also decouples the many-electron problem allowing efficient implementation on parallel processing machines. Furthermore, the computation associated with each of the resulting subproblems is very well suited to vector processing machines. Timing tests are reported for the CRAY 1 and CDC Cyber 205 vector processors, for a 1 processor implementation on the CRAY X-MP/48 and the ETA-10E, and for a 4 processor implementation on the Cray X-MP/48.

A new iterative scheme for obtaining eigenvectors of large, real-symmetric matrices

A rapidly convergent block-iterative scheme for the computation of a few of the lowest eigenvalues and eigenvectors of a large dense real symmetric matrix is proposed. The method is especially applicable to matrix eigenvalue problems that arise from the discretization of self-adjoint partial differential equations. One such application to certain symmetric matrices that arise in solid-state band structure calculations is considered in detail. The most timeconsuming parts of the present algorithm are a matrix multiplication and a Gauss-Siedel relaxation step which are performed on each iteration. These two parts can, however, be very efficiently implemented on a vector or parallel processing computer.

Structure of a simple molecular dynamics FORTRAN program optimized for CRAY vector processing computers

A program structure for efficient vectorization of molecular dynamics FORTRAN programs on CRAY vector processing computers is described. Though coded for a very simple pure atomic fluid in a cubic cell with periodic boundary conditions the program can easily be modified to handle more complicated systems. A detailed analysis shows that the present program is faster by 36% for N = 256 particles and faster by more than a factor of 3 for N = 2048 compared with a fully vectorized molecular dynamics program written for the CYBER 205 vector processing machine. In comparison with a link cell MD program also written for a CRAY the program described here runs three times faster for a large particle number N = 6912. This factor increases with decreasing N to 6.3 for N = 1372. The speedup is achieved by 1. i) long vectors in inner loops wherever possible; 2. ii) limiting the number of arithmetic operations in inevitably short loops as much as possible; 3. iii) appropriate library routines; 4. iv) integer index vector neighbour lists.

A vectorization technique for the conjugate gradient method by using element matrices.

With the wider use of vector-processing computers, much attention has come to be paid to the conjugate gradient method to solve a linear system of algebraic equations. However, in the case of finite element methods, it is rather difficult to tune up the computer program of the conjugate gradient method so as to make a big reduction in CPU time, because a coefficient matrix appearing in the finite element analysis is sparse and the nonzero components are irregularly located. In this paper, we present a vectorization technique for the conjugate gradient method by using element matrices only. The loop length attained by the present technique is as long as the number of elements and the rate of acceleration is very high.

Solving a large nonlinear programming problem on a vector processing computer

In an earlier paper, the authors formulated an acoustic antenna array design problem as a nonlinear program. Computation times for large problems (about 400 sensors) were in the range of 8 to 10 hours on a Vax 11/780. Most of this time was spent in computing the objective function and its gradient. This paper describes how these computations (and these only) were recoded to exploit the vector processing capabilities of a Cray 1-M computer. Run times are reduced to less than one minute. The results have implications for many nonlinear programs whose function evaluations are very time consuming.

Nonlinear Structural Analysis of Complex Electronic and Electromechanical Assemblies

The finite-element technique, which is particularly well suited to problems with complex geometries, has been extensively used in structural mechanics. For the highly nonlinear problem, however, the technique has been limited to problems with a modest number of elements for want of a solution scheme that efficiently uses computer resources. We have found that the conjugate gradient method, taken from the field of optimization, provides the required solution technique. The technique has been adapted in a finite-element formulation for structural mechanics and configured to take advantage of vector processing computer hardware, resulting in the development of two computer programs, the two-dimensional JAC1 and the three-dimensional JAC3D.2 These programs provide the analyst with powerful, efficient tools to solve for the nonlinear quasi-static response of solids. In this paper, we describe the development and capabilities of JAC and JAC3D. Following this, we present a selected group of applications relating to the structural analysis of electronic and electromechanical subsystems.

A Monte Carlo FORTRAN 200 programme for the determination of static properties of liquids vectorized to run on the CYBER 205 vector processing computer

We present a Monte Carlo programme version written in Vector-FORTRAN 200 which allows a fast computation of thermodynamic properties of dense model fluids on the CYBER 205 vector processing computer. A comparison of the execution speed of this programme, a scalar version and a vectorized molecular dynamics programme showed the following: (i) the vectorized form of the Monte Carlo programme runs about a factor of 8 faster on the CYBER 205 than the scalar version on the conventional computer CYBER 855; (ii) for small ensembles of 32–108 particles, the Monte Carlo programme is of about the velocity as the molecular dynamics one. However, for larger numbers of particles, the molecular dynamics programme is vastly faster executed on the CYBER 205 than the Monte Carlo programme, particularly when neighbour tables are used. We propose a technique to accelerate the Monte Carlo programme for larger ensembles.

Microcrack-induced damage accumulation in brittle rock under dynamic loading

Abstract This paper is concerned with the development of a computational constitutive model which simulates the dynamic fracture behavior of brittle rock. The essential feature of this model is the treatment of the dynamic fracture process in rock as a continuous accrual of damage, where the damage mechanism has been attributed to the microcracking in the rock medium. Detailed descriptions of the model and its numerical implementation are given. A method is presented for determining the three additional material constants governing the damage accumulation. All of the numerical algorithms are amenable to vectorization for a vector processing computer and the model is computationally very efficient. A numerical example of an oil-shale blasting experiment is presented and compared to field observations. Results demonstrate that the model is applicable to the prediction of dynamic rock-fracture behavior.

Efficient recursive computation of molecular integrals over Cartesian Gaussian functions

Recurrence expressions are derived for various types of molecular integrals over Cartesian Gaussian functions by the use of the recurrence formula for three-center overlap integrals. A number of characteristics inherent in the recursive formalism allow an efficient scheme to be developed for molecular integral computations. With respect to electron repulsion integrals and their derivatives, the present scheme with a significant saving of computer time is found superior to other currently available methods. A long innermost loop incorporated in the present scheme facilitates a fast computation on a vector processing computer.

Restrained structure-factor least-squares refinement of protein structures using a vector processing computer

A least-squares refinement program RESTRAIN has been developed, which is capable of refining macromolecular structures using structure amplitudes, phases from isomorphous replacement or anomalous scattering and pseudo-energy restraints. In addition to positional parameters and isotropic temperature factors, anisotropic mean-square displacements may be refined either as individual atomic U tensors or as TLS tensors applied to groups of atoms. Anharmonic effects may be handled by coupling together occupancies to enable the electron density of an atomic group to be distributed over more than one subsite. A novel way of restraining groups of atoms to be planar has been developed that does not require dummy atoms and does not restrain the plane to lie in its current orientation.