A bibliography on parallel and vector numerical algorithms

This is a bibliography of numerical methods. It also includes a number of other references on machine architecture, programming language, and other topics of interest to scientific computing. Certain conference proceedings and anthologies which have been published in book form are listed also

Group implicit concurrent algorithms in nonlinear structural dynamics

During the 70's and 80's, considerable effort was devoted to developing efficient and reliable time stepping procedures for transient structural analysis. Mathematically, the equations governing this type of problems are generally stiff, i.e., they exhibit a wide spectrum in the linear range. The algorithms best suited to this type of applications are those which accurately integrate the low frequency content of the response without necessitating the resolution of the high frequency modes. This means that the algorithms must be unconditionally stable, which in turn rules out explicit integration. The most exciting possibility in the algorithms development area in recent years has been the advent of parallel computers with multiprocessing capabilities. So, this work is mainly concerned with the development of parallel algorithms in the area of structural dynamics. A primary objective is to devise unconditionally stable and accurate time stepping procedures which lend themselves to an efficient implementation in concurrent machines. Some features of the new computer architecture are summarized. A brief survey of current efforts in the area is presented. A new class of concurrent procedures, or Group Implicit algorithms is introduced and analyzed. The numerical simulation shows that GI algorithms hold considerable promise for application in coarse grain as well as medium grain parallel computers

Report from the MPP Working Group to the NASA Associate Administrator for Space Science and Applications

NASA's Office of Space Science and Applications (OSSA) gave a select group of scientists the opportunity to test and implement their computational algorithms on the Massively Parallel Processor (MPP) located at Goddard Space Flight Center, beginning in late 1985. One year later, the Working Group presented its report, which addressed the following: algorithms, programming languages, architecture, programming environments, the way theory relates, and performance measured. The findings point to a number of demonstrated computational techniques for which the MPP architecture is ideally suited. For example, besides executing much faster on the MPP than on conventional computers, systolic VLSI simulation (where distances are short), lattice simulation, neural network simulation, and image problems were found to be easier to program on the MPP's architecture than on a CYBER 205 or even a VAX. The report also makes technical recommendations covering all aspects of MPP use, and recommendations concerning the future of the MPP and machines based on similar architectures, expansion of the Working Group, and study of the role of future parallel processors for space station, EOS, and the Great Observatories era

Streamlining of the state-dependent Riccati equation controller algorithm for an embedded implementation

In many practical control problems the dynamics of the plant to be controlled are nonlinear. However, in most cases the controller design is based on a linear approximation of the dynamics. One of the reasons for this is that, in general, nonlinear control design methods are difficult to apply to practical problems. The State Dependent Riccati Equation (SDRE) control approach is a relatively new practical approach to nonlinear control that has the simplicity of the classical Linear Quadratic control method. This approach has been recently applied to control experimental autonomous air vehicles with relative success. To make the SDRE approach practical in applications where the computational resources are limited and where the dynamic models are more complex it would be necessary to re-examine and streamline this control algorithm. The main objective of this work is to identify improvements that can be made to the implementation of the SDRE algorithm to improve its performance. This is accomplished by analyzing the structure of the algorithm and the underlying functions used to implement it. At the core of the SDRE algorithm is the solution, in real time, of an Algebraic Riccati Equation. The impact of the selection of a suitable algorithm to solve the Riccati Equation is analyzed. Three different algorithms were studied. Experimental results indicate that the Kleinman algorithm performs better than two other algorithms based on Newton’s method. This work also demonstrates that appropriately setting a maximum number of iterations for the Kleinman approach can improve the overall system performance without degrading accuracy significantly. Finally, a software implementation of the SDRE algorithm was developed and benchmarked to study the potential performance improvements of a hardware implementation. The test plant was an inverted pendulum simulation based on experimental hardware. Bottlenecks in the software implementation were identified and a possible hardware design to remove one such bottleneck was developed

A high-accuracy optical linear algebra processor for finite element applications

Optical linear processors are computationally efficient computers for solving matrix-matrix and matrix-vector oriented problems. Optical system errors limit their dynamic range to 30-40 dB, which limits their accuray to 9-12 bits. Large problems, such as the finite element problem in structural mechanics (with tens or hundreds of thousands of variables) which can exploit the speed of optical processors, require the 32 bit accuracy obtainable from digital machines. To obtain this required 32 bit accuracy with an optical processor, the data can be digitally encoded, thereby reducing the dynamic range requirements of the optical system (i.e., decreasing the effect of optical errors on the data) while providing increased accuracy. This report describes a new digitally encoded optical linear algebra processor architecture for solving finite element and banded matrix-vector problems. A linear static plate bending case study is described which quantities the processor requirements. Multiplication by digital convolution is explained, and the digitally encoded optical processor architecture is advanced

Solution of partial differential equations on vector and parallel computers

The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed

Application of constrained optimisation techniques in electrical impedance tomography

A Constrained Optimisation technique is described for the reconstruction of temporal resistivity images. The approach solves the Inverse problem by optimising a cost function under constraints, in the form of normalised boundary potentials. Mathematical models have been developed for two different data collection methods for the chosen criterion. Both of these models express the reconstructed image in terms of one dimensional (I-D) Lagrange multiplier functions. The reconstruction problem becomes one of estimating these 1-D functions from the normalised boundary potentials. These models are based on a cost criterion of the minimisation of the variance between the reconstructed resistivity distribution and the true resistivity distribution. The methods presented In this research extend the algorithms previously developed for X-ray systems. Computational efficiency is enhanced by exploiting the structure of the associated system matrices. The structure of the system matrices was preserved in the Electrical Impedance Tomography (EIT) implementations by applying a weighting due to non-linear current distribution during the backprojection of the Lagrange multiplier functions. In order to obtain the best possible reconstruction it is important to consider the effects of noise in the boundary data. This is achieved by using a fast algorithm which matches the statistics of the error in the approximate inverse of the associated system matrix with the statistics of the noise error in the boundary data. This yields the optimum solution with the available boundary data. Novel approaches have been developed to produce the Lagrange multiplier functions. Two alternative methods are given for the design of VLSI implementations of hardware accelerators to improve computational efficiencies. These accelerators are designed to implement parallel geometries and are modelled using a verification description language to assess their performance capabilities