Garry H. Rodrigue

Approximating the inverse of a matrix for use in iterative algorithms on vector processors

Most iterative techniques for solving the symmetric positive-definite systemAx=b involve approximating the matrixA by another symmetric positive-definite matrixM and then solving a system of the formMz=d at each iteration. On a vector machine such as the CDC-STAR-100, the solution of this new system can be very time consuming. If, however, an approximationM−1 can be given toA−1, the solutionz=M−1d can be computed rapidly by matrix multiplication, a fast operation on the STAR. Approximations using the Neumann expansion of the inverse ofA give reasonable forms forM−1 and are presented. Computational results using the conjugate gradient method for the “5-point” matrixA are given.ZusammenfassungDie meisten iterativen Methoden zur Lösung des symmetrischen positiv-definitiven SystemsAx=b enthalten die Näherung der MatrixA durch eine andere symmetrische positiv-definitive MatrixM und anschließend daran die Lösung eines Systems der ArtMz=d bei jeder Wiederholung. Auf einer Vektor-Maschine wie der CDC-STAR-100 kann die Lösung dieses neuen Systems sehr zeitraubend sein. Wenn jedoch eine NäherungM−1 zuA−1 gegeben werden kann, so kann die Lösungz=M−1d sehr schnell durch Matrixmultiplikation errechnet werden. Diese Kalkulation kann auf dem STAR schnell ausgeführt werden. Näherungen, bei denen die Neumann-Entwicklung der Inversen vonA verwendet wird, ergeben angemessene Ausdrücke fürM−1. Diese Ausdrücke sind angeführt. Die mit Hilfe der Konjugierten-Gradienten-Methode errechneten Resultate für die „5-Punk”-MatrixA sind angegeben.

High-order symplectic integration methods for finite element solutions to time dependent Maxwell equations

In this paper, we motivate the use of high-order integration methods for finite element solutions of the time dependent Maxwell equations.. In particular, we present a symplectic algorithm for the integration of the coupled first-order Maxwell equations for computing the time dependent electric and magnetic fields. Symplectic methods have the benefit a conserving total electromagnetic field energy and are, therefore, preferred over dissipative methods (such as traditional Runge-Kutta) in applications that require high-accuracy and energy conservation over long periods of time integration. We show that in the context or symplectic methods, several popular schemes can be elegantly cast in a single algorithm. We conclude with some numerical examples which demonstrate the superior performance of high-order time integration methods.

A Vector Finite Element Time-Domain Method for Solving Maxwell's Equations on Unstructured Hexahedral Grids

In this paper the vector finite element time-domain (VFETD) method is derived, analyzed, and validated. The VFETD method uses edge vector finite elements as a basis for the electric field and face vector finite elements as a basis for the magnetic flux density. The Galerkin method is used to convert Maxwells equations to a coupled system of ordinary differential equations. The leapfrog method is used to advance the fields in time. The method is shown to be stable and to conserve energy and charge for arbitrary hexahedral grids. A numerical dispersion analysis shows the method to be second order accurate on distorted hexahedral grids. Several computational experiments are performed to determine the accuracy and efficiency of the method.

Matrix multiplication by diagonals on a vector/parallel processor☆

The advent of vector and parallel computers has forced the reexamination, reformulation, and 9 rethinking of essentially all of the basic mathematical algorithms. In this paper, we will bonsider the seemingly straightforward process of matrix multiplication. We will be primarily concerned with this process on a vector processor, the CDC STAR-100. For large full matrices, matrix *multiplication is easily “vectorized” when the matrix is stored by columns in the typical Fortran fashion. However, there are at least two disadvantages to this normal approach. First, it becomes quite inefficient for banded matrices with relatively narrow bandwidths. Second, when a matrix is stored by columns (or rows), the transpose of the matrix is not as readily available for use in a vector form (this is particularly a problem on the STAR-100). The purpose of this paper is to present a new algorithm for matrix multiplication which: is readily “‘vectorizetl”, is very efficient for narrow banded matrices, and allows for the transpose to be easily accessible in a vector form.

A generalized mass lumping technique for vector finite-element solutions of the time-dependent Maxwell equations

Time-domain finite-element solutions of Maxwells equations require the solution of a sparse linear system involving the mass matrix at every time step. This process represents the bulk of the computational effort in time-dependent simulations. As such, mass lumping techniques in which the mass matrix is reduced to a diagonal or block-diagonal matrix are very desirable. In this paper, we present a special set of high order 1-form (also known as curl-conforming) basis functions and reduced order integration rules that, together, allow for a dramatic reduction in the number of nonzero entries in a vector finite element mass matrix. The method is derived from the Nedelec curl-conforming polynomial spaces and is valid for arbitrary order hexahedral basis functions for finite-element solutions to the second-order wave equation for the electric (or magnetic) field intensity. We present a numerical eigenvalue convergence analysis of the method and quantify its accuracy and performance via a series of computational experiments.

Vector quantization of ECG wavelet coefficients

An improved wavelet compression algorithm for ECG signals has been developed with the use of vector quantization on wavelet coefficients. Vector quantization on scales of long duration and low dynamic range retains feature integrity of the ECG with a very low bit-per-sample rate. Preliminary results indicate that the proposed method excels over standard techniques for high fidelity compression.<<ETX>>

Inner/outer iterative methods and numerical Schwarz algorithms

Abstract Variants of the numerical Schwarz algorithms for solving elliptic partial differential equations on multiprocessing systems are described and analyzed. the methods are described in terms of domain decomposition techniques and mathematically cast into an inner/outer iterative form. It is shown that under certain matrix nonnegativity conditions that the convergence rate of the global iteration is invariant to the amount of overlap of the subdomains.

Improved conditioning of finite element matrices using new high-order interpolatory bases

The condition number of finite element matrices constructed from interpolatory bases will grow as the polynomial degree of the basis functions is increased. The worst case scenario for this growth rate is exponential and in this paper we demonstrate through computational example that the traditional set of uniformly distributed interpolation points yields this behavior. We propose a set of nonuniform interpolation points which yield a much improved polynomial growth rate of condition number. These points can be used to construct several types of popular hexahedral basis functions including the 0-form (standard Lagrangian), 1-form (Curl conforming), and 2-form (Divergence conforming) varieties. We demonstrate through computational example the benefits of using these new interpolatory bases in finite element solutions to Maxwells equations in both the frequency and time domain.

Odd-Even Reduction for Banded Linear Equations

The method of odd-even reduction for tridiagonal systems is generalized to banded systems. The method is developed so that it can be easily implemented on a vector processor such as the CDC STAR-100. Results are presented which describe when this odd-even reduction can be performed on a pentadiagonal system. A computational example is given. 1 table.

Benchmarks and models for time-dependent grey radiation transport with material temperature in binary stochastic media

Abstract We present benchmark calculations for radiation transport coupled to a material temperature equation in a slab geometry binary random media. The mixing statistics are taken to be homogeneous Markov statistics where the material chunk sizes are described by Poisson distribution functions. The material opacities are taken to be constant. Benchmark values for time evolution of the ensemble average values of material temperature energy density and radiation transmission are computed via a Monte Carlo-type method. These benchmarks are used as a basis for comparison with three other approximate methods of solution. One of these approximate methods is simple atomic mix which is seen to consistently over absorb resulting in lower steady-state radiation transmission and material temperature. The second approximate model is an adaptation of what is commonly called the Levermore–Pomraning model and which we refer to as the standard model. It is shown to consistently under absorb resulting in higher steady-state radiation transmission and material temperature. We show that recasting the temperature coupling as a type of effective scattering can be useful in formulating the third approximate model, an adaptation of a model due to Su and Pomraning which attempts to account for the effects of scattering in a stochastic context. We show this last adaptation shows consistent improvement over both the atomic mix and standard models.