THE PRECONDITIONED CONJUGATE GRADIENT METHOD ON THE CONNECTION MACHINE

Computer Solution of Large Linear Systems

There has been a recent emergence of many interesting and highly efficient hierarchical (multilevel) algorithms (e.g. multigrid, domain decomposition, wavelets, multilevel preconditioning, the fast multipole algorithms, etc.) for solving numerical problems in scientific computing. These algorithms derive their efficiency from using a hierarchical approach to capture the sharing of global information which is inherent in the physical processes being modelled. In addition to being computationally efficient, these algorithms also possess relatively high degrees of parallelism. I therefore argue that the architectures of parallel computers (especially massively parallel ones) should be designed to support hierarchical communication and synchronization needs of these algorithms. Hierarchical architectures are also more universal because they go beyond supporting a particular class of algorithms to supporting the underlying physical processes being modelled. Finally, I propose that algorithm designers take a critical look at traditional kernel algorithms and re-examine their cost-effectiveness in a massively parallel computing environment.

The purpose of the article is to present and evaluate usefulness of a new preconditioning technique (for the Gauss-Seidel algorithm), namely incomplete WZ factorization, for iterative solving of sparse and singular linear equations systems, which arise during modeling with Markov chains. The incomplete WZ factorization proposed here will be compared with the incomplete LU factorization in respect of amount of fill-in (newly created non-zeros) and in respect of the accuracy improvement of preconditioned algorithms in relation to not preconditioned ones. In the paper, the results of some numerical experiments will be presented, which were conducted for various matrices representing Markov chains. The experiments show that the incomplete WZ factorization can be a real alternative - because it is faster than incomplete LU factorization and the fill-in generated in the process is smaller (the output matrices are sparser).

CHAPTER 7 - Iterative Solution of Finite Element Equations

We consider the problem of vectorizing the recursive calculations found in modified incomplete factorizations and SSOR preconditioners for the conjugate gradient method. We examine matrix problems derived from partial differential equations which are discretized on regular 2-D and 3-D grids, where the grid nodes are ordered in the natural ordering. By performing data dependency analyses of the computations, we show that there is concurrency in both the factorization and the forward and backsolves. The computations may be performed with an average vector length of $O(n)$ on an $n^2 $ or $n^3 $ grid in two and three dimensions. Numerical studies on four model problems show that the conjugate gradient method using these vectorized implementations of the modified incomplete factorizations and SSOR preconditioners achieves overall speeds approaching 100 megaflops on a Cray X-MP/24 vector computer. Furthermore, these methods require considerably less overall execution time than diagonal scaling and no-fill red-black incomplete factorization preconditioners, both of which allow full vectorization but are not as convergent.

When solving large linear systems of equations arising from the discretization of elliptic boundary value problems, a combination of iterative methods and preconditioners based on incomplete LU factorizations is frequently used. Given a model problem with variable coefficients, we investigate a class of incomplete LU factorizations depending on a relaxation parameter. We show that the associated preconditioner and the factorization itself both are numerically stable. The theoretical results are complemented by numerical experiments.

This paper investigates parallel iterative solvers with preconditioning for the computation of magnetic and electric fields on a printed circuit board (PCB) containing a magnetic film. Special scaled symmetric quasi-minimal residual (QMR) methods and a bi-conjugate gradient stabilized (BICGSTAB) method, preconditioned by an incomplete LU factorization with threshold (ILUT), are presented. The results show that these solvers are well suited for PCB simulations on parallel computers, requiring both real and complex arithmetic.

A highly efficient and robust scheme is proposed for analyzing electromagnetic scattering from electrically large arbitrary shaped conductors in a half space. This scheme is based on the electric field integral equation (EFIE) with a half-space Green’s function. The precorrected fast Fourier transform (p-FFT) is first extended to a half space for general three-dimensional scattering problems. A novel enhanced dual threshold incomplete LU factorization (ILUT) is then constructed as an effective preconditioner to improve the convergence of the half-space EFIE. Inspired by the idea of the improved electric field integral operator (IEFIO), the geometrical-optics current/principle value term of the magnetic field integral equation is used as a physical perturbation to stabilize the traditional ILUT perconditioning matrix. The high accuracy of EFIE is maintained, yet good calculating efficiency comparable to the combined field integral equation (CFIE) can be achieved. Furthermore, this approach can be applied to arbitrary geometrical structures including open surfaces and requires no extra types of Sommerfeld integrals needed in the half-space CFIE. Numerical examples are presented to demonstrate the high performance of the proposed solver among several other approaches in typical half-space problems.

Motivated by applications of distributed linear estimation, distributed control, and distributed optimization, we consider the question of designing linear iterative algorithms for computing the average of numbers in a network. Specifically, our interest is in designing such an algorithm with the fastest rate of convergence given the topological constraints of the network. As the main result of this paper, we design an algorithm with the fastest possible rate of convergence using a nonreversible Markov chain on the given network graph. We construct such a Markov chain by transforming the standard Markov chain, which is obtained using the Metropolis-Hastings method. We call this novel transformation <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pseudo-lifting</i> . We apply our method to graphs with geometry, or graphs with doubling dimension. Specifically, the convergence time of our algorithm (equivalently, the mixing time of our Markov chain) is proportional to the diameter of the network graph and hence optimal. As a byproduct, our result provides the fastest mixing Markov chain given the network topological constraints, and should naturally find their applications in the context of distributed optimization, estimation and control.

The application of parallel computers to reservoir simulation is studied with the aid of two recently introduced Multiple Instruction, Multiple Data (MIMD) computers. The problems posed by reservoir simulators are shown to be particularly well suited for such parallel computers* Fundamental parallel programming techniques using Fortran are presented for the Hetrogeneous Element Processor (HEP) produced by Denelcor and the iPSC Hypercube produced by Intel. These techniques are then applied to two of the most time consuming tasks in reservoir simulation: forming the matrix coefficients and producing a sparse matrix solution. The problem of parallel formation of sparse matrix coefficients is addressed for the single and multiphase cases using both black oil and compositional fluid models. Parallel sparse matrix solution is addressed by considering D4 ordered Gaussian elimination, and the multigrid, conjugate gradient, and Successive Over-Relaxation (SOR) methods. Three parallel algorithms based on SOR are developed to illustrate the possible diversity of parallel algorithms. The red-black and multicolored SOR methods which are often used on vector machines are shown to be easily adaptable to parallel MIMD machine. Matrix partitioning and a new method of iteration pipelining are also presented as parallel SOR methods. The various SOR algorithms, in parallel form, are mapped onto the HEP to illustrate the speedups currently possible on an MIMD machine. Projections are made on the future importance of these types of machines to reservoir simulation.

An efficient method for solving the chemical nonequilibrium Navier-Stokes equations including all the species conservation and energy production equations are developed. The algorithm is embodied in a fully coupled, implicit, large block structure. Van Leer flux splitting for invisid terms and central differencing for viscous terms in the explicit operators are applied in the numerical algorithm. The fully-coupled system is solved implicitly and the Bi-Conjugate Gradient Stable (Bi-CGSTAB) method with a preconditioner of incomplete LU-factorization (ILU) is used for solving large block-structure and diagonal dominate matrix equations. The computations are performed for the hypersonic inflow over blunt bodies including half cylinder, double ellipse, and blunt nose with cylindrical portion following 15 deg inclined plane. Computational results in the present study are compared with other calculated data and exhibit good agreement. Convergence histories of the mean flow variables and species equations demonstrate that the fast convergent rate can be achieved by the preconditioned Bi-CGSTAB method. (Author)

This article presents an innovative research project (sponsored by the National Science Foundation, the American Iron and Steel Institute, and the American Institute of Steel Construction) where computationally elegant algorithms based on the integration of a novel connectionist computing model, mathematical optimization, and a massively parallel computer architecture are used to automate the complex process of engineering design.

Virtual earthquake engineering laboratory with physics-based degrading materials on parallel computers

Cardiomyopathy refers to gradual weakening of the muscular walls of the cardiac chambers. Due to the hypertrophic condition of the muscular walls, damage and stretching of the muscle may lead to arrhythmias, which is detectable using the ECG. In the past, any deviations from a healthy rhythm provide cardiologists with accurate information regarding the heart condition. However, cardiologists are prone to making inaccurate interpretation from the visual observation, leading to erroneous diagnosis. Hence, this paper proposes a computerized method for accurate analysis and detection of cardiomyopathy disease using MLP network. Data for normal, cardiomyopathy, and other arrhythmias were obtained from the PTB Diagnostic ECG database. The raw signals were preprocessed for high-frequency noise removal using median and moving average filters. Baseline corrections were conducted using two-stage polynomial fitting method. Nine time-based features were extracted from the three bipolar limb leads. A total of 600 beats were used to train, validate and test five different MLP network structures. Four different learning algorithms were implemented to obtain the best classification accuracy and fastest convergence rate. Results show that the Levenberg-Marquardt algorithm shows the highest average classification accuracy of 98.9% for the different structures with the fastest average convergence rate of 12 epochs.

The use of spectral methods now has a long history in global atmospheric modelling wherein the attractive properties of Fourier series on spheres, including higher-order convergence rates and efficient implementation via the transform method, have proven advantageous. Partially offsetting these advantages, however, are several competing disadvantages. Two of these, the appearance of Gibbs oscillations for localized processes (e.g., orographic interactions) and the difficulty of mapping spectral techniques onto parallel computer architectures, are inherent to the global nature of these techniques. A third drawback, the restriction of these methods to regular geometries, has severely limited their application to the modelling of the large-scale ocean circulation. We describe a global circulation model that has, in principle, none of these limitations. The model utilizes the spectral element method that combines the geometrical flexibility of traditional finite element methods with the rapid convergence rates of spectral approximation techniques. Simple test problems drawn from both oceanic and atmospheric modelling are used to demonstrate that the resulting model is exponentially convergent, yet allows effective representation of irregular geometry and efficient grid refinement in regions of dynamical interest. Lastly, performance characteristics on the nCUBE/2 and Cray T3D architectures confirm that the element model is ideally suited to the parallel computing environment.