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An iterative algorithm for the solution of the Helmholtz equation is developed. The algorithm is based on a preconditioned conjugate gradient iteration for the normal equations. The preconditioning is based on an SSOR sweep for the discrete Laplacian. Numerical results are presented for a wide variety of problems of physical interest and demonstrate the effectiveness of the algorithm.
Region growing, often given as a classical example of the recursive control structures used in image processing which are often awkward to implement in hardware where the intent is the segmentation of an image at raster scan rates, is addressed in light of the postulate that any computation which can be performed recursively can be performed easily and efficiently by iteration coupled with association. Attention is given to an algorithm and hardware structure able to perform region labeling iteratively at scan rates. Every pixel is individually labeled with an identifier which signifies the region to which it belongs. Difficulties otherwise requiring recursion are handled by maintaining an equivalence table in hardware transparent to the computer, which reads the labeled pixels. A simulation of the associative memory has demonstrated its effectiveness.
A method of solving the boundary-layer equations that arise in singular-perturbation analysis of flightpath optimization problems is presented. The method is based on Picard iterations of the integrated form of the equations and does not require iteration to find unknown boundary conditions. As an example, the method is used to develop a solution algorithm for the zero-order boundary-layer equations of the aircraft minimum-time-to-climb problem.
In this paper, two acceleration techniques for Euler calculations are investigated. The first technique is an extrapolation procedure based on the Power Method; it is applicable when the iterative matrix has dominant eigenvalues. Both real and complex conjugate roots are allowed. The second technique is a generalization of the Minimal Residual Method, where the extrapolation step consists of a weighted combination of the corrections at different iteration levels and the weights are chosen to minimize the Lz norm of the residual. Numerical results, using Jameson's Runge-Kutta Multigrid Code, are presented. The extra computational work to apply either technique is negligible and the extra storage is not a problem on current supercomputers.
The calculation of sensitivity derivatives of solutions of iteratively solved systems of algebraic equations is investigated. A modified finite difference procedure is presented which improves the accuracy of the calculated derivatives. The procedure is demonstrated for a simple algebraic example as well as an element-by-element preconditioned conjugate gradient iterative solution technique applied to truss examples.
Morrison's iterative noise removal method is studied by characterizing its effect upon systems of differing noise level and response function. The nature of data acquired from a linear shift invariant instrument is discussed so as to define the relationship between the input signal, the instrument response function, and the output signal. Fourier analysis is introduced, along with several pertinent theorems, as a tool to more thorough understanding of the nature of and difficulties with deconvolution. In relation to such difficulties the necessity of a noise removal process is discussed. Morrison's iterative noise removal method and the restrictions upon its application are developed. The nature of permissible response functions is discussed, as is the choice of the response functions used.
The applicability of static data flow architectures to the iterative solution of sparse linear systems of equations is investigated. An analytic performance model of a static data flow computation is developed. This model includes both spatial parallelism, concurrent execution in multiple PE's, and pipelining, the streaming of data from array memories through the PE's. The performance model is used to analyze a row partitioned iterative algorithm for solving sparse linear systems of algebraic equations. Based on this analysis, design parameters for the static data flow architecture as a function of matrix sparsity and dimension are proposed.
The particular problems investigated in the present study arise from nonlinear transient heat conduction. One of two types of nonlinearities considered is related to a material temperature dependence which is frequently needed to accurately model behavior over the range of temperature of engineering interest. The second nonlinearity is introduced by radiation boundary conditions. The finite element equations arising from the solution of nonlinear transient heat conduction problems are formulated. The finite element matrix equations are temporally discretized, and a nonlinear iterative solution algorithm is proposed. Algorithms for solving the linear problem are discussed, taking into account the form of the matrix equations, Gaussian elimination, cost, and iterative techniques. Attention is also given to approximate factorization, implementational aspects, and numerical results.
The incorporation of a photorefractive gain element into a confocal Fabry-Perot processor, with a photorefractive BaTiO3 crystal in the feedback path providing the coherent image amplification, is discussed. The implementation of an iterative algorithm for matrix inversion is described, and its experimental implementation is demonstrated. It is shown that the BaTiO3 amplifier provides loss compensation and coherence restoration of the feedback signal, thereby increasing the effective number of iterations in the algorithm. Thus, the technique makes it possible to implement slowly converging (as well as rapidly converging) algorithms.
Morrison's iterative method of noise removal, or Morrison's smoothing, is applied in a simulation to noise-added data sets of various noise levels to determine its optimum use. Morrison's smoothing is applied for noise removal alone, and for noise removal prior to deconvolution. For the latter, an accurate method is analyzed to provide confidence in the optimization. The method consists of convolving the data with an inverse filter calculated by taking the inverse discrete Fourier transform of the reciprocal of the transform of the response of the system. Various length filters are calculated for the narrow and wide Gaussian response functions used. Deconvolution of non-noisy data is performed, and the error in each deconvolution calculated. Plots are produced of error versus filter length; and from these plots the most accurate length filters determined. The statistical methodologies employed in the optimizations of Morrison's method are similar. A typical peak-type input is selected and convolved with the two response functions to produce the data sets to be analyzed. Both constant and ordinate-dependent Gaussian distributed noise is added to the data, where the noise levels of the data are characterized by their signal-to-noise ratios. The error measures employed in the optimizations are the L1 and L2 norms. Results of the optimizations for both Gaussians, both noise types, and both norms include figures of optimum iteration number and error improvement versus signal-to-noise ratio, and tables of results. The statistical variation of all quantities considered is also given.
Regular mesh-connected arrays are shown to be isomorphic to a class of so-called regular iterative algorithms. For a wide variety of problems it is shown how to obtain appropriate iterative algorithms and then how to translate these algorithms into arrays in a systematic fashion. Several 'systolic' arrays presented in the literature are shown to be specific cases of the variety of architectures that can be derived by the techniques presented here. These include arrays for Fourier Transform, Matrix Multiplication, and Sorting.
The computational efficiency of subspace iteration is addressed relative to the data structures adopted for the very large and generally sparse coefficient matrices. The frequent triangulations and matrix multiplications demand that access to the terms in the coefficient matrices be unbiased. Reliance on virtual memory (paging) operating systems with no special considerations for localized data access is not adequate. Specific data structures must be designed that accommodate the needs of the numerical algorithm yet eliminate unnecessary paging. An implementation of the subspace iteration method using hypermatrix data structures is presented. Use of hypermatrices is shown to provide unbiased and localized data access. The various modifications to the conventional formulation are described and an example problem illustrates the potential benefits of the hypermatrix formulation. Possibilities for adapting hypermatrix data structures to new supercomputer architectures are discussed.
A numerical method is presented for design sensitivity analysis, using an iterative-method reanalysis of the structure generated by a small perturbation in the design variable; a forward-difference scheme is then employed to obtain the approximate sensitivity. Algorithms are developed for displacement and stress sensitivity, as well as for eignevalues and eigenvector sensitivity, and the iterative schemes are modified so that the coefficient matrices are constant and therefore decomposed only once.
An iterative algorithm for the efficient solution of systems of nonlinear hyperbolic equations is presented. Parallelism is evident at several levels. In the formation of the iteration, the equations are decoupled, thereby providing large grain parallelism. Parallelism may also be exploited within the solves for each equation. Convergence of the interation is established via a bounding function argument. Experimental results in two-dimensions are presented.
The numerical solution of stiff initial value problems, which lead to the problem of solving large systems of mildly nonlinear equations are considered. For many problems derived from engineering and science, a solution is possible only with methods derived from iterative linear equation solvers. A common approach to solving the nonlinear equations is to employ an approximate solution obtained from an explicit method. The error is examined to determine how it is distributed among the stiff and non-stiff components, which bears on the choice of an iterative method. The conclusion is that error is (roughly) uniformly distributed, a fact that suggests the Chebyshev method (and the accompanying Manteuffel adaptive parameter algorithm). This method is described, also commenting on Richardson's method and its advantages for large problems. Richardson's method and the Chebyshev method with the Mantueffel algorithm are applied to the solution of the nonlinear equations by Newton's method.
As three-dimensional models are gaining importance, iterative methods will become almost mandatory. Among these, preconditioned Krylov subspace methods have been viewed as the most efficient and reliable, when solving linear as well as nonlinear systems of equations. There has been several different approaches taken to adapt iterative methods for supercomputers. Some of these approaches are discussed and the methods that deal more specifically with general unstructured sparse matrices, such as those arising from finite element methods, are emphasized.
A brief introduction to the basic principles of an eigen method used to obtain the optimum source location of lightning is presented. The location of the optimum source is obtained by using multiple direction finders (DF's) on a spherical surface. An improvement of this method, which takes the distance of source-DF's as a constant, is presented. It is pointed out that using a weight factor of signal strength is not the most ideal method because of the inexact inverse signal strength-distance relation and the inaccurate signal amplitude. An iterative calculation method is presented using the distance from the source to the DF as a weight factor. This improved method has higher accuracy and needs only a little more calculation time. Some computer simulations for a 4DF system are presented to show the improvement of location through use of the iterative method.
The spatially constrained clustering (SCC) iterative parallel region-growing technique is applied to image analysis. The SCC algorithm is implemented on the massively parallel processor at NASA Goddard. Most previous region-growing approaches have the drawback that the segmentation produced depends on the order in which portions of the image are processed. The ideal solution to this problem (merging only the single most similar pair of spatially adjacent regions in the image in each iteration) becomes impractical except for very small images, even on a massively parallel computer. The SCC algorithm overcomes these problems by performing, in parallel, the best merge within each of a set of local, possibly overlapping, subimages. A region-splitting stage is also incorporated into the algorithm, but experiments show that region splitting generally does not improve segmentation results. The SCC algorithm has been tested on various imagery data, and test results for a Landsat TM image are summarized.