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NASA NTRS1985

GMRES acceleration of computational fluid dynamics codes

The generalized minimal residual algorithm (GMRES) is a conjugate-gradient like method that applies directly to nonsymmetric linear systems of equations. In this paper, GMRES is modified to handle nonlinear equations characteristic of computational fluid dynamics. Attention is devoted to the concept of preconditioning and the role it plays in assuring rapid convergence. A formulation is developed that allows GMRES to be preconditioned by the solution procedures already built into existing computer codes. Examples are provided that demonstrate the ability of GMRES to greatly improve the robustness and rate of convergence of current state-of-the-art fluid dynamics codes. Theoretical aspects of GMRES are presented that explain why it works. Finally, the advantage GMRES enjoys over related methods such as conjugate gradients are discussed.

Wigton, L. B.↗

NASA NTRS1986

Multi-color incomplete Cholesky conjugate gradient methods for vector computers

In this research, we are concerned with the solution on vector computers of linear systems of equations, Ax = b, where A is a larger, sparse symmetric positive definite matrix. We solve the system using an iterative method, the incomplete Cholesky conjugate gradient method (ICCG). We apply a multi-color strategy to obtain p-color matrices for which a block-oriented ICCG method is implemented on the CYBER 205. (A p-colored matrix is a matrix which can be partitioned into a pXp block matrix where the diagonal blocks are diagonal matrices). This algorithm, which is based on a no-fill strategy, achieves O(N/p) length vector operations in both the decomposition of A and in the forward and back solves necessary at each iteration of the method. We discuss the natural ordering of the unknowns as an ordering that minimizes the number of diagonals in the matrix and define multi-color orderings in terms of disjoint sets of the unknowns. We give necessary and sufficient conditions to determine which multi-color orderings of the unknowns correpond to p-color matrices. A performance model is given which is used both to predict execution time for ICCG methods and also to compare an ICCG method to conjugate gradient without preconditioning or another ICCG method. Results are given from runs on the CYBER 205 at NASA's Langley Research Center for four model problems.

Poole, E. L.↗

NASA NTRS1986

Analytic computation of energy derivatives - Relationships among partial derivatives of a variationally determined function

Formulas are presented relating Taylor series expansion coefficients of three functions of several variables, the energy of the trial wave function (W), the energy computed using the optimized variational wave function (E), and the response function (lambda), under certain conditions. Partial derivatives of lambda are obtained through solution of a recursive system of linear equations, and solution through order n yields derivatives of E through order 2n + 1, extending Puley's application of Wigner's 2n + 1 rule to partial derivatives in couple perturbation theory. An examination of numerical accuracy shows that the usual two-term second derivative formula is less stable than an alternative four-term formula, and that previous claims that energy derivatives are stationary properties of the wave function are fallacious. The results have application to quantum theoretical methods for the computation of derivative properties such as infrared frequencies and intensities.

King, H. F.↗

NASA NTRS1986

Acceleration of convergence of vector sequences

A general approach to the construction of convergence acceleration methods for vector sequence is proposed. Using this approach, one can generate some known methods, such as the minimal polynomial extrapolation, the reduced rank extrapolation, and the topological epsilon algorithm, and also some new ones. Some of the new methods are easier to implement than the known methods and are observed to have similar numerical properties. The convergence analysis of these new methods is carried out, and it is shown that they are especially suitable for accelerating the convergence of vector sequences that are obtained when one solves linear systems of equations iterative. A stability analysis is also given, and numerical examples are provided. The convergence and stability properties of the topological epsilon algorithm are likewise given.

Sidi, A.↗

NASA NTRS1986

Fourier analysis of the SOR iteration

The SOR iteration for solving linear systems of equations depends upon an overrelaxation factor omega. It is shown that for the standard model problem of Poisson's equation on a rectangle, the optimal omega and corresponding convergence rate can be rigorously obtained by Fourier analysis. The trick is to tilt the space-time grid so that the SOR stencil becomes symmetrical. The tilted grid also gives insight into the relation between convergence rates of several variants.

Leveque, R. J.↗

NASA NTRS1986

Analysis of the SOR iteration for the 9-point Laplacian

The SOR iteration for solving linear systems of equations depends upon an overrelaxation factor omega. A theory for determining omega was given by Young (1950) for consistently ordered matrices. Here the optimal omega is determined for the 9-point stencil for the model problem of Laplace's equation on a square. Several orderings of the equations are considered, including the natural rowwise and multicolor orderings, all of which lead to non-consistently ordered matrices, and two equivalence classes of orderings are found with different convergence behavior and optimal omega's. The results for the natural rowwise ordering are compared to those of Garabedian (1956) and it is explained why both results are, in a sense, correct, even though they differ. Also analyzed is a pseudo SOR method for the model problem and it is shown that it is not as effective as the SOR methods. Finally, the point SOR methods are compared to known results for line SOR methods for this problem.

Loyce M. Adams↗

NASA NTRS1987

Solving very large, sparse linear systems on mesh-connected parallel computers

The implementation of Pan and Reif's Parallel Nested Dissection (PND) algorithm on mesh connected parallel computers is described. This is the first known algorithm that allows very large, sparse linear systems of equations to be solved efficiently in polylog time using a small number of processors. How the processor bound of PND can be matched to the number of processors available on a given parallel computer by slowing down the algorithm by constant factors is described. Also, for the important class of problems where G(A) is a grid graph, a unique memory mapping that reduces the inter-processor communication requirements of PND to those that can be executed on mesh connected parallel machines is detailed. A description of an implementation on the Goodyear Massively Parallel Processor (MPP), located at Goddard is given. Also, a detailed discussion of data mappings and performance issues is given.

Opsahl, Torstein↗

NASA NTRS1987

Optimal pre-scheduling of problem remappings

A large class of scientific computational problems can be characterized as a sequence of steps where a significant amount of computation occurs each step, but the work performed at each step is not necessarily identical. Two good examples of this type of computation are: (1) regridding methods which change the problem discretization during the course of the computation, and (2) methods for solving sparse triangular systems of linear equations. Recent work has investigated a means of mapping such computations onto parallel processors; the method defines a family of static mappings with differing degrees of importance placed on the conflicting goals of good load balance and low communication/synchronization overhead. The performance tradeoffs are controllable by adjusting the parameters of the mapping method. To achieve good performance it may be necessary to dynamically change these parameters at run-time, but such changes can impose additional costs. If the computation's behavior can be determined prior to its execution, it can be possible to construct an optimal parameter schedule using a low-order-polynomial-time dynamic programming algorithm. Since the latter can be expensive, the performance is studied of the effect of a linear-time scheduling heuristic on one of the model problems, and it is shown to be effective and nearly optimal.

Nicol, David M.↗

NASA NTRS1988

Newton solution of inviscid and viscous problems

The application of Newton iteration to inviscid and viscous airfoil calculations is examined. Spatial discretization is performed using upwind differences with split fluxes. The system of linear equations which arises as a result of linearization in time is solved directly using either a banded matrix solver or a sparse matrix solver. In the latter case, the solver is used in conjunction with the nested dissection strategy, whose implementation for airfoil calculations is discussed. The boundary conditions are also implemented in a fully implicit manner, thus yielding quadratic convergence. Complexities such as the ordering of cell nodes and the use of a far field vortex to correct freestream for a lifting airfoil are addressed. Various methods to accelerate convergence and improve computational efficiency while using Newton iteration are discussed. Results are presented for inviscid, transonic nonlifting and lifting airfoils and also for laminar viscous cases.

Venkatakrishnan, V.↗

NASA NTRS1988

On the constrained Chebyshev approximation problem on ellipses

Constrained Chebyshev approximation problems of the type with minimum (p is an element of Pi(sub n):p(c)=1) and maximum (z is an element of E) with /p(z)/ are considered. Here Pi(sub n) denotes the set of all complex polynomials of degree at most n, E is any ellipse in the complex plane, and c is an element of C/E. Such approximation problems arise in the context of optimizing semi-iterative methods for the solution of large, sparse systems of linear equations Ax=b with complex non-Hermitian coefficient matrices A. The problem of obtaining optimal polynomial preconditioners for conjugate gradient type methods for Ax=b also leads to problems of this type. A new family of polynomials -- q(sub n)(z;c), n is an element of N, and c is an element of C/E -- are introduced as the polynomials which are optimal for a modified version of the Chebyshev approximation problem with Pi(sub n) replaced by a certain subfamily. Some simple properties of q(sub n) are also listed. A necessary and sufficient condition for q(sub n) to be the extremal polynomial for the approximation problem is then derived. Finally, it is shown that q(sub n) is indeed optimal for the problem for all fixed n whenever the distance between c and E is sufficiently large. Results of some numerical tests are presented.

Freund, Roland↗

NASA NTRS1989

Thin-plate spline quadrature of geodetic integrals

Thin-plate spline functions (known for their flexibility and fidelity in representing experimental data) are especially well-suited for the numerical integration of geodetic integrals in the area where the integration is most sensitive to the data, i.e., in the immediate vicinity of the evaluation point. Spline quadrature rules are derived for the contribution of a circular innermost zone to Stoke's formula, to the formulae of Vening Meinesz, and to the recursively evaluated operator L(n) in the analytical continuation solution of Molodensky's problem. These rules are exact for interpolating thin-plate splines. In cases where the integration data are distributed irregularly, a system of linear equations needs to be solved for the quadrature coefficients. Formulae are given for the terms appearing in these equations. In case the data are regularly distributed, the coefficients may be determined once-and-for-all. Examples are given of some fixed-point rules. With such rules successive evaluation, within a circular disk, of the terms in Molodensky's series becomes relatively easy. The spline quadrature technique presented complements other techniques such as ring integration for intermediate integration zones.

Vangysen, Herman↗

NASA NTRS1991

Conjugate Gradient Algorithms For Manipulator Simulation

Report discusses applicability of conjugate-gradient algorithms to computation of forward dynamics of robotic manipulators. Rapid computation of forward dynamics essential to teleoperation and other advanced robotic applications. Part of continuing effort to find algorithms meeting requirements for increased computational efficiency and speed. Method used for iterative solution of systems of linear equations.

Fijany, Amir↗

NASA NTRS1991

Analysis of lightning field changes produced by Florida thunderstorms

A new method is introduced for inferring the charges deposited in a lightning flash. Lightning-caused field changes (delta E's) are described by a more general volume charge distribution than is defined on a large cartesian grid system centered above the measuring networks. It is shown that a linear system of equations can be used to relate delta E's at the ground to the values of charge on this grid. It is possible to apply more general physical constraints to the charge solutions, and it is possible to access the information content of the delta E data. Computer-simulated delta E inversions show that the location and symmetry of the charge retrievals are usually consistent with the known test sources.

Koshak, William John↗

NASA NTRS1990

Implementation of two projection methods on a shared memory multiprocessor - DEC VAX 6240

The relative performance of two iterative schemes, based on projection techniques, is compared on a shared memory multiprocessor - VAX 6240. The CG accelerated Block-SSOR method and the CG accelerated Symmetric-Kaczmarz method are considered for the solution of large sparse nonsymmetric systems of linear equations. It is shown that the regular structure of many matrices can be exploited by the CG-accelerated Block-SSOR method to provide good speedup in a multiprocessing environment. However, the CG accelerated Symmetric-Kaczmarz method, while being a viable alternative on a scalar machine, is unable to benefit from multiprocessing.

Kamath, C.↗

NASA NTRS1991

Efficient use of direct solvers for the calculation of compressible flows

While the direct solution of systems of linear equations resulting from fluid dynamic problems has generally not been practical in the past, it is presently demonstrated that the direct method is often more efficient than the most popular iterative schemes when constructed in such a way as to take advantage of presently available vector processing capabilities and large memory. The vertical line Gauss-Seidel algorithm was chosen as the iterative method to be compared with the direct method. It is fond that the direct method becomes efficient only when large residual reductions are desired.

Riggins, David W.↗

NASA NTRS1991

Efficient multitasking of Choleski matrix factorization on CRAY supercomputers

A Choleski method is described and used to solve linear systems of equations that arise in large scale structural analysis. The method uses a novel variable-band storage scheme and is structured to exploit fast local memory caches while minimizing data access delays between main memory and vector registers. Several parallel implementations of this method are described for the CRAY-2 and CRAY Y-MP computers demonstrating the use of microtasking and autotasking directives. A portable parallel language, FORCE, is used for comparison with the microtasked and autotasked implementations. Results are presented comparing the matrix factorization times for three representative structural analysis problems from runs made in both dedicated and multi-user modes on both computers. CPU and wall clock timings are given for the parallel implementations and are compared to single processor timings of the same algorithm.

Overman, Andrea L.↗

NASA NTRS1992

An incremental strategy for calculating consistent discrete CFD sensitivity derivatives

In this preliminary study involving advanced computational fluid dynamic (CFD) codes, an incremental formulation, also known as the 'delta' or 'correction' form, is presented for solving the very large sparse systems of linear equations which are associated with aerodynamic sensitivity analysis. For typical problems in 2D, a direct solution method can be applied to these linear equations which are associated with aerodynamic sensitivity analysis. For typical problems in 2D, a direct solution method can be applied to these linear equations in either the standard or the incremental form, in which case the two are equivalent. Iterative methods appear to be needed for future 3D applications; however, because direct solver methods require much more computer memory than is currently available. Iterative methods for solving these equations in the standard form result in certain difficulties, such as ill-conditioning of the coefficient matrix, which can be overcome when these equations are cast in the incremental form; these and other benefits are discussed. The methodology is successfully implemented and tested in 2D using an upwind, cell-centered, finite volume formulation applied to the thin-layer Navier-Stokes equations. Results are presented for two laminar sample problems: (1) transonic flow through a double-throat nozzle; and (2) flow over an isolated airfoil.

Korivi, Vamshi Mohan↗

NASA NTRS1991

Parallel variable-band Choleski solvers for computational structural analysis applications on vector multiprocessor supercomputers

A Choleski method used to solve linear systems of equations that arise in large scale structural analyses is described. The method uses a novel variable-band storage scheme and is structured to exploit fast local memory caches while minimizing data access delays between main memory and vector registers. Several parallel implementations of this method are described for the CRAY-2 and CRAY Y-MP computers demonstrating the use of microtasking and autotasking directives. A portable parallel language, FORCE, is also used for two different parallel implementations, demonstrating the use of CRAY macrotasking. Results are presented comparing the matrix factorization times for three representative structural analysis problems from runs made in both dedicated and multi-user modes on both the CRAY-2 and CRAY Y-MP computers. CPU and wall clock timings are given for the various parallel methods and are compared to single processor timings of the same algorithm. Computation rates over 1 GIGAFLOP (1 billion floating point operations per second) on a four processor CRAY-2 and over 2 GIGAFLOPS on an eight processor CRAY Y-MP are demonstrated as measured by wall clock time in a dedicated environment. Reduced wall clock times for the parallel methods relative to the single processor implementation of the same Choleski algorithm are also demonstrated for runs made in multi-user mode.

Poole, E. L.↗