Fast Direct Methods Research Articles

Fast direct Fourier methods, based on one- and two-pass coordinate transformations, yield accurate reconstructions of x-ray CT clinical images

The conversion from polar to Cartesian coordinates can be carried out with two-pass algorithms. The paper describes two different methods based on concentric square frames and octagonal frames and their results, obtained with accurate interpolations based on the `moving window Shannon reconstruction' (MWSR). The embedding of these algorithms in direct Fourier methods (DFMs) of tomographic reconstruction is discussed. With respect to one-pass methods and to the use of octagonal frames, the square frame method makes it possible to carry out the first pass, a radial resampling, in the direct space, before computing 1D Fourier transforms (FTs) of projections. Reconstructions of clinical images from the raw data of a third-generation x-ray tomograph are presented and compared with those obtained with one-pass DFMs and with the convolution back-projection method (CBPM) performed by the instrument. The simple algorithm using square frames yields results in complete agreement with other DFM protocols and the CBPM. On a general-purpose computer, the execution of DFM protocols based on one-pass and two-pass coordinate transformations is 35 to 55 times faster than the CBPM and make the algorithms attractive for modern instrumentation.

Hybrid Extended Equal-Area Criterion for Fast Transient Stability Assessment with Detailed Power System Models

The use of fast direct methods to evaluate the stability of power systems can benefit planning studies and operation planning. The existing Extended Equal-Area Criterion (EEAC), developed in 1988, enabled calculation much faster than timedomain simulations, but failed to deal with large and detailed dynamic systems. Based on it, we propose a new method, which is a Hybrid Extended Equal-Area Criterion (HEEAC), where time-domain simulations are used with a stability assessment provided by EEAC. HEEAC discriminates critical from non-critical machines, and yields a set of stability indicators such as critical clearing times, power transfer limits or power generation limits. The method has been applied to the French EHV system, with detailed generator and load models. The perfonnances are good both in tenns of computing speed and accuracy.

Matrix Decomposition Algorithms in Orthogonal Spline Collocation for Separable Elliptic Boundary Value Problems

Fast direct methods are presented for the solution of linear systems arising in high-order, tensor-product orthogonal spline collocation applied to separable, second order, linear, elliptic partial...

Fast direct methods of solving finite-element grid schemes with bicubic elements for the Poisson equation

Fast direct methods of solving finite-element grid schemes with bicubic elements for the Poisson equation

Fast Direct Solvers for Piecewise Hermite Bicubic Orthogonal Spline Collocation Equations

Fast direct methods are proposed for the solution of linear systems arising when orthogonal spline collocation with piecewise Hermite bicubics is employed for the approximate solution of Poisson’s equation in a rectangle. The new methods, which are matrix decomposition algorithms involving fast Fourier transforms, require $O(N^2 \log N)$ arithmetic operations on an $N \times N$ uniform partition and are highly parallel in nature. The treatment of more general elliptic problems is also discussed and the results of numerical experiments using parallel implementations of the new algorithms on several shared memory machines are presented.

FFTs and three-dimensional Poisson solvers for hypercubes

This paper investigates the implementation of fast direct methods for solving the three-dimensional Poisson equation on loosely coupled hypercube multiprocessors. As a preliminary step, the problem of computing multiple FFTs is considered and two different algorithms are compared. These algorithms are then used to implement two FFT-based fast Poisson solvers. Proceeding both with experiments and with performance models, these two solvers are studied and compared. No single algorithm is superior in all ranges of the parameters and the best choice depends upon problem size, number of processors and communication costs. In closing, two additional parallel algorithms for solving the Poisson equation are proposed, both of which appear to be competitive.

A historical overview of iterative methods

The object of this paper is to present a historical overview of the development of iterative methods for the solution of large sparse systems of linear equations. The emphasis is on methods which are applicable to linear systems arising in the numerical solution of partial differential equations. Aspects to be covered including the methods of L.F. Richardson and of Liebmann as well as relaxation methods used by Southwell and others; the SOR method and extensions such as block SOR methods, and p-cyclic matrices; Chebyshev polynomial methods; alternating direction implicit methods; the SSOR method; approximate matrix factorization methods including the strongly implicit method (SIP) and the incomplete Cholesky method (ICC); fast direct methods; conjugate gradient methods; adaptive methods for the automatic determination of iteration parameters; multigrid methods; methods for nonsymmetric systems; and iterative software. Future developments will be discussed with emphasis on the use of vector and parallel processors.

Recent research in iterative methods at Boeing

We present a brief summary of two research projects at Boeing, both addressing the solution of very large sparse systems of linear equations with preconditioned iterative methods. The first project is concerned with pratical numerical solution of PDE's arising from the modeling of complex geometries for aerospace applications. Current research has led to development of a powerful combination of fast direct methods and general iterative methods. This novel combination of standard technologies has succeeded in solving previously intractable problems. The second project is an investigation of the use of hyperbolic transformations for generating incomplete Cholesky factorizations. A natural application is to the preconditioning problem for matrices arising from domain decomposition.

AN APPLICATION OF FAST ALGORITHMS TO NUMERICAL ELECTROMAGNETIC MODELING*

ABSTRACTNumerical electromagnetic modeling by the finite‐difference or finite‐element methods leads to a large sparse system of linear algebraic equations. Fast direct methods, requiring an order of at most q log q arithmetic operations to solve a system of q equations, cannot easily be applied to such a system. This paper describes the iterative application of a fast method, namely cyclic reduction, to the numerical solution of the Helmholtz equation with a piecewise constant imaginary coefficient of the absolute term in a plane domain. By means of numerical tests the advantages and limitations of the method compared with classical direct methods are discussed. The iterative application of the cyclic reduction method is very efficient if one can exploit a known solution of a similar (e.g., simpler) problem as the initial approximation. This makes cyclic reduction a powerful tool in solving the inverse problem by trial‐and‐error.

An Exterior Poisson Solver Using Fast Direct Methods and Boundary Integral Equations with Applications to Nonlinear Potential Flow

A general method is developed combining fast direct methods and boundary integral equation methods to solve Poisson’s equation on irregular exterior regions. The method requires $O(N\log N)$operations where N is the number of grid points. Error estimates are given that hold for regions with corners and other boundary irregularities. Computational results are given in the context of computational aerodynamics for a two-dimensional lifting airfoil. Solutions of boundary integral equations for lifting and nonlifting aerodynamic configurations using preconditioned conjugate gradient are examined for varying degrees of thinness.

Preconditioning by Fast Direct Methods for Nonself-Adjoint Nonseparable Elliptic Equations

We consider the use of fast direct methods as preconditioners for iterative methods for computing the numerical solution of nonself-adjoint elliptic boundary value problems. We derive bounds on convergence rates that are independent of discretization mesh size. For two-dimensional problems on rectangular domains, discretized on an $n \times n$ grid, these bounds lead to asymptotic operation counts of $O(n^2 \log n\log \varepsilon ^{ - 1} )$ to achieve relative error $\varepsilon $ and $O(n^2 (\log n)^2 )$ to reach truncation error.

A new method for computing solenoidal vector fields on arbitrary regions

AbstractWe develop a new method for the efficient calculation of solenoidal vector fields on general regions. The method takes advantage of fast direct methods and uses boundary integral equations to satisfy boundary conditions. For the latter we give an effective scheme for computing far‐field boundary influences (based on discrete charges). Examples and numerical results are given. The method is applicable to incompressible Navier‐Stokes calculations.

On fast direct methods for solving elliptic equations over nonrectangular regions

In this paper, two methods based on the symmetric marching technique (SMT) are presented for the solution of elliptic equations over nonrectangular regions. Method I illustrates the direct adaptation of the SMT to irregular geometries. In method II, an efficient implementation of the capacitance matrix method has been considered using SMT. The favorable characteristics of the SMT for solving the Poisson equation with several right hand side functions and different boundary conditions without extra computational effort have been exploited for the last generation of the capacitance matrix. The successful application of the SMT combined with quasilinearization to solve mildly nonlinear elliptic equations is also described. Several test examples have been solved to demonstrate the efficiency of the proposed methods.

High-Order, Fast-Direct Methods for Separable Elliptic Equations

The Rayleigh–Ritz–Galerkin method with tensor product B-splines yields high-order discretiz-ations for elliptic partial differential equations. For smooth problems the resulting linear system of equations is both smaller and denser than the corresponding systems for lower order discretizations. However, several fast direct methods are known for solving these low order systems when the partial differential equation is separable. In this paper we show how to extend the matrix decomposition technique to yield a fast direct method for high-order, finite-element discretizations.

Fast elliptic solvers—an overview

The purpose of this paper is to present a brief survey of fast direct methods for solving elliptic boundary-value problems. The methods reviewed are based on Fourier analysis, block reduction techniques, and marching algorithms. First, the Poisson equation with Dirichlet and mixed boundary conditions are considered. Then we go to more general elliptic problems and irregular regions.

The direct solution of periodic parabolic problems by boundary-value techniques

In this paper fast direct methods are presented for the solution of the parabolic boundary value problem. The algorithms used are block variants of the cyclic factorisation algorithm previously presented in Evans [1]

Use of fast direct methods for mildly nonlinear elliptic difference equations

We consider in a rectangle the dirichelet mildly nonlinear elliptic boundary value problem. The numerical solution by finite-difference equations leads to the problem of solving special nonlinear systems. We propose a numerical technique founded on the application of a fast double-sweep method to the linear system induced by modified Picard iteration. Computer results are given and a comparison is made with other methods which illustrate the effectiveness of the method.

On efficient multigrid software for elliptic problems on rectangular domains

Emphasis has been laid on software development concerning ‘non-standard’ multigrid techniques as introduced by Foerster, Stüben and Trottenberg 1. The advantages of these methods result in very efficient code for the solution of elliptic problems. In comparison with fast direct methods, the multigrid solver MGOO is as good as direct solvers for model equations on rectangular domains, even better in special cases, and more generally applicable (variable coefficients).

Families of High Order Accurate Discretizations of Some Elliptic Problems

In this paper we construct and analyze high order finite difference discretizations of a class of elliptic partial differential equations. In particular, two one-parameter families of fourth order HODIE discretizations of the Helmholtz equation are derived and a discretization optimal with respect to a certain norm of the truncation error is identified. The use of compact nine-point formulas of positive type admits both fast direct methods and standard iterative methods for the solution of the resulting systems of linear equations. Extensions yielding sixth order accuracy for the Helmholtz equation and fourth order accuracy for a more general operator are given. Finally, numerical results demonstrating the effectiveness of the discretizations for a wide range of problems are presented.

High-Order Fast Elliptic Equation Solvers

An algorithm for approximating certain classes of elliptic partial differential equations on a rectangle is presented. The algorithm uses high-order 9-point difference approximations to the Helmholtz-type (fourth-order) or Poisson (sixth-order) equations and the fast Fourier transform. Compared to efficient second-order fast direct, methods for smooth problems, the execution time is reduced by a large factor, typically 50 for the Helmholtz-type equations and over 100 for the Poisson problem. Comparisons with two high-order fast direct methods indicate the superiority of the algorithm.