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1980

Volume 1, Issue 4, pp. 401-526


Minimum Covering Ellipses

B. W. Silverman and D. M. Titterington

SIAM J. Sci. and Stat. Comput. 1, pp. 401-409 (9 pages) | Cited 10 times

Online Publication Date: July 14, 2006

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With the aid of a duality relation originally obtained in the theory of statistical experimental design, an exact terminating algorithm is developed for finding the ellipse of smallest area covering a given plane point set. Some applications and related problems are discussed. Empirical timings show the algorithm to be highly efficient, particularly for large sets of points.

A Finite Element-Capacitance Matrix Method for the Neumann Problem for Laplace’s Equation

Włodzimierz Proskurowski and Olof Widlund

SIAM J. Sci. and Stat. Comput. 1, pp. 410-425 (16 pages) | Cited 12 times

Online Publication Date: July 14, 2006

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Capacitance matrix methods extend the usefulness of fast Poisson solvers to an important family of elliptic problems on arbitrary bounded regions. New algorithms of this kind are introduced for finite element approximations. The use of these methods for the Neumann problem for Laplace’s equation and important issues of implementation are discussed in some detail. It is shown that the new methods offer considerable advantages in comparison with finite difference–capacitance matrix methods previously employed.

On Improving the $2-4$ Two-Dimensional Leap-Frog Scheme

Saul Abarbanel and David Gottlieb

SIAM J. Sci. and Stat. Comput. 1, pp. 426-430 (5 pages)

Online Publication Date: July 14, 2006

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The present paper shows how to modify the Kreiss–Oliger $2-4$ two-dimensional leap-frog scheme so that the allowable time step may be doubled while the computational complexity remains about the same.

Generation of Body-Fitted Coordinates Using Hyperbolic Partial Differential Equations

Joseph L. Steger and Denny S. Chaussee

SIAM J. Sci. and Stat. Comput. 1, pp. 431-437 (7 pages) | Cited 19 times

Online Publication Date: July 14, 2006

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Grid generation equations formulated as hyperbolic partial differential equations are solved numerically to generate body conforming meshes. This grid generation procedure can be efficiently used to generate smoothly varying grids in which the user has good control of the grid clustering. Two-dimensional results are presented for typical external aerodynamic applications.

Distribution of Quadratic Forms in Normal Random Variables—Evaluation by Numerical Integration

S. O. Rice

SIAM J. Sci. and Stat. Comput. 1, pp. 438-448 (11 pages) | Cited 17 times

Online Publication Date: July 14, 2006

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The problem of calculating the distribution function of a general quadratic form in normal random variables is examined. Two numerical integration methods for inverting the characteristic function are presented. Both make use of paths of integration that pass through, or near to, a suitable saddle-point. It is assumed that a computer is available for the calculation of functions of complex variables and for the performance of various matrix computations. Approximations for special cases are stated and examples are given.

A Numerical Method for Computing the Shape of a Vertical Slender Jet

John Strikwerda and James Geer

SIAM J. Sci. and Stat. Comput. 1, pp. 449-466 (18 pages) | Cited 2 times

Online Publication Date: July 14, 2006

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A numerical method is presented for computing the shape of a vertical slender jet of fluid falling steadily under the force of gravity. The problem to be solved is fomulated as a nonlinear free boundary value problem for the cross-sectional shape of the jet. The numerical method of solution treats the boundary conditions of the problem as a pair of nonlinear hyperbolic pseudo-differential equations to be integrated in the stream-wise direction. The original differential equation appears as an auxiliary condition. This formulation is shown to be well-posed. The numerical method is found to be stable and second-order accurate. Computations are presented for jets issuing from several different orifice shapes. The numerical method of solution appears to be new and may be applicable to other nonlinear free boundary value problems.

Solving Finite Difference Approximations to Nonlinear Two-Point Boundary Value Problems by a Homotopy Method

Layne T. Watson

SIAM J. Sci. and Stat. Comput. 1, pp. 467-480 (14 pages) | Cited 11 times

Online Publication Date: July 14, 2006

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The Chow–Yorke algorithm is a homotopy method that has been proved globally convergent for Brouwer fixed point problems, classes of zero finding, nonlinear programming, and two-point boundary value problems. The method is numerically stable, and has been successfully applied to several practical nonlinear optimization and fluid dynamics problems. Previous application of the homotopy method to two-point boundary value problems has been based on shooting, which is inappropriate for fluid dynamics problems with sharp boundary layers. Here the Chow–Yorke algorithm is proved globally convergent for a class of finite difference approximations to nonlinear two-point boundary value problems. The numerical implementation of the algorithm is briefly sketched, and computational results are given for two fairly difficult fluid dynamics boundary value problems.

Fitting Empirical Data by Positive Sums of Exponentials

Axel Ruhe

SIAM J. Sci. and Stat. Comput. 1, pp. 481-498 (18 pages) | Cited 8 times

Online Publication Date: July 14, 2006

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Least squares and maximum likelihood fitting of a positive sum of exponentials to an empirical data series is discussed. A characterization in terms of a convex moment cone is used, to develop a globally convergent self-starting algorithm. The sensitivity of the results to errors in the data and during the computations is also discussed. Numerical tests are reported.

Equidistributing Meshes with Constraints

J. Kautsky and N. K. Nichols

SIAM J. Sci. and Stat. Comput. 1, pp. 499-511 (13 pages) | Cited 27 times

Online Publication Date: July 14, 2006

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Adaptive methods which “equidistribute” a given positive weight function are now used fairly widely for selecting discrete meshes. The disadvantage of such schemes is that the resulting mesh may not be smoothly varying. In this paper a technique is developed for equidistributing a function subject to constraints on the ratios of adjacent steps in the mesh. Given a weight function $f \geqq 0$ on an interval $[a,b]$ and constants $c$ and $K$, the method produces a mesh with points $x_0 = a,x_{j + 1} = x_j + h_j ,j = 0,1, \cdots ,n - 1$ and $x_n = b$ such that\[ \int_{xj}^{x_{j + 1} } {f \leqq c\quad {\text{and}}\quad \frac{1} {K}} \leqq \frac{{h_{j + 1} }} {{h_j }} \leqq K\quad {\text{for}}\, j = 0,1, \cdots ,n - 1 . \] A theoretical analysis of the procedure is presented, and numerical algorithms for implementing the method are given. Examples show that the procedure is effective in practice. Other types of constraints on equidistributing meshes are also discussed.
The principal application of the procedure is to the solution of boundary value problems, where the weight function is generally some error indicator, and accuracy and convergence properties may depend on the smoothness of the mesh. Other practical applications include the regrading of statistical data.

A Stable Algorithm for Solving the Multifacility Location Problem Involving Euclidean Distances

P. H. Calamai and A. R. Conn

SIAM J. Sci. and Stat. Comput. 1, pp. 512-526 (15 pages) | Cited 18 times

Online Publication Date: July 14, 2006

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There is a rapidly growing interdisciplinary interest in the application of location models to real life problems. Unfortunately, the current methods used to solve the most popular minisum and minimax location problems are computationally inadequate. A more unified and numerically stable approach for solving these problems is proposed. Detailed analysis is done for the linearly constrained Euclidean distance minisum problem for facilities located in a plane. Preliminary computational experience suggests that this approach compares favourably with other methods.
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