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SIAM J. on Scientific Computing

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1988

Volume 9, Issue 6, pp. 963-1132


Effectively Well-Conditioned Linear Systems

Tony F. Chan and David E. Foulser

SIAM J. Sci. and Stat. Comput. 9, pp. 963-969 (7 pages) | Cited 21 times

Online Publication Date: July 13, 2006

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When solving the linear system ${\bf A}x = {\bf b}$, the condition number $K(A) \equiv \| A \| \| A^{ - 1} \|$ is a useful, albeit often overly conservative, measure of the sensitivity of the solution ${\bf x}$ under perturbations $\Delta A$ and $\Delta {\bf b}$ to $A$ and ${\bf b}$. We demonstrate how the projection of ${\bf b}$ onto the range space of $A$, in addition to $K(A)$, can strongly affect the sensitivity of ${\bf x}$ in specific problem instances. Two practical cases are presented in which the sensitivity of ${\bf x}$ can be substantially smaller than that predicted by $K(A)$ alone. In the first example, we characterize a class of Vandermonde matrices and right-hand sides for which accurate algorithms can exist. For the second example, we show that a (fast Fourier transform-) FFT-based fast Poisson solver can produce very accurate results for smooth right-hand sides. Computational examples on the fast Poisson solver are included to illustrate these concepts.

On the Numerical Approximation of an Optimal Correction Problem

M. C. Bancora-Imbert, P. L. Chow, and J. L. Menaldi

SIAM J. Sci. and Stat. Comput. 9, pp. 970-991 (22 pages) | Cited 1 time

Online Publication Date: July 13, 2006

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The numerical solution of an optimal correction problem for a damped random linear oscillator is studied. A numerical algorithm for the discretized system of the associated dynamic programming equation is given. To initiate the computation, we adopt a numerical scheme derived from the deterministic version of the problem. Next, a correction-type algorithm based on a discrete maximum principle is introduced to ensure the convergence of the iteration procedure.

Mode-Dependent Finite-Difference Discretization of Linear Homogeneous Differential Equations

C. -C. Jay Kuo and Bernard C. Levy

SIAM J. Sci. and Stat. Comput. 9, pp. 992-1015 (24 pages)

Online Publication Date: July 13, 2006

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A new methodology utilizing the spectral analysis of local differential operators is proposed to design and analyze mode-dependent finite-difference schemes for linear homogeneous ordinary and partial differential equations. We interpret the finite-difference method as a procedure for approximating exactly a local differential operator over a finite-dimensional space of test functions called the coincident space, and show that the coincident space is basically determined by the nullspace of the local differential operator. Since local operators are linear and approximately with constant coefficients, we introduce a transform domain approach to perform the spectral analysis. For the case of boundary-value ordinary differential equations (ODEs), a mode-dependent finite-difference scheme can be systematically obtained. For boundary-value partial differential equations (PDEs), mode-dependent 5-point, rotated 5-point, and 9-point stencil discretizations for the Laplace, Helmholtz, and convection-diffusion equations are developed. The effectiveness of the resulting schemes is shown analytically, as well as by considering several numerical examples.

Absorbing Boundary Conditions for Rayleigh Waves

Alain Bamberger, Bruno Chalindar, Patrick Joly, Jean Elizabeth Roberts, and Jean Luc Teron

SIAM J. Sci. and Stat. Comput. 9, pp. 1016-1049 (34 pages) | Cited 3 times

Online Publication Date: July 13, 2006

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The first-order absorbing boundary conditions for elastic waves are transparent for $P$ and $S$ waves at normal incidence, but give rise to parasitic reflections of Rayleigh waves. To treat these phenomena, a solution of geometric type is proposed that eliminates these parasitic waves but causes others to appear, which, while less important, are still troublesome. A second solution is proposed by constructing a new condition of second-order type, transparent for $P$ and $S$ waves at normal incidence as well as for Rayleigh waves. This condition is analyzed mathematically and its good behavior is demonstrated with regard to reflection phenomena.

Computing the Inverse of the Chebyshev Collocation Derivative

Daniele Funaro

SIAM J. Sci. and Stat. Comput. 9, pp. 1050-1057 (8 pages) | Cited 3 times

Online Publication Date: July 13, 2006

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Various numerical techniques for inverting the matrix $S$ of the pseudospectral derivative in one spatial dimension at the Chebyshev nodes are compared in terms of accuracy and CPU time. An explicit formula for the entries of $S^{ - 1} $ is presented.

A Modified Remes Algorithm

Yi-Ling F. Chiang

SIAM J. Sci. and Stat. Comput. 9, pp. 1058-1072 (15 pages)

Online Publication Date: July 13, 2006

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The frequently used Remes algorithm, which finds the best approximation to a continuous function in a finite interval, may not always converge. This iterative algorithm requires the error function of the intermediate approximation at every iteration to have equal magnitude with alternating signs at a specified number of points. When this requirement cannot be fulfilled, the algorithm fails to converge. In this paper, a property (called property $y$) and a modified Remes algorithm are defined such that the convergence of the new algorithm is guaranteed if the initial approximation has property $y$. Numerical examples of best approximations in various forms are given to show the use and convergence of the new algorithm.

Total-Variation-Diminishing Time Discretizations

Chi-Wang Shu

SIAM J. Sci. and Stat. Comput. 9, pp. 1073-1084 (12 pages) | Cited 177 times

Online Publication Date: July 13, 2006

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In the computation of conservation laws $u_t + f(u)_x = 0$, total-variation-diminishing (TVD) schemes have been very successful. Many TVD schemes are of method-of-lines form (i.e., discretized in spatial variables only); hence time discretizations that keep TVD and have other properties (e.g., large CFL numbers for steady state calculations, or high-order accuracy for time-dependent problems) are desirable. In this paper we present a class of $m$-step Runge–Kutta-type TVD time discretizations with large CFL number $m$, suitable for steady state calculations, and a class of multilevel type TVD high-order time discretizations suitable for time-dependent problems. Some preliminary numerical results are also given.

On the Parallel Implementation of Implicit Runge–Kutta Methods

Ohannes A. Karakashian and William Rust

SIAM J. Sci. and Stat. Comput. 9, pp. 1085-1090 (6 pages) | Cited 2 times

Online Publication Date: July 13, 2006

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We implement and test a parallel version of an implicit Runge–Kutta method as a representative of a general class of methods. These schemes are simple to program and allow speedups very close to the number of processors available.

Multivariate Stratification with Size Constraints

Robert S. Jewett and David R. Judkins

SIAM J. Sci. and Stat. Comput. 9, pp. 1091-1097 (7 pages)

Online Publication Date: July 13, 2006

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Multivariate stratification of sampling units is an important facet of large-scale sample surveys. Traditional univariate methods do not generalize easily. Much progress has been made using clustering algorithms. This paper discusses an algorithm that is useful when constraints are placed on the sizes of the clusters. The Bureau of the Census applied this algorithm for the stratification of primary sampling units during the recent redesign of its current demographic surveys.

Absolute Bounds on the Mean and Standard Deviation of Transformed Data for Constant-Sign-Derivative Transformations

Neil C. Rowe

SIAM J. Sci. and Stat. Comput. 9, pp. 1098-1113 (16 pages) | Cited 5 times

Online Publication Date: July 13, 2006

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Absolute bounds (or inequalities) on statistical quantities are often a desirable feature of statistical packages since, as contrasted with estimates of those same quantities, they can avoid distributional assumptions and can often be calculated very fast. We investigate bounds on the mean and standard deviation of transformed data values, given only a few statistics (e.g., mean, standard deviation, minimum, maximum, and median) on the original data values. Our work applies to transformation functions with constant-sign derivatives (e.g., logarithm, antilog, square root, and reciprocal). We can often get surprisingly tight bounds with simple closed-form expressions, so that confidence intervals are unnecessary. Most of the results of this paper seem to be new, though they are straightforward to derive by geometrical arguments and analytical optimization methods.

Discrete Event Simulations and Parallel Processing: Statistical Properties

Philip Heidelberger

SIAM J. Sci. and Stat. Comput. 9, pp. 1114-1132 (19 pages) | Cited 9 times

Online Publication Date: July 13, 2006

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This paper addresses statistical issues that arise when discrete event, or Monte Carlo, simulations are run on parallel processing computers. In particular, the statistical properties of estimators obtained by running parallel independent replications on a multiple processor computing system are considered. Because of the effects of parallelism, care must be taken in order to obtain estimators with the proper statistical properties. A variety of estimators are considered. The convergence properties, including strong laws, central limit theorems and bias expansions of these estimators are derived. It is shown that some of the more obvious estimators are guaranteed to converge to the wrong quantity as the number of processors increases. Strong laws and central limit theorems for completion times of the estimators are also given. The application of results from reliability and scheduling theory yields bounds on expected completion times under a variety of distributional assumptions. Based on these results, it does not appear possible to obtain a strongly consistent estimate in finite expected time as the number of processors increases, unless the computational time to complete a single replication is bounded.
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