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1993

Volume 14, Issue 6, pp. 1259-1503


Block-Cyclic Dense Linear Algebra

Woody Lichtenstein and S. Lennart Johnsson

SIAM J. Sci. Comput. 14, pp. 1259-1288 (30 pages) | Cited 7 times

Online Publication Date: July 13, 2006

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Block-cyclic order elimination algorithms for LU and OR factorization and solve routines are described for distributed memory architectures with processing nodes configured as two-dimensional arrays of arbitrary shape. The cyclic-order elimination, together with a consecutive data allocation, yields good load balance for both the factorization and solution phases for the solution of dense systems of equations by LU and OR decomposition. Blocking may offer a substantial performance enhancement on architectures for which the level-2 or level-3 BLAS (basic linear algebra subroutines) are ideal for operations local to a node. High-rank updates local to a node may have a performance that is a factor of four or more higher than a rank-1 update.
This paper shows that in many parallel implementations, the $O(N^2 )$ work in the factorization may be of the same significance as the $O(N^3 )$ work, even for large matrices. The $O(N^2 )$ work is poorly load balanced in two-dimensional nodal arrays, which are shown to be optimal with respect to communication for consecutive data allocation, block-cyclic order elimination, and a simple, but fairly general, communications model.
In this Connection Machine system CM-200 implementation, the peak performance for LU factorization is about 9.4 Gflops/s in 64-bit precision and 16 Gflops/s in 32-bit precision. Blocking offers an overall performance enhancement of an approximate factor of two. The broadcast-and-reduce operations fully utilize the bandwidth available in the Boolean cube network interconnecting the nodes along each axis of the two-dimensional nodal array embedded in the cube network.

Computing the Exact Least Median of Squares Estimate and Stability Diagnostics in Multiple Linear Regression

Arnold J. Stromberg

SIAM J. Sci. Comput. 14, pp. 1289-1299 (11 pages) | Cited 18 times

Online Publication Date: July 13, 2006

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The difficulty in computing the least median of squares (LMS) estimate in multiple linear regression is due to the nondifferentiability and many local minima of the objective function. Several approximate, but not exact, algorithms have been suggested. This paper presents a method for computing the exact value of the LMS estimate in multiple linear regression. The LMS estimate is a special case of the least quantile of squares (LQS) estimate, which minimizes the $q$th smallest squared residual for a given data set. For LMS, $q = [n/2] + [(p + 1)/2]$ where $[ \, ]$ is the greatest integer function, $n$ is the sample size, and $p$ is the number of columns in the $X$ matrix. The algorithm can compute a range of exact LQS estimates in multiple linear regression by considering $\left( {\begin{array}{*{20}c} n \\ {p + 1} \\ \end{array} } \right)$ possible $\theta $ values. It is based on the fact that each LQS estimate is the Chebyshev (or minimax) fit to some $q$ element subset of the data. This yields a surprisingly easy algorithm for computing the exact LQS estimates. These and other estimates are used to study the stability of the LMS estimate in several examples.

Numerical and Asymptotic Solutions for Peristaltic Motion of Nonlinear Viscous Flows with Elastic Free Boundaries

Dalin Tang and Samuel Rankin

SIAM J. Sci. Comput. 14, pp. 1300-1319 (20 pages) | Cited 4 times

Online Publication Date: July 13, 2006

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A mathematical model for peristaltic motion of nonlinear viscous flows with elastic free boundaries is introduced. An iterative numerical method is used to solve the free boundary problem. Long wave asymptotic expansion is developed and the zeroth order approximation is used as the numerical initial condition. The existence and uniqueness of the solution for the free boundary equation derived from the long wave expansion are proved. Computations were conducted to study the long wave approximation, the numerical solutions for the exact equations, and the influences of the parameters on the solutions.

A Parallel Algorithm for Reducing Symmetric Banded Matrices to Tridiagonal Form

Bruno Lang

SIAM J. Sci. Comput. 14, pp. 1320-1338 (19 pages) | Cited 6 times

Online Publication Date: July 13, 2006

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An algorithm is presented for reducing symmetric banded matrices to tridiagonal form via Householder transformations. The algorithm is numerically stable and is well suited to parallel execution on distributed memory multiple instruction multiple data (MIMD) computers. Numerical experiments on the iPSC/860 hypercube show that the new method yields nearly full speedup if it is run on multiple processors. In addition, even on a single processor the new method usually will be several times faster than the corresponding EISPACK and LAPACK routines.

Computing Periodic Gravity Waves on Water by Using Moving Composite Overlapping Grids

N. Anders Petersson

SIAM J. Sci. Comput. 14, pp. 1339-1358 (20 pages) | Cited 1 time

Online Publication Date: July 13, 2006

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The composite overlapping grid method is applied to compute periodic gravity waves on water of finite constant depth. One component grid is made to follow the free surface while the remaining components are independent of the location of the surface. A pseudo-arclength continuation method is used to compute the solution as function of the phase velocity of the wave. The type of equation associated with some grid points and the number of equations in the discretized problem will change when the surface moves. The author expounds a stable way of switching the composite grid during the continuation procedure that works close to limit points. An adaptive technique is also developed to efficiently resolve the solution where sharp gradients develop.
Numerical examples are presented that show very good agreement with existing results.

A Modified Broyden Update with Interpolation

Miguel F. Anjos

SIAM J. Sci. Comput. 14, pp. 1359-1367 (9 pages)

Online Publication Date: July 13, 2006

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A new class of Hessian updates is introduced for quasi-Newton methods obtained by modifying the Broyden class of updates so that it interpolates function values. In general, the members of this new class do not satisfy the quasi-Newton equation. Through the use of a weak inverse updating strategy, a specific choice of update within this class is then introduced. Numerical results are given to illustrate the behaviour of this new update and to compare it with other updating sequences.

Fast Fourier Transforms for Nonequispaced Data

A. Dutt and V. Rokhlin

SIAM J. Sci. Comput. 14, pp. 1368-1393 (26 pages) | Cited 132 times

Online Publication Date: July 13, 2006

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A group of algorithms is presented generalizing the fast Fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval $[ - \pi ,\pi ]$. The schemes of this paper are based on a combination of certain analytical considerations with the classical fast Fourier transform and generalize both the forward and backward FFTs. Each of the algorithms requires $O(N\cdot \log N + N\cdot \log (1/\varepsilon ))$ arithmetic operations, where $\varepsilon $ is the precision of computations and $N$ is the number of nodes. The efficiency of the approach is illustrated by several numerical examples.

Numerical Solution of the Riemann Problem for Two-Dimensional Gas Dynamics

Carsten W. Schulz-Rinne, James P. Collins, and Harland M. Glaz

SIAM J. Sci. Comput. 14, pp. 1394-1414 (21 pages) | Cited 69 times

Online Publication Date: July 13, 2006

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The Riemann problem for two-dimensional gas dynamics with isentropic or polytropic gas is considered. The initial data is constant in each quadrant and chosen so that only a rarefaction wave, shock wave, or slip line connects two neighboring constant initial states. With this restriction sixteen (respectively, fifteen) genuinely different wave combinations for isentropic (respectively, polytropic) gas exist. For each configuration the numerical solution is analyzed and illustrated by contour plots. Additionally, the required relations for the initial data and the symmetry properties of the solutions are given. The chosen calculations correspond closely to the cases studied by T. Zhang and Y. Zheng [SIAM J. Math. Anal., 21 (1990), pp. 593–630], so that the analytical theory can be directly compared to our numerical study.

Construction of $K$-Dimensional Delaunay Triangulations Using Local Transformations

Barry Joe

SIAM J. Sci. Comput. 14, pp. 1415-1436 (22 pages) | Cited 2 times

Online Publication Date: July 13, 2006

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In [SIAM J. Sci. Statist. Comput.,10 (1989), pp. 718–741] and [Comput. Aided Geom. Des., 8 (1991), pp. 123–142] the author presented algorithms that use local transformations to construct a Delaunay triangulation of a set of $n$ three-dimensional points. This paper proves that local transformations can be used to construct a Delaunay triangulation of a set of $n$$k$-dimensional points for any $k \geq 2$, and presents algorithms using this approach. The empirical time complexities of these algorithms are discussed for sets of random points from the uniform distribution as well as worst-case time complexities. These time complexities are about the same or better than those of other algorithms for constructing $k$-dimensional Delaunay triangulations (when $k \geq 3$).

A Method for Devising Efficient Multigrid Smoothers for Complicated PDE Systems

Irad Yavneh

SIAM J. Sci. Comput. 14, pp. 1437-1463 (27 pages) | Cited 7 times

Online Publication Date: July 13, 2006

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A systematic approach is developed for gaining insight into complicated systems of partial differential equations (PDEs) in order to construct efficient smoothers for multigrid solvers. The method is derived from local mode (smoothing) analysis and employs algebraic graph theory, but it requires no knowledge of either in its implementation. It is applied to several problems, for which finding the best approach without such an analysis can be quite challenging.

Computing the Generalized Singular Value Decomposition

Zhaojun Bai and James W. Demmel

SIAM J. Sci. Comput. 14, pp. 1464-1486 (23 pages) | Cited 17 times

Online Publication Date: July 13, 2006

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A variation of Paige’s algorithm is presented for computing the generalized singular value decomposition (GSVD) of two matrices $A$ and $B$. There are two innovations. The first is a new preprocessing step which reduces $A$ and $B$ to upper triangular forms satisfying certain rank conditions. The second is a new $2 \times 2$ triangular GSVD algorithm, which constitutes the inner loop of Paige’s algorithm. Proofs of stability and high accuracy of the $2 \times 2$ GSVD algorithm are presented and are demonstrated using examples on which all previous algorithms fail.

The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems

Per Christian Hansen and Dianne Prost O’Leary

SIAM J. Sci. Comput. 14, pp. 1487-1503 (17 pages) | Cited 183 times

Online Publication Date: July 13, 2006

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Regularization algorithms are often used to produce reasonable solutions to ill-posed problems. The L-curve is a plot—for all valid regularization parameters—of the size of the regularized solution versus the size of the corresponding residual. Two main results are established. First a unifying characterization of various regularization methods is given and it is shown that the measurement of “size” is dependent on the particular regularization method chosen. For example, the 2-norm is appropriate for Tikhonov regularization, but a 1-norm in the coordinate system of the singular value decomposition (SVD) is relevant to truncated SVD regularization. Second, a new method is proposed for choosing the regularization parameter based on the L-curve, and it is shown how this method can be implemented efficiently. The method is compared to generalized cross validation and this new method is shown to be more robust in the presence of correlated errors.
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