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2003

Volume 24, Issue 6, pp. 1839-2198


Detecting and Locating Near-Optimal Almost-Invariant Sets and Cycles

Gary Froyland and Michael Dellnitz

SIAM J. Sci. Comput. 24, pp. 1839-1863 (25 pages) | Cited 13 times

Online Publication Date: July 25, 2006

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The behaviors of trajectories of nonlinear dynamical systems are notoriously hard to characterize and predict. Rather than characterizing dynamical behavior at the level of trajectories, we consider following the evolution of sets. There are often collections of sets that behave in a very predictable way, in spite of the fact that individual trajectories are entirely unpredictable. Such special collections of sets are invisible to studies of long trajectories. We describe a global set-oriented method to detect and locate these large dynamical structures. Our approach is a marriage of new ideas in modern dynamical systems theory and the novel application of graph dissection algorithms.

A Fast, Two-Dimensional Panel Method

Prabhu Ramachandran, S. C. Rajan, and M. Ramakrishna

SIAM J. Sci. Comput. 24, pp. 1864-1878 (15 pages) | Cited 2 times

Online Publication Date: July 25, 2006

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A new two-dimensional panel technique has been developed to solve Laplacian flows, which eliminates the edge effect present in traditional panel methods. Such a method is very useful for applications where the velocity induced by the panels is required at arbitrary locations. Particle based flow solvers are a prime example. The method, however, requires considerably more computational effort. In this paper the method is modified to improve computational efficiency by adapting the fast multipole algorithm for the panel method. Significant improvement in computational efficiency is obtained while ensuring that the edge effects are eliminated.

Multi-Adaptive Galerkin Methods for ODEs I

Anders Logg

SIAM J. Sci. Comput. 24, pp. 1879-1902 (24 pages) | Cited 11 times

Online Publication Date: July 25, 2006

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We present multi-adaptive versions of the standard continuous and discontinuous Galerkin methods for ODEs. Taking adaptivity one step further, we allow for individual time-steps, order and quadrature, so that in particular each individual component has its own time-step sequence. This paper contains a description of the methods, an analysis of their basic properties, and a posteriori error analysis. In the accompanying paper [A. Logg, SIAM J. Sci. Comput., submitted], we present adaptive algorithms for time-stepping and global error control based on the results of the current paper.

Fast Computation of MD-DCT-IV/MD-DST-IV by MD-DWT or MD-DCT-II

Yonghong Zeng, Zhiping Lin, Guoan Bi, and Lizhi Cheng

SIAM J. Sci. Comput. 24, pp. 1903-1918 (16 pages)

Online Publication Date: July 25, 2006

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This paper reveals relationships between the type-IV multidimensional discrete cosine transform (MD-DCT-IV) and the type-II multidimensional discrete cosine transform (MD-DCT-II) or the multidimensional discrete W transform (MD-DWT). Based on the relationships, MD-DCT-IV can be efficiently computed by using the fast algorithms for MD-DCT-II or MD-DWT-III or MD-DWT-III. Efficient method for implementation of the post-addition or pre-addition process is also presented. Compared to the widely used row-column method, considerable savings on the number of arithmetic operations are achieved. For example, the number of multiplications for computing an r-dimensional DCT-IV is only about $\frac{1}{r}$ times that needed by the row-column method, and the number of additions is also reduced. Numerical experiments are made to test the computational time and roundoff errors. They show that the proposed fast algorithms save time.

Numerical Computations of Viscous, Incompressible Flow Problems Using a Two-Level Finite Element Method

Faisal Fairag

SIAM J. Sci. Comput. 24, pp. 1919-1929 (11 pages) | Cited 3 times

Online Publication Date: July 25, 2006

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We consider two-level finite element discretization methods for the stream function formulation of the Navier--Stokes equations. The two-level method consists of solving a small nonlinear system on the coarse mesh and then solvinga linear system on the fine mesh. The basic result states that the errors between the coarse and fine meshes are related superlinearly. This paper demonstrates that the two-level method can be implemented to approximate efficiently solutions to the Navier--Stokes equations. Two fluid flow calculations are considered to test problems which have a known solution and the driven cavity problem. Stream function contours are displayed showing the main features of the flow.

Optimal Magnetic Shield Design with Second-Order Cone Programming

Takashi Sasakawa and Takashi Tsuchiya

SIAM J. Sci. Comput. 24, pp. 1930-1950 (21 pages) | Cited 8 times

Online Publication Date: July 25, 2006

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In this paper, we consider a continuous version of theconvex network flow problem which involves the integral of the Euclidean norm of the flow and its square in the objective function. A discretized version of this problem can be cast as a second-order cone program, for which efficient primal-dual interior-point algorithms have been developed recently. An optimal magnetic shielding design problem of the MAGLEV train, a new bullet train under development in Japan, is formulated as the continuous convex network flow problem and is solved with the primal-dual interior-point algorithm. Taking advantage of its efficiency and stability, we further apply the algorithm to robust design of the magnetic shielding.

Verlet-I/R-RESPA/Impulse is Limited by Nonlinear Instabilities

Qun Ma, Jesús A. Izaguirre, and Robert D. Skeel

SIAM J. Sci. Comput. 24, pp. 1951-1973 (23 pages) | Cited 19 times

Online Publication Date: July 25, 2006

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This paper shows that in molecular dynamics (MD) when constant- energy (NVE) simulations of Newton's equations of motion are attempted using the multiple time stepping (MTS) integrator Verlet-I/r-RESPA/Impulse, there are nonlinear instabilities when the longest step size is a third or possibly a fourth of the period(s) of the fastest motion(s) in the system. This is demonstrated both through a thorough set of computer experiments and through the analysis of a nonlinear model problem. The numerical experiments include not only the unconstrained dynamics simulation of a droplet of flexible water and a flexible protein, but also the constrained dynamics simulation of a solvated protein, representing a range of simulation protocols commonly in use by biomolecular modelers. The observed and predicted instabilities match exactly. Previous work has identified and explained a linear instability for Verlet-I/r-RESPA/Impulse at around half the period of the fastest motion. Mandziuk and Schlick discovered nonlinear resonances in single time stepping MD integrators, but unstable nonlinear resonances for MTS integrators are reported here for the first time. This paper also offers an explanation on the instability of MTS constrained molecular dynamics simulations of explicitly solvated proteins. More aggressive multiple step sizes are possible with mild Langevin coupling or targeted Langevin coupling, and its combination with the mollified Impulse method permits step sizes 3 to 4 times larger than Verlet-I/r-RESPA/Impulse while still retaining some accuracy.

Extrapolation for Finite Volume Approximations

Xiuling Ma, Dong Mao, and Aihui Zhou

SIAM J. Sci. Comput. 24, pp. 1974-1993 (20 pages) | Cited 3 times

Online Publication Date: July 25, 2006

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In this paper, the extrapolation technique is applied to a finite volume method. Based on some asymptotic error expansions of finite volume solutions established in this paper, several extrapolation schemes are proposed for finite volume approximations to elliptic problems.

A Newton-GMRES Approach for the Analysis of the Postbuckling Behavior of the Solutions of the von Kármán Equations

Kokou Dossou and Roger Pierre

SIAM J. Sci. Comput. 24, pp. 1994-2012 (19 pages) | Cited 8 times

Online Publication Date: July 25, 2006

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We propose a Newton-GMRES--type algorithm to solve the discrete von Kármán problem. We show that this algorithm is efficient both in memory and computation time and robust in the neighborhood of the singular points of the bifurcation diagrams. Placing ourselves in the context of the Schaeffer and Golubitsky theory, we use this algorithm to study the postbuckling behavior of a rectangular plate clamped and compressed along its four sides.

Fast Summation at Nonequispaced Knots by NFFT

Daniel Potts and Gabriele Steidl

SIAM J. Sci. Comput. 24, pp. 2013-2037 (25 pages) | Cited 12 times

Online Publication Date: July 25, 2006

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We develop a new algorithm for the fast computation of discrete sums $f(y_j) := \sum_{k=1}^N \alpha_k K(y_j-x_k)$ (j =1, . . ., M) based on the recently developed fast Fourier transform (FFT) at nonequispaced knots. Our algorithm, in particular our regularization procedure, is simply structured and can be easily adapted to different kernels K. Our method utilizes the widely known FFT and can consequently incorporate advanced FFT implementations. In summary, it requires ${\cal O} (N \log N +M)$ arithmetic operations. We prove error estimates to obtain clues about the choice of the involved parameters and present numerical examples in one and two dimensions.

An A Posteriori Error Estimator for the FEM in Nonlinear Elastostatics

Jens Georg Schmidt

SIAM J. Sci. Comput. 24, pp. 2038-2057 (20 pages) | Cited 2 times

Online Publication Date: July 25, 2006

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In this paper we deal with static problems of geometrically nonlinear elasticity. For the more challenging problems of this type, adaptive finite element methods are needed. A crucial ingredient for these methods is a robust and efficient a posteriori error estimator. In most implementations the well-known error estimators for linear elasticity are applied to a linearization of the nonlinear problems under consideration. But these linearized estimators work only for mildly nonlinear problems.
In geometrically nonlinear elastostatics a coarse grid approximation can be used to resolve the main features of the solution.Using that fact and Verfürth's error estimator analysis we develop a new estimator, which uses coarse-grid projections of the operators. Our estimator is shown to be efficient, even in the presence of strong nonlinearities, especially around simple limit points, where the linearized estimators break down. To illustrate its performance we present numerical results derived from a planar elasticity problem containing simple limit points.

A Mixed Finite-Element Discretization of the Energy-Transport Model for Semiconductors

Stefan Holst, Ansgar Jungel, and Paolo Pietra

SIAM J. Sci. Comput. 24, pp. 2058-2075 (18 pages) | Cited 11 times

Online Publication Date: July 25, 2006

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Energy-transport models describe the flow of electrons through a semiconductor device, influenced by diffusive, electrical, and thermal effects. They consist of the continuity equations for the mass and energy, coupled with Poisson's equation for the electrostatic potential. The energy-transport model can be written in a drift-diffusion formulation which is used for the numerical approximation. The stationary equations are discretized with an exponential fitting mixed finite-element method in two space dimensions. Numerical simulations of a ballistic diode are performed and numerical convergence rates are computed. Furthermore, a two-dimensional metal-semiconductor field-effect transistor device with parabolic band structure is simulated.

On Solving Block-Structured Indefinite Linear Systems

Gene H. Golub and Chen Greif

SIAM J. Sci. Comput. 24, pp. 2076-2092 (17 pages) | Cited 31 times

Online Publication Date: July 25, 2006

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We consider 2 ×2 block inde finite linear systems whose (2 ,2)block is zero.Such systems arise in many applications.We discuss two techniques that are based on modifying the (1 ,1)block in a way that makes the system easier to solve.The main part of the paper focuses on an augmented Lagrangian approach:a technique that modi fies the (1,1)block without changing the system size.The choice of the parameter involved,the spectrum of the linear system,and its condition number are discussed,and some analytical observations are provided.A technique of de flating the (1,1)block is then introduced.Finally,numerical experiments that validate the analysis are presented.

Solution ofthe Discontinuous P1 Equations in Two-Dimensional Cartesian Geometry with Two-Level Preconditioning

J. S. Warsa, T. A. Wareing, and J. E. Morel

SIAM J. Sci. Comput. 24, pp. 2093-2124 (32 pages) | Cited 2 times

Online Publication Date: July 25, 2006

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We present a new bilinear discontinuous (Galerkin) finite element discretization of the P1 (spherical harmonics) equations, a first order system of equations used for describing neutral particle radiation transport or modeling radiative transfer problems. The discrete equations are described for two-dimensional rectangular meshes; we solve the linear system with Krylov iterative methods. We have developed a novel, two-level preconditioner to improve convergence of the Krylov solvers that is based on a linear continuous finite element discretization of the diffusion equation, solved with a conjugate gradient iteration, preceded and followed by one of several different smoothing relaxations. A Fourier analysis shows that our approach is very effective over a wide range of problems. Numerical experiments confirm the results of the Fourier analysis. Computations for a realistic problem show that the preconditioner is effective and the solution method is efficient in practice.

Spartan Gibbs Random Field Models for Geostatistical Applications

Dionissios T. Hristopulos

SIAM J. Sci. Comput. 24, pp. 2125-2162 (38 pages) | Cited 20 times

Online Publication Date: July 25, 2006

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The inverse problem of determining the spatial dependence of random fields (inference of the spatial model) from experimental samples is a central issue in geostatistics. We propose a computationally efficient approach based on Spartan Gibbs random fields. Their probability density function depends on a small set of parameters that can be determined by matching sample constraints with corresponding model constraints based on the stochastic moments. We investigate a specific Spartan probability density with spatial dependence derived from generalized gradient and Laplacian operators, and we derive permissibility conditions for the model parameters. The optimal values of the model parameters are determined by minimizing a normalized metric measuring the "distance" between stochastic moments and the respective sample constraints. The computational complexity of the minimization depends on the number of parameters but not on the sample size. The computational complexity of the constraint calculation increases linearly with the sample size. In contrast, the classical variogram calculation is considerably less efficient and its complexity increases as the square of the sample size. We propose a numerical approach for inferring the Spartan model parameters and illustrate it using simulated (synthetic) samples for regular (lattice) and irregular sample distributions. Based on our analysis, Spartan Gibbs random fields provide computationally efficient spatial models, which are especially useful if the sample size is large or reliable estimation of the variogram is not possible. Estimation of the field values at unsampled positions, conditional simulations, anisotropic spatial dependence, and non-Gaussian probability densities are briefly discussed.

Duality Estimates and Multigrid Analysis for Saddle Point Problems Arising from Mortar Discretizations

Christian Wieners and Barbara I. Wohlmuth

SIAM J. Sci. Comput. 24, pp. 2163-2184 (22 pages) | Cited 3 times

Online Publication Date: July 25, 2006

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We present an abstract framework for the analysis of multigrid methods for a saddle point problem arising from mortar finite element discretizations. In contrast to other approaches, the iterates do not have to be in the positive definite subspace. Moreover, our approach also covers the case of nonnested Lagrange multiplier spaces. We apply the multigrid method to different mortar settings including dual Lagrange multipliers, linear elasticity, and a rotating geometry. Numerical results in two dimensions and three dimensions demonstrate the flexibility, efficiency, and reliability of our multigrid method.

Finite Difference WENO Schemes with Lax--Wendroff-Type Time Discretizations

Jianxian Qiu and Chi-Wang Shu

SIAM J. Sci. Comput. 24, pp. 2185-2198 (14 pages) | Cited 11 times

Online Publication Date: July 25, 2006

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In this paper we develop a Lax--Wendroff time discretization procedure for high order finite difference weighted essentially nonoscillatory schemes to solve hyperbolic conservation laws. This is an alternative method for time discretization to the popular TVD Runge--Kutta time discretizations. We explore the possibility in avoiding the local characteristic decompositions or even the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining nonoscillatory properties for problems with strong shocks. As a result, the Lax--Wendroff time discretization procedure is more cost effective than the Runge--Kutta time discretizations for certain problems including two-dimensional Euler systems of compressible gas dynamics.
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