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2008

Volume 30, Issue 5, pp. 2207-2708


Asymptotic Sampling Distribution for Polynomial Chaos Representation from Data: A Maximum Entropy and Fisher Information Approach

Sonjoy Das, Roger Ghanem, and James C. Spall

SIAM J. Sci. Comput. 30, pp. 2207-2234 (28 pages) | Cited 2 times

Online Publication Date: June 11, 2008

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A procedure is presented for characterizing the asymptotic sampling distribution of estimators of the polynomial chaos (PC) coefficients of a second-order nonstationary and non-Gaussian random process by using a collection of observations. The random process represents a physical quantity of interest, and the observations made over a finite denumerable subset of the indexing set of the random process are considered to form a set of realizations of a random vector $\mathcal{Y}$ representing a finite-dimensional projection of the random process. The Karhunen–Loève decomposition and a scaling transformation are employed to produce a reduced-order model $\mathcal{Z}$ of $\mathcal{Y}$. The PC expansion of $\mathcal{Z}$ is next determined by having recourse to the maximum-entropy principle, the Metropolis–Hastings Markov chain Monte Carlo algorithm, and the Rosenblatt transformation. The resulting PC expansion has random coefficients, where the random characteristics of the PC coefficients can be attributed to the limited data available from the experiment. The estimators of the PC coefficients of $\mathcal{Y}$ obtained from that of $\mathcal{Z}$ are found to be maximum likelihood estimators as well as consistent and asymptotically efficient. Computation of the covariance matrix of the associated asymptotic normal distribution of estimators of the PC coefficients of $\mathcal{Y}$ requires knowledge of the Fisher information matrix (FIM). The FIM is evaluated here by using a numerical integration scheme as well as a sampling technique. The resulting confidence interval on the PC coefficient estimators essentially reflects the effect of incomplete information (due to data limitation) on the characterization of the stochastic process. This asymptotic distribution is significant as its characteristics can be propagated through predictive models for which the stochastic process in question describes uncertainty on some input parameters.

Multilevel Adaptive Aggregation for Markov Chains, with Application to Web Ranking

H. De Sterck, Thomas A. Manteuffel, Stephen F. McCormick, Quoc Nguyen, and John Ruge

SIAM J. Sci. Comput. 30, pp. 2235-2262 (28 pages) | Cited 6 times

Online Publication Date: June 11, 2008

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A multilevel adaptive aggregation method for calculating the stationary probability vector of an irreducible stochastic matrix is described. The method is a special case of the adaptive smoothed aggregation and adaptive algebraic multigrid methods for sparse linear systems and is also closely related to certain extensively studied iterative aggregation/disaggregation methods for Markov chains. In contrast to most existing approaches, our aggregation process does not employ any explicit advance knowledge of the topology of the Markov chain. Instead, adaptive agglomeration is proposed that is based on the strength of connection in a scaled problem matrix, in which the columns of the original problem matrix at each recursive fine level are scaled with the current probability vector iterate at that level. The strength of connection is determined as in the algebraic multigrid method, and the aggregation process is fully adaptive, with optimized aggregates chosen in each step of the iteration and at all recursive levels. The multilevel method is applied to a set of stochastic matrices that provide models for web page ranking. Numerical tests serve to illustrate for which types of stochastic matrices the multilevel adaptive method may provide significant speedup compared to standard iterative methods. The tests also provide more insight into why Google's PageRank model is a successful model for determining a ranking of web pages.

A Numerical Method for the Generalized Regularized Long Wave Equation Using a Reproducing Kernel Function

Shusen Xie, Seokchan Kim, Gyungsoo Woo, and Sucheol Yi

SIAM J. Sci. Comput. 30, pp. 2263-2285 (23 pages)

Online Publication Date: June 11, 2008

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A new numerical method for solving the generalized regularized long wave equation is devised and analyzed. By using a reproducing kernel function, the numerical solution at each discrete time step is obtained by an explicit integral expression even though the scheme is truly implicit, and, hence, the computation is fully parallel. The error estimates are given and some numerical results are presented.

Factorization Techniques for Nodal Spectral Elements in Curved Domains

Jörg Stiller and Uwe Fladrich

SIAM J. Sci. Comput. 30, pp. 2286-2301 (16 pages)

Online Publication Date: July 02, 2008

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Spectral element methods on tetrahedra with symmetric collocation points can be accelerated by factorizing the discrete operators according to Hesthaven and Teng [SIAM J. Sci. Comput., 21 (2000), pp. 2352–2380]. While these authors focused on first-order conservation laws, the present paper provides an extension to second-order problems. Though factorization is easily accomplished for planar elements, difficulties arise from the presence of variable metric coefficients in curved tetrahedra. Two approaches are considered to cope with this peculiarity: (i) approximation of the metric terms by collocation projection, (ii) Gauss quadrature based on axisymmetric point sets. The first method achieves a separation of the metric terms such that the discrete operators can be reduced to factorizable standard matrices. As a consequence, the performance is comparable to the planar case, whereas the accuracy is limited by the projection step. The second approach maintains accuracy since all terms are evaluated individually in the quadrature points. Nonetheless, complete factorization is achieved by exploiting the symmetry in the quadrature points. Performance analysis shows that the curved element operator is less than three times as costly as the planar element counterpart.

Balanced Incomplete Factorization

Rafael Bru, José MarÍn, José Mas and M. TŮMA

SIAM J. Sci. Comput. 30, pp. 2302-2318 (17 pages) | Cited 1 time

Online Publication Date: July 02, 2008

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In this paper we present a new incomplete factorization of a square matrix into triangular factors in which we get standard $LU$ or $LDL^T$ factors (direct factors) and their inverses (inverse factors) at the same time. Algorithmically, we derive this method from the approach based on the Sherman–Morrison formula [R. Bru, J. Cerdán, J. Marín, and J. Mas, SIAM J. Sci. Comput., 25 (2003), pp. 701–715]. In contrast to the robust incomplete decomposition (RIF) algorithm [M. Benzi and M. Tůma, Numer. Linear Algebra Appl., 10 (2003), pp. 385–400] the direct and inverse factors here directly influence each other throughout the computation. Consequently, the algorithm to compute the approximate factors may mutually balance dropping in the factors and control their conditioning in this way. For the symmetric positive definite case, we derive the theory and present an algorithm for computing the incomplete $LDL^T$ factorization, and we discuss experimental results. We call this new approximate $LDL^T$ factorization the balanced incomplete factorization (BIF). Our experimental results confirm that this factorization is very robust and may be useful in solving difficult ill conditioned problems by preconditioned iterative methods. Moreover, the internal coupling of the computation of direct and inverse factors results in much shorter setup times (times to compute approximate decomposition) than RIF, a method of a similar and very high level of robustness. We also derive and present the theory for the general nonsymmetric case, but do not discuss its implementation.

Mixed Multiscale Finite Element Methods for Stochastic Porous Media Flows

J. E. Aarnes and Y. Efendiev

SIAM J. Sci. Comput. 30, pp. 2319-2339 (21 pages) | Cited 4 times

Online Publication Date: July 02, 2008

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In this paper, we propose a stochastic mixed multiscale finite element method. The proposed method solves the stochastic porous media flow equation on the coarse grid using a set of precomputed basis functions. The precomputed basis functions are constructed based on selected realizations of the stochastic permeability field, and furthermore the solution is projected onto the finite-dimensional space spanned by these basis functions. We employ multiscale methods using limited global information since the permeability fields do not have apparent scale separation. The proposed approach does not require any interpolation in stochastic space and can easily be coupled with interpolation-based approaches to predict the solution on the coarse grid. Numerical results are presented for permeability fields with normal and exponential variograms.

Efficient Calculation of Bounds on Spectra of Hessian Matrices

M. Mönnigmann

SIAM J. Sci. Comput. 30, pp. 2340-2357 (18 pages) | Cited 1 time

Online Publication Date: July 02, 2008

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We introduce a new method for the calculation of bounds on eigenvalues of Hessian matrices $\nabla^2 \varphi(x)$ of twice continuously differentiable functions $\varphi:U\subseteq\mathbb{R}^n\rightarrow \mathbb{R}$. The computational complexity of the new approach is shown to be of order ${\cal O}(n)\,N(\varphi)$ where $N(\varphi)$ is the number of operations necessary to evaluate the function $\varphi(x)$ at a point in its domain. This result is surprising, since the complexity of the calculation of the Hessian itself is of order ${\cal O}(n^2)\,N(\varphi)$ if the same method is used as in the proposed eigenvalue bounding approach. The favorable complexity of the new approach results because the eigenvalue bounds can be found without ever calculating the Hessian matrix.

Transparent Boundary Conditions for Time-Dependent Problems

Daniel Ruprecht, Achim Schädle, Frank Schmidt, and Lin Zschiedrich

SIAM J. Sci. Comput. 30, pp. 2358-2385 (28 pages) | Cited 1 time

Online Publication Date: July 02, 2008

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A new approach to derive transparent boundary conditions (TBCs) for dispersive wave, Schrödinger, heat, and drift-diffusion equations is presented. It relies on the pole condition and distinguishes between physically reasonable and unreasonable solutions by the location of the singularities of the Laplace transform of the exterior solution. Here the Laplace transform is taken with respect to a generalized radial variable. To obtain a numerical algorithm, a Möbius transform is applied to map the Laplace transform onto the unit disc. In the transformed coordinate the solution is expanded into a power series. Finally, equations for the coefficients of the power series are derived. These are coupled to the equation in the interior and yield transparent boundary conditions. Numerical results are presented in the last section, showing that the error introduced by the new approximate TBCs decays exponentially in the number of coefficients.

Efficient Solution of Anisotropic Lattice Equations by the Recovery Method

I. Babuška and S. A. Sauter

SIAM J. Sci. Comput. 30, pp. 2386-2404 (19 pages) | Cited 1 time

Online Publication Date: July 03, 2008

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In a recent paper, the authors introduced the recovery method (local energy matching principle) for solving large systems of lattice equations. The idea is to construct a partial differential equation along with a finite element discretization such that the arising system of linear equations has equivalent energy as the original system of lattice equations. Since a vast variety of efficient solvers is available for solving large systems of finite element discretizations of elliptic PDEs, these solvers may serve as preconditioners for the system of lattice equations. In this paper, we will focus on both the theoretical and the numerical dependence of the method on various mesh-dependent parameters, which can be easily computed and monitored during the solution process. Systematic parameter tests have been performed which underline (a) the robustness and the efficiency of the recovery method and (b) the reliability of the control parameters, which are computed in a preprocessing step to predict the performance of the preconditioner based on the recovery method.

BPCONT: An Auto Driver for the Continuation of Branch Points of Algebraic and Boundary-Value Problems

Fabio Dercole

SIAM J. Sci. Comput. 30, pp. 2405-2426 (22 pages)

Online Publication Date: July 03, 2008

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BPcont, a driver for the software package Auto for the numerical continuation of simple branch points of algebraic and boundary-value problems, is described in detail. Simple branch points are points in the continuation space where two solution branches intersect transversally. Detection and accurate location of branch points along the continuation of a solution branch and switching to the continuation of the other branch are implemented in Auto and other software packages, but branch point continuation has not been fully supported. BPcont fills this gap. It handles both generic problems, where branch point continuation is performed in two extra parameters, and nongeneric cases, where one extra parameter is typically involved due to problem-specific symmetries. This paper presents the extended algebraic and boundary-value problems which define branch point continuation, discusses their initialization at branch point detection and the symmetry-breaking technique used to automatically handle nongeneric cases, and describes the BPcont implementation. Several examples, including generic and nongeneric algebraic problems, generic and nongeneric periodic boundary-value problems, and a nongeneric nonperiodic boundary-value problem, are also presented. (A corrected version of this paper has been appended to the originally posted pdf.)

Exact de Rham Sequences of Spaces Defined on Macro-Elements in Two and Three Spatial Dimensions

Joseph E. Pasciak and Panayot S. Vassilevski

SIAM J. Sci. Comput. 30, pp. 2427-2446 (20 pages)

Online Publication Date: July 03, 2008

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This paper proposes new finite element spaces that can be constructed for agglomerates of standard elements that have certain regular structure. The main requirement is that the agglomerates share faces that have closed boundaries composed of 1-d edges. The spaces resulting from the agglomerated elements are subspaces of the original de Rham sequence of $H^1$-conforming, $H(\mathbf{curl})$-conforming, $H(\mathbf{div})$-conforming, and piecewise constant spaces associated with an unstructured “fine” mesh. The procedure can be recursively applied so that a sequence of nested de Rham complexes can be constructed. As an illustration we generate coarser spaces from the sequence corresponding to the lowest-order Nédélec spaces, lowest-order Raviart–Thomas spaces, and for piecewise linear $H^1$-conforming spaces, all in three dimensions. The resulting $V$-cycle multigrid methods used in preconditioned conjugate gradient iterations appear to perform similar to those of the geometrically refined case.

Norm Preconditioners for Discontinuous Galerkin $hp$-Finite Element Methods

Emmanuil H. Georgoulis and Daniel Loghin

SIAM J. Sci. Comput. 30, pp. 2447-2465 (19 pages)

Online Publication Date: July 18, 2008

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We consider a norm-preconditioning approach for the solution of discontinuous Galerkin finite element discretizations of second order partial differential equations with a nonnegative characteristic form. Our solution method is a norm-preconditioned three-term GMRES routine. We find that for symmetric positive-definite diffusivity tensors the convergence of our solver is independent of discretization, while for the semidefinite case both theory and experiment indicate dependence on both $h$ and $p$. Numerical results are included to illustrate performance on several test cases.

On the Time Splitting Spectral Method for the Complex Ginzburg–Landau Equation in the Large Time and Space Scale Limit

Pierre Degond, Shi Jin, and Min Tang

SIAM J. Sci. Comput. 30, pp. 2466-2487 (22 pages)

Online Publication Date: July 18, 2008

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We are interested in the numerical approximation of the complex Ginzburg–Landau equation in the large time and space limit. There are two interesting regimes in this problem, one being the large space time limit, and one being the nonlinear Schrödinger limit. These limits have been studied analytically in, for example, T. Colin and A. Soyeur, Asymptot. Anal., 13 (1996), pp. 361–372; F. H. Lin, Comm. Pure Appl. Math., 51 (1998), pp. 385–441; F. H. Lin and J. X. Xin, Comm. Math. Phys., 200 (1999), pp. 249–274. We study a time splitting spectral method for this problem. In particular, we are interested in whether such a scheme is asymptotic preserving (AP) with respect to these two limits. Our results show that the scheme is AP for the first limit but not the second one. For the large space time limit, our numerical experiments show that the scheme can capture the correct physical behavior without resolving the small scale dynamics, even for transitional problems, where small and large scales coexist.

Energy-Consistent CoRotational Schemes for Frictional Contact Problems

P. Hauret, J. Salomon, A. A. Weiss, and B. I. Wohlmuth

SIAM J. Sci. Comput. 30, pp. 2488-2511 (24 pages)

Online Publication Date: July 23, 2008

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In this paper, we consider the unilateral frictional contact problem of a hyperelastic body in the case of large displacements and small strains. In order to retain the linear elasticity framework, we decompose the deformation into a large global rotation and a small elastic displacement. This corotational approach is combined with a primal-dual active set strategy to tackle the contact problem. The resulting algorithm preserves both energy and angular momentum.

A Fast Iterative Method for Eikonal Equations

Won-Ki Jeong and Ross T. Whitaker

SIAM J. Sci. Comput. 30, pp. 2512-2534 (23 pages) | Cited 4 times

Online Publication Date: July 23, 2008

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In this paper we propose a novel computational technique to solve the Eikonal equation efficiently on parallel architectures. The proposed method manages the list of active nodes and iteratively updates the solutions on those nodes until they converge. Nodes are added to or removed from the list based on a convergence measure, but the management of this list does not entail an extra burden of expensive ordered data structures or special updating sequences. The proposed method has suboptimal worst-case performance but, in practice, on real and synthetic datasets, runs faster than guaranteed-optimal alternatives. Furthermore, the proposed method uses only local, synchronous updates and therefore has better cache coherency, is simple to implement, and scales efficiently on parallel architectures. This paper describes the method, proves its consistency, gives a performance analysis that compares the proposed method against the state-of-the-art Eikonal solvers, and describes the implementation on a single instruction multiple datastream (SIMD) parallel architecture.

An Efficient and Robust Method for Simulating Two-Phase Gel Dynamics

Grady B. Wright, Robert D. Guy, and Aaron L. Fogelson

SIAM J. Sci. Comput. 30, pp. 2535-2565 (31 pages) | Cited 1 time

Online Publication Date: August 01, 2008

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We develop a computational method for simulating models of gel dynamics where the gel is described by two phases: a networked polymer and a fluid solvent. The models consist of transport equations for the two phases, two coupled momentum equations, and a volume-averaged incompressibility constraint, which we discretize with finite differences/volumes. The momentum and incompressibility equations present the greatest numerical challenges since (i) they involve partial derivatives with variable coefficients that can vary quite significantly throughout the domain (when the phases separate), and (ii) their approximate solution requires the “inversion” of a large linear system of equations. For solving this system, we propose a box-type multigrid method to be used as a preconditioner for the generalized minimum residual (GMRES) method. Through numerical experiments of a model problem, which exhibits phase separation, we show that the computational cost of the method scales nearly linearly with the number of unknowns and performs consistently well over a wide range of parameters. For solving the transport equation, we use a conservative finite-volume method for which we derive stability bounds.

The Aitken-Like Acceleration of the Schwarz Method on Nonuniform Cartesian Grids

J. Baranger, M. Garbey, and F. Oudin-Dardun

SIAM J. Sci. Comput. 30, pp. 2566-2586 (21 pages)

Online Publication Date: August 01, 2008

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In this paper, we present a family of domain decomposition based on an Aitken-like acceleration of the Schwarz method seen as an iterative procedure with a linear rate of convergence. This paper is a generalization of the method first introduced at the 12th International Conference on Domain Decomposition that was restricted to regular Cartesian grids. The potential of this method to provide scalable parallel computing on a geographically broad grid of parallel computers was demonstrated for some linear and nonlinear elliptic problems discretized by finite differences on a Cartesian mesh. The main purpose of this paper is to present a generalization of the method to nonuniform Cartesian meshes. The salient feature of the method consists of accelerating the sequence of traces on the artificial interfaces generated by the Schwarz procedure using a good approximation of the main eigenvectors of the trace transfer operator. For linear separable elliptic operators, our solver is a direct solver. For nonlinear operators, we use an approximation of the eigenvectors of the Jacobian of the trace transfer operator. The acceleration is then applied to the sequence generated by the Schwarz algorithm applied directly to the nonlinear operator.

A Positive Preserving High Order VFRoe Scheme for Shallow Water Equations: A Class of Relaxation Schemes

Christophe Berthon and Fabien Marche

SIAM J. Sci. Comput. 30, pp. 2587-2612 (26 pages) | Cited 1 time

Online Publication Date: August 01, 2008

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The VFRoe scheme has been recently introduced by Buffard, Gallouët, and Hérard [Comput. Fluids, 29 (2000), pp. 813–847] to approximate the solutions of the shallow water equations. One of the main interests of this method is to be easily implemented. As a consequence, such a scheme appears as an interesting alternative to other more sophisticated schemes. The VFRoe methods perform approximate solutions in good agreement with the expected ones. However, the robustness of this numerical procedure has not been proposed. Following the ideas introduced by Jin and Xin [Comm. Pure Appl. Math., 45 (1995), pp. 235–276], a relevant relaxation method is derived. The interest of this relaxation scheme is twofold. In the first hand, the relaxation scheme is shown to coincide with the considered VFRoe scheme. In the second hand, the robustness of the relaxation scheme is established, and thus the nonnegativity of the water height obtained involving the VFRoe approach is ensured. Following the same idea, a family of relaxation schemes is exhibited. Next, robust high order slope limiter methods, known as MUSCL reconstructions, are proposed. The final scheme is obtained when considering the hydrostatic reconstruction to approximate the topography source terms. Numerical experiments are performed to attest the interest of the procedure.

Postprocessing of the Linear Sampling Method by Means of Deformable Models

R. Aramini, M. Brignone, J. Coyle, and M. Piana

SIAM J. Sci. Comput. 30, pp. 2613-2634 (22 pages)

Online Publication Date: August 01, 2008

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The linear sampling method is a qualitative procedure for the visualization of both impenetrable and inhomogeneous scatterers, which requires the regularized solution of a linear ill-posed integral equation of the first kind. An open issue in this technique is the one of determining the optimal scatterer profile from the visualization maps in an automatic manner. In the present paper this problem is addressed in two steps. First, linear sampling is optimized by using a new regularization algorithm for the solution of the integral equation, which provides more accurate maps for different levels of the noise affecting the data. Then an edge detection technique based on active contours is applied to the optimized maps. Our computation exploits a recently introduced implementation of the linear sampling method, which enhances both the accuracy and the numerical effectiveness of the approach.

Constructing Sobol Sequences with Better Two-Dimensional Projections

Stephen Joe and Frances Y. Kuo

SIAM J. Sci. Comput. 30, pp. 2635-2654 (20 pages) | Cited 2 times

Online Publication Date: August 01, 2008

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Direction numbers for generating Sobol$'$ sequences that satisfy the so-called Property A in up to 1111 dimensions have previously been given in Joe and Kuo [ACM Trans. Math. Software, 29 (2003), pp. 49–57]. However, these Sobol$'$ sequences may have poor two-dimensional projections. Here we provide a new set of direction numbers alleviating this problem. These are obtained by treating Sobol$'$ sequences in $d$ dimensions as $(t,d)$-sequences and then optimizing the $t$-values of the two-dimensional projections. Our target dimension is 21201.

Iterative Algorithms Based on Decoupling of Deblurring and Denoising for Image Restoration

You-Wei Wen, Michael K. Ng, and Wai-Ki Ching

SIAM J. Sci. Comput. 30, pp. 2655-2674 (20 pages)

Online Publication Date: August 06, 2008

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In this paper, we propose iterative algorithms for solving image restoration problems. The iterative algorithms are based on decoupling of deblurring and denoising steps in the restoration process. In the deblurring step, an efficient deblurring method using fast transforms can be employed. In the denoising step, effective methods such as the wavelet shrinkage denoising method or the total variation denoising method can be used. The main advantage of this proposal is that the resulting algorithms can be very efficient and can produce better restored images in visual quality and signal-to-noise ratio than those by the restoration methods using the combination of a data-fitting term and a regularization term. The convergence of the proposed algorithms is shown in the paper. Numerical examples are also given to demonstrate the effectiveness of these algorithms.

Bottom-Up Construction and 2:1 Balance Refinement of Linear Octrees in Parallel

Hari Sundar, Rahul S. Sampath, and George Biros

SIAM J. Sci. Comput. 30, pp. 2675-2708 (34 pages) | Cited 5 times

Online Publication Date: August 06, 2008

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In this article, we propose new parallel algorithms for the construction and 2:1 balance refinement of large linear octrees on distributed memory machines. Such octrees are used in many problems in computational science and engineering, e.g., object representation, image analysis, unstructured meshing, finite elements, adaptive mesh refinement, and N-body simulations. Fixed-size scalability and isogranular analysis of the algorithms using an MPI-based parallel implementation was performed on a variety of input data and demonstrated good scalability for different processor counts (1 to 1024 processors) on the Pittsburgh Supercomputing Center's TCS-1 AlphaServer. The results are consistent for different data distributions. Octrees with over a billion octants were constructed and balanced in less than a minute on 1024 processors. Like other existing algorithms for constructing and balancing octrees, our algorithms have $\mathcal{O}(N\log N)$ work and $\mathcal{O}(N)$ storage complexity. Under reasonable assumptions on the distribution of octants and the work per octant, the parallel time complexity is $\mathcal{O}(\frac{N}{n_p}\log(\frac{N}{n_p})+n_p\log n_p)$, where $N$ is the size of the final linear octree and $n_p$ is the number of processors.
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