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Algebraic Systems (algebraic + system)
Selected AbstractsOptimizing process allocation of parallel programs for heterogeneous clustersCONCURRENCY AND COMPUTATION: PRACTICE & EXPERIENCE, Issue 4 2009Shuichi Ichikawa Abstract The performance of a conventional parallel application is often degraded by load-imbalance on heterogeneous clusters. Although it is simple to invoke multiple processes on fast processing elements to alleviate load-imbalance, the optimal process allocation is not obvious. Kishimoto and Ichikawa presented performance models for high-performance Linpack (HPL), with which the sub-optimal configurations of heterogeneous clusters were actually estimated. Their results on HPL are encouraging, whereas their approach is not yet verified with other applications. This study presents some enhancements of Kishimoto's scheme, which are evaluated with four typical scientific applications: computational fluid dynamics (CFD), finite-element method (FEM), HPL (linear algebraic system), and fast Fourier transform (FFT). According to our experiments, our new models (NP-T models) are superior to Kishimoto's models, particularly when the non-negative least squares method is used for parameter extraction. The average errors of the derived models were 0.2% for the CFD benchmark, 2% for the FEM benchmark, 1% for HPL, and 28% for the FFT benchmark. This study also emphasizes the importance of predictability in clusters, listing practical examples derived from our study. Copyright © 2008 John Wiley & Sons, Ltd. [source] Application of the additive Schwarz method to large scale Poisson problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2004K. M. Singh Abstract This paper presents an application of the additive Schwarz method to large scale Poisson problems on parallel computers. Domain decomposition in rectangular blocks with matching grids on a structured rectangular mesh has been used together with a stepwise approximation to approximate sloping sides and complicated geometric features. A seven-point stencil based on central difference scheme has been used for the discretization of the Laplacian for both interior and boundary grid points, and this results in a symmetric linear algebraic system for any type of boundary conditions. The preconditioned conjugate gradient method has been used as an accelerator for the additive Schwarz method, and three different methods have been assessed for the solution of subdomain problems. Numerical experiments have been performed to determine the most suitable set of subdomain solvers and the optimal accuracy of subdomain solutions; to assess the effect of different decompositions of the problem domain; and to evaluate the parallel performance of the additive Schwarz preconditioner. Application to a practical problem involving complicated geometry is presented which establishes the efficiency and robustness of the method. Copyright © 2004 John Wiley & Sons, Ltd. [source] Non-linear additive Schwarz preconditioners and application in computational fluid dynamicsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2002Xiao-Chuan Cai Abstract The focus of this paper is on the numerical solution of large sparse non-linear systems of algebraic equations on parallel computers. Such non-linear systems often arise from the discretization of non-linear partial differential equations, such as the Navier,Stokes equations for fluid flows, using finite element or finite difference methods. A traditional inexact Newton method, applied directly to the discretized system, does not work well when the non-linearities in the algebraic system become unbalanced. In this paper, we study some preconditioned inexact Newton algorithms, including the single-level and multilevel non-linear additive Schwarz preconditioners. Some results for solving the high Reynolds number incompressible Navier,Stokes equations are reported. Copyright © 2002 John Wiley & Sons, Ltd. [source] Null-field approach for Laplace problems with circular boundaries using degenerate kernelsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2009Jeng-Tzong Chen Abstract In this article, a semi-analytical method for solving the Laplace problems with circular boundaries using the null-field integral equation is proposed. The main gain of using the degenerate kernels is to avoid calculating the principal values. To fully utilize the geometry of circular boundary, degenerate kernels for the fundamental solution and Fourier series for boundary densities are incorporated into the null-field integral equation. An adaptive observer system is considered to fully employ the property of degenerate kernels in the polar coordinates. A linear algebraic system is obtained without boundary discretization. By matching the boundary condition, the unknown coefficients can be determined. The present method can be seen as one kind of semianalytical approaches since error only attributes to the truncated Fourier series. For the eccentric case, vector decomposition technique for the normal and tangential directions is carefully considered in implementing the hypersingular equation in mathematical essence although we transform it to summability to divergent series. The five advantages, well-posed linear algebraic system, principal value free, elimination of boundary-layer effect, exponential convergence, and mesh free, are achieved. Several examples involving infinite, half-plane, and bounded domains with circular boundaries are given to demonstrate the validity of the proposed method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source] New Zonal, Spectral Solutions for the Navier-Stokes Layer and Their ApplicationsPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003A. Nastase Prof. Dr.-Ing., Dr. Math. New zonal, spectral forms for the axial, lateral and vertical velocity's components, density function and absolute temperature inside of compressible three-dimensional Navier-Stokes layer (NSL) over flattened, flying configurations (FC), are here proposed. The inviscid flow over the FC, obtained after the solidification of the NSL, is here used as outer flow. If the spectral forms of the velocity's components are introduced in the partial differential equations of the NSL and the collocation method is used, the spectral coefficients are obtained by the iterative solving of an equivalent quadratical algebraic system with slightly variable coefficients. [source] Scalable Numerical Tools for Flow and Pressure Drop Computation in Fibrous Filter MediaCHEMICAL ENGINEERING & TECHNOLOGY (CET), Issue 5 2009F. Strauß Abstract The prediction of flow behavior and pressure drop in fibrous filter media is challenging due to the complexity of the nonuniform fiber structure. Numerical calculation tools can considerably contribute to pressure drop determination for inhomogeneous filter structures. A numerical solution approach based on the finite element method to simulate 2D and 3D filter structures is considered. As numerical examples, computer designed homogeneous and inhomogeneous 2D cases where the numerical approach is validated by analytical models are investigated. Furthermore, the capability of the numerical method to simulate real 3D structures corresponding to more than 25 million degrees of freedom of the related algebraic system is demonstrated. The large systems involved require the use of dedicated techniques related to high performance computing. [source] Adaptive preconditioning of linear stochastic algebraic systems of equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 11 2007Y. T. Feng Abstract This paper proposes an adaptively preconditioned iterative method for the solution of large-scale linear stochastic algebraic systems of equations with one random variable that arise from the stochastic finite element modelling of linear elastic problems. Firstly, a Rank-one posteriori preconditioner is introduced for a general linear system of equations. This concept is then developed into an effective adaptive preconditioning scheme for the iterative solution of the stochastic equations in the context of a modified Monte Carlo simulation approach. To limit the maximum number of base vectors used in the scheme, a simple selection criterion is proposed to update the base vectors. Finally, numerical experiments are conducted to assess the performance of the proposed adaptive preconditioning strategy, which indicates that the scheme with very few base vectors can improve the convergence of the standard Incomplete Cholesky preconditioning up to 50%. Copyright © 2006 John Wiley & Sons, Ltd. [source] A relation between the logarithmic capacity and the condition number of the BEM-matricesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2007W. Dijkstra Abstract We establish a relation between the logarithmic capacity of a two-dimensional domain and the solvability of the boundary integral equation for the Laplace problem on that domain. It is proved that when the logarithmic capacity is equal to one the boundary integral equation does not have a unique solution. A similar result is derived for the linear algebraic systems that appear in the boundary element method. As these systems are based on the boundary integral equation, no unique solution exists when the logarithmic capacity is equal to one. Hence, the system matrix is ill-conditioned. We give several examples to illustrate this and investigate the analogies between the Laplace problem with Dirichlet and mixed boundary conditions. Copyright © 2006 John Wiley & Sons, Ltd. [source] Quadratic form of stable sub-manifold for power systemsINTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 9-10 2004Daizhan Cheng Abstract The stable sub-manifold of type-1 unstable equilibrium point is fundamental in determining the region of attraction of a stable working point for power systems, because such sub-manifolds form the boundary of the region (IEEE Trans. Automat. Control 1998; 33(1):16,27; IEEE Trans. Circuit Syst. 1988; 35(6):712,728). The quadratic approximation has been investigated in some recent literatures (Automatica 1997; 33(10):1877,1883; IEEE Trans. Power Syst. 1997; 12(2):797,802). First, the paper reports our recent result: a precise formula is obtained, which provides the unique quadratic approximation with the error of 0(,,x,,3). Then the result is applied to differential,algebraic systems. The real form of practical large scale power systems are of this type. A detailed algorithm is obtained for the quadratic approximation of the stable sub-manifold of type-1 unstable equilibrium points of such systems. Some examples are presented to illustrate the algorithm and the application of the approximation to stability analysis of power systems. Copyright © 2004 John Wiley & Sons, Ltd. [source] Categorical abstract algebraic logic: The Diagram and the Reduction Operator LemmasMLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 2 2007George Voutsadakis Abstract The study of structure systems, an abstraction of the concept of first-order structures, is continued. Structure systems have algebraic systems as their algebraic reducts and their relational component consists of a collection of relation systems on the underlying functors. An analog of the expansion of a first-order structure by constants is presented. Furthermore, analogs of the Diagram Lemma and the Reduction Operator Lemma from the theory of equality-free first-order structures are provided in the framework of structure systems. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] A new multilevel algebraic preconditioner for the diffusion equation in heterogeneous mediaNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2010Yu Kuznetsov Abstract We develop and analyze a new multilevel preconditioner for algebraic systems arising from the finite volume discretization of 3D diffusion,reaction problems in highly heterogeneous media. The system matrices are assumed to be symmetric M -matrices. The preconditioner is based on a special coarsening algorithm and the inner Chebyshev iterative procedure. The condition number of the preconditioned matrix does not depend on the coefficients in the diffusion operator. Numerical experiments confirm theoretical results and reveal the competitiveness of the new preconditioner with respect to a well-known algebraic multigrid preconditioner. Copyright © 2010 John Wiley & Sons, Ltd. [source] Projected Schur complement method for solving non-symmetric systems arising from a smooth fictitious domain approachNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 9 2007J. Haslinger Abstract This paper deals with a fast method for solving large-scale algebraic saddle-point systems arising from fictitious domain formulations of elliptic boundary value problems. A new variant of the fictitious domain approach is analyzed. Boundary conditions are enforced by control variables introduced on an auxiliary boundary located outside the original domain. This approach has a significantly higher convergence rate; however, the algebraic systems resulting from finite element discretizations are typically non-symmetric. The presented method is based on the Schur complement reduction. If the stiffness matrix is singular, the reduced system can be formulated again as another saddle-point problem. Its modification by orthogonal projectors leads to an equation that can be efficiently solved using a projected Krylov subspace method for non-symmetric operators. For this purpose, the projected variant of the BiCGSTAB algorithm is derived from the non-projected one. The behavior of the method is illustrated by examples, in which the BiCGSTAB iterations are accelerated by a multigrid strategy. Copyright © 2007 John Wiley & Sons, Ltd. [source] Analysis of algebraic systems arising from fourth-order compact discretizations of convection-diffusion equationsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2002Ashvin Gopaul Abstract We study the properties of coefficient matrices arising from high-order compact discretizations of convection-diffusion problems. Asymptotic convergence factors of the convex hull of the spectrum and the field of values of the coefficient matrix for a one-dimensional problem are derived, and the convergence factor of the convex hull of the spectrum is shown to be inadequate for predicting the convergence rate of GMRES. For a two-dimensional constant-coefficient problem, we derive the eigenvalues of the nine-point matrix, and we show that the matrix is positive definite for all values of the cell-Reynolds number. Using a recent technique for deriving analytic expressions for discrete solutions produced by the fourth-order scheme, we show by analyzing the terms in the discrete solutions that they are oscillation-free for all values of the cell Reynolds number. Our theoretical results support observations made through numerical experiments by other researchers on the non-oscillatory nature of the discrete solution produced by fourth-order compact approximations to the convection-diffusion equation. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 155,178, 2002; DOI 10.1002/num.1041 [source] An adaptive displacement/pressure finite element scheme for treating incompressibility effects in elasto-plastic materialsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2001Franz, Theo SuttmeierArticle first published online: 13 AUG 200 Abstract In this article, a mixed finite element formulation is described for coping with (nearly) incompressible behavior in elasto-plastic problems. In addition to the displacements, an auxiliary variable, playing the role of a pressure, is introduced resulting in Stokes-like problems. The discretization is done by a stabilized conforming Q1/Q1 -element, and the corresponding algebraic systems are solved by an adaptive multigrid scheme using a smoother of block Gauss,Seidel type. The adaptive algorithm is based on the general concept of using duality arguments to obtain weighted a posteriori error bounds. This procedure is carried out here for the described discretization of elasto-plastic problems. Efficiency and reliability of the proposed adaptive method is demonstrated at (plane strain) model problems. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:369,382, 2001 [source] | ||||||||||||||||