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multigrid algorithms FOR hp-DISCONTINUOUS GALERKIN DISCRETIZATIONS OF ELLIPTIC PROBLEMS

SIAM JOURNAL ON NUMERICAL ANALYSIS 2015年第1期53卷 598-618页

作者： Antonietti, Paola F. Sarti, Marco Verani, Marco Politecn Milan Dipartimento Matemat MOX Lab Modeling & Sci Comp I-20133 Milan Italy

We present W-cycle h-, p-, and hp-multigrid algorithms for the solution of the linear system of equations arising from a wide class of hp-version discontinuous Galerkin discretizations of elliptic problems. Starting from a classical framework in geometric multigrid analysis, we define a smoothing and an approximation property, which are used to prove uniform convergence of the W-cycle scheme with respect to the discretization parameters and the number of levels, provided the number of smoothing steps is chosen of order p(2+mu), where p is the polynomial approximation degree and mu = 0, 1. A discussion on the effects of employing inherited or noninherited sublevel solvers is also presented. Numerical experiments confirm the theoretical results.

关键词： hp-version discontinuous Galerkin multigrid algorithms elliptic problems geometric multigrid

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multigrid algorithms ON THE HYPERCUBE MULTIPROCESSOR

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IEEE TRANSACTIONS ON COMPUTERS 1986年第11期35卷 969-977页

作者： CHAN, TF SAAD, Y Department of Computer Science Yale University

This paper examines several ways of implementing multigrid algorithms on the hypercube multiprocessor. We consider both the standard multigrid algorithms and a concurrent version proposed by Gannon and Van Rosendale. We present several mappings of the mesh points onto the nodes of the cube. The main property of these mappings, which are based on binary reflected Gray codes, is that the distance between neighboring grid points remains constant from one grid level to another. This results in a communication effective implementation of multigrid algorithms on the hypercube multiprocessor.

关键词： Gray codes hypercube multiprocessor multigrid algorithms parallel computing partial differential equations

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multigrid algorithms FOR THE SOLUTION OF LINEAR COMPLEMENTARITY-PROBLEMS ARISING FROM FREE-BOUNDARY PROBLEMS

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SIAM JOURNAL ON SCIENTIFIC AND STATISTICAL COMPUTING 1983年第4期4卷 655-684页

作者： BRANDT, A CRYER, CW UNIV WISCONSIN DEPT COMP SCIMADISONWI 53706 UNIV WISCONSIN MATH RES CTRMADISONWI 53706

Several free boundary problems (including saturated-unsaturated flow through porous dams, elastic-plastic torsion and cavitating journal bearings) can be formulated as linear complementarity problems of the following type: Find a nonnegative function u which satisfies prescribed boundary conditions on a given domain and which, furthermore, satisfies a linear elliptic equation at each point of the domain where u is greater than zero. We show that the multigrid FAS algorithm, which was developed by Brandt to solve boundary value problems for elliptic partial differential equations, can easily be adapted to handle linear complementarity problems. For large problems, the resulting algorithm, PFAS (projected full approximation scheme) is significantly faster than previous single-grid algorithms, since the computation time is proportional to the number of grid points on the finest grid.

关键词： 35J65 35R35 65N99 90C33 multigrid algorithms free boundary problems linear complementarity problems

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multigrid algorithms for optimization and inverse problems

Multigrid algorithms for optimization and inverse problems

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Conference on Computational Imaging

作者： Oh, S Milstein, AB Bouman, CA Webb, KJ Purdue Univ Sch Elect & Comp Engn W Lafayette IN 47907 USA

ISBN: (纸本)0819448168

A variety of new imaging modalities, such as optical diffusion tomography, require the inversion of a forward problem that is modeled by the solution to a three-dimensional partial differential equation. For these applications, image reconstruction can be formulated as the solution to a non-quadratic optimization problem. In this paper, we discuss the use of nonlinear multigrid methods as both tools for optimization and algorithms for the solution of difficult inverse problems. In particular, we review some existing methods for directly formulating optimization algorithm in a multigrid framework, and we introduce a new method for the solution of general inverse problems which we call multigrid inversion. These methods work by dynamically adjusting the cost functionals at different scales so that they axe consistent with, and ultimately reduce, the finest scale cost functional. In this way, the multigrid optimization methods can efficiently compute the solution to a desired fine scale optimization problem. Importantly, the multigrid inversion algorithm can greatly reduce computation because both the forward and inverse problems axe more coarsely discretized at lower resolutions. An application of our method to optical diffusion tomography shows the potential for very large computational savings.

关键词： multigrid algorithms inverse problems optimization optical diffusion tomography multiresolution

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APPLICATIONS OF multigrid algorithms TO FINITE DIFFERENCE SCHEMES FOR ELLIPTIC EQUATIONS WITH VARIABLE COEFFICIENTS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2014年第3期36卷 A1140-A1162页

作者： Jia, Rong-Qing Univ Alberta Dept Math & Stat Sci Edmonton AB T6G 2G1 Canada

This paper is devoted to a study of multigrid algorithms applied to finite difference schemes. If the elliptic equation has variable coefficients, the analysis of multigrid algorithms in the existent literature only gave a convergence rate depending on the number of levels. In this paper, for multigrid algorithms applied to finite difference schemes for elliptic equations with variable coefficients, we establish a convergence rate independent of the number of levels. Our convergence analysis does not require any additional regularity of the solution and is valid for commonly used smoothing operators including the standard Gauss-Seidel method. Under guidance of the general theory, we give details of implementation of the inherited multigrid V(1,1) algorithm. Furthermore, we will provide numerical examples to illustrate the general theory and demonstrate that the inherited multigrid algorithm is efficient for numerical solutions of elliptic equations with variable coefficients. In particular, we will consider elliptic equations on an L-shaped domain whose solutions do not have full regularity and show that the multigrid V(1,1) algorithm performs well in such situations.

关键词： multigrid algorithms finite difference schemes multilevel approximation elliptic equations

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On the Role of Ill-conditioning: Biharmonic Eigenvalue Problem and multigrid algorithms

On the Role of Ill-conditioning: Biharmonic Eigenvalue Probl...

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作者： Kasey Bray University of Kentucky

学位级别：博士

Very fine discretizations of differential operators often lead to large, sparse matrices A, where the condition number of A is large. Such ill-conditioning has well known effects on both solving linear systems and eigenvalue computations, and, in general, computing solutions with relative accuracy independent of the condition number is highly desirable. This dissertation is divided into two parts. In the first part, we discuss a method of preconditioning, developed by Ye, which allows solutions of Ax=b to be computed accurately. This, in turn, allows for accurate eigenvalue computations. We then use this method to develop discretizations that yield accurate computations of the smallest eigenvalue of the biharmonic operator across several domains. Numerical results from the various schemes are provided to demonstrate the performance of the methods. In the second part we address the role of the condition number of A in the context of multigrid algorithms. Under various assumptions, we use rigorous Fourier analysis on 2- and 3-grid iteration operators to analyze round off errors in floating point arithmetic. For better understanding of general results, we provide detailed bounds for a particular algorithm applied to the 1-dimensional Poisson equation. Numerical results are provided and compared with those obtained by the schemes discussed in part 1.

关键词： Accuracy biharmonic operator eigenvalue problem preconditioning multigrid algorithms rigorous Fourier analysis roundoff errors

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A refined convergence analysis of multigrid algorithms for elliptic equations

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ANALYSIS AND APPLICATIONS 2015年第3期13卷 255-290页

作者： Chang, Qianshun Jia, Rong-Qing Jiangsu Normal Univ Sch Math & Stat Xuzhou 221116 Peoples R China Chinese Acad Sci Acad Math & Syst Sci Beijing Peoples R China Univ Alberta Dept Math & Stat Sci Edmonton AB T6G 2G1 Canada

multigrid algorithms, in particular, multigrid V-cycles, are investigated in this paper. We establish a general theory for convergence of the multigrid algorithm under certain approximation conditions and smoothing conditions. Our smoothing conditions are satisfied by commonly used smoothing operators including the standard Gauss-Seidel method. Our approximation conditions are verified for finite element approximation to numerical solutions of elliptic partial differential equations without any requirement of additional regularity of the solution. Our convergence analysis of multigrid algorithms can be applied to a wide range of problems. Numerical examples are also provided to illustrate the general theory.

关键词： multigrid algorithms elliptic equations multilevel approximation approximation conditions smoothing conditions finite element approximation

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Convergence of multigrid algorithms for Interior Penalty Methods

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Applied Numerical Analysis & Computational Mathematics 2005年第1期2卷

作者： Susanne C. Brenner Jie Zhao Department of Mathematics University of South Carolina Columbia SC 29208 USA Current Address: Washington Mutual Bank 1301 5th Ave. RBB 1919 Seattle WA 98101 USA

V -cycle, F -cycle and W -cycle multigrid algorithms for interior penalty methods for second order elliptic boundary value problems are studied in this paper. It is shown that these algorithms converge uniformly with respect to all grid levels if the number of smoothing steps is sufficiently large, and that the contraction numbers decrease as the number of smoothing steps increases, at a rate determined by the elliptic regularity of the problem. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

关键词： multigrid algorithms interior penalty methods second order problems

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Automatic coarsening of three dimensional anisotropic unstructured meshes for multigrid applications

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APPLIED MATHEMATICS AND COMPUTATION 2012年第21期218卷 10500-10519页

作者： Mesri, Youssef Guillard, Herve Coupez, Thierry IFP Energies Nouvelles F-92852 Rueil Malmaison France Inria F-06902 Sophia Antipolis France Univ Nice Sophia Antipolis Lab Jean Akexandre Dieudonne F-06108 Nice France CNRS UMR 7635 Ecole Mines Paris CEMEF F-06904 Sophia Antipolis France

This paper describes an algorithm designed for the automatic coarsening of three-dimensional unstructured simplicial meshes. This algorithm can handle very anisotropic meshes like the ones typically used to capture the boundary layers in CFD with Low Reynolds turbulence modeling that can have aspect ratio as high as 10(4). It is based on the concept of mesh generation governed by metrics and on the use of a natural metric mapping the initial (fine) mesh into an equilateral one. The paper discusses and compares several ways to define node based metric from element based metric. Then the semi-coarsening algorithm is described. Several application examples are presented, including a full three-dimensional complex model of an aircraft with extremely high anisotropy. (C) 2012 Elsevier Inc. All rights reserved.

关键词： Mesh generation Tetrahedral meshes Coarsening Anisotropy multigrid algorithms CFD Boundary layers

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Reliable performance prediction for multigrid software on distributed memory systems

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ADVANCES IN ENGINEERING SOFTWARE 2011年第5期42卷 247-258页

作者： Romanazzi, Giuseppe Jimack, Peter K. Goodyer, Christopher E. Univ Coimbra Dept Matemat CMUC P-3001454 Coimbra Portugal Univ Leeds Sch Comp Leeds LS2 9JT W Yorkshire England

We propose a model for describing and predicting the parallel performance of a broad class of parallel numerical software on distributed memory architectures. The purpose of this model is to allow reliable predictions to be made for the performance of the software on large numbers of processors of a given parallel system, by only benchmarking the code on small numbers of processors. Having described the methods used, and emphasized the simplicity of their implementation, the approach is tested on a range of engineering software applications that are built upon the use of multigrid algorithms. Despite their simplicity, the models are demonstrated to provide both accurate and robust predictions across a range of different parallel architectures, partitioning strategies and multigrid codes. In particular, the effectiveness of the predictive methodology is shown for a practical engineering software implementation of an elastohydrodynamic lubrication solver. (C) 2010 Civil-Comp Ltd and Elsevier Ltd. All rights reserved.

关键词： Performance prediction Parallel engineering software multigrid algorithms Partial differential equations Distributed memory architectures Parallel distributed algorithms

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