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22
An Algorithm for Coarsening Unstructured Meshes
 Numer. Math
, 1996
"... . We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the lin ..."
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Cited by 47 (5 self)
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. We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the linear systems arising from finite element discretizations of elliptic partial differential equations. We show that such iterative methods perform as well as those developed for the usual case of structured, locally refined meshes. In particular, we show that the generalized condition numbers for such iterative methods are of order J 2 , where J is the number of hierarchical basis levels. Key words. Finite element, hierarchical basis, multigrid, unstructured mesh. AMS subject classifications. 65F10, 65N20 1. Introduction. Iterative methods using the hierarchical basis decomposition have proved to be among the most robust for solving broad classes of elliptic partial differential equations, ...
The Adaptive Multilevel Finite Element Solution of the PoissonBoltzmann Equation on Massively Parallel Computers
 J. COMPUT. CHEM
, 2000
"... Using new methods for the parallel solution of elliptic partial differential equations, the teraflops computing power of massively parallel computers can be leveraged to perform electrostatic calculations on large biological systems. This paper describes the adaptive multilevel finite element soluti ..."
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Cited by 43 (14 self)
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Using new methods for the parallel solution of elliptic partial differential equations, the teraflops computing power of massively parallel computers can be leveraged to perform electrostatic calculations on large biological systems. This paper describes the adaptive multilevel finite element solution of the PoissonBoltzmann equation for a microtubule on the NPACI IBM Blue Horizon supercomputer. The microtubule system is 40 nm in length and 24 nm in diameter, consists of roughly 600,000 atoms, and has a net charge of1800 e. PoissonBoltzmann calculations are performed for several processor configurations and the algorithm shows excellent parallel scaling.
The Incomplete Factorization Multigraph Algorithm
 SIAM J. SCI. COMPUT
"... We present a new family of multigraph algorithms, ILUMG, based upon an incomplete sparse matrix factorization using a particular ordering and allowing a limited amount of fillin. While much of the motivation for multigraph comes from multigrid ideas, ILUMG is distinctly different from algebraic ..."
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Cited by 23 (4 self)
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We present a new family of multigraph algorithms, ILUMG, based upon an incomplete sparse matrix factorization using a particular ordering and allowing a limited amount of fillin. While much of the motivation for multigraph comes from multigrid ideas, ILUMG is distinctly different from algebraic multilevel methods. The graph of the sparse matrix A is recursively coarsened by eliminating vertices using a graph model similar to Gaussian elimination. Incomplete factorizations are obtained by allowing only the fillin generated by the vertex parents associated with each vertex. Multigraph is numerically compared with algebraic multigrid on some examples arising from discretizations of partial differential equations on unstructured grids.
Multilevel ILU Decomposition
, 1997
"... . In this paper, the multilevel ILU (MLILU) decomposition is introduced. During an incomplete Gaussian elimination process new matrix entries are generated such that a special ordering strategy yields distinct levels. On these levels, some smoothing steps are computed. The MLILU decomposition exists ..."
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Cited by 21 (2 self)
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. In this paper, the multilevel ILU (MLILU) decomposition is introduced. During an incomplete Gaussian elimination process new matrix entries are generated such that a special ordering strategy yields distinct levels. On these levels, some smoothing steps are computed. The MLILU decomposition exists and the corresponding iterative scheme converges for all symmetric and positive definite matrices. Convergence rates independent of the number of unknowns are shown numerically for several examples. Many numerical experiments including unsymmetric and anisotropic problems, problems with jumping coefficients as well as realistic problems are presented. They indicate a very robust convergence behavior of the MLILU method. Key words. Algebraic multigrid, filter condition, ILU decomposition, iterative method, partial differential equation, robustness, test vector. AMS subject classifications. 65F10, 65N55. 1 Introduction In this paper, we consider iterative algorithms u (i+1) = u (i) +M...
Diagonal Threshold Techniques in Robust MultiLevel ILU Preconditioners for General Sparse Linear Systems
 NUMER. LINEAR ALGEBRA APPL
, 1998
"... This paper introduces techniques based on diagonal threshold tolerance when developing multielimination and multilevel incomplete LU (ILUM) factorization preconditioners for solving general sparse linear systems. Existing heuristics solely based on the adjacency graph of the matrices have been ..."
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Cited by 14 (10 self)
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This paper introduces techniques based on diagonal threshold tolerance when developing multielimination and multilevel incomplete LU (ILUM) factorization preconditioners for solving general sparse linear systems. Existing heuristics solely based on the adjacency graph of the matrices have been used to find independent sets and are not robust for matrices arising from certain applications in which the matrices may have small or zero diagonals. New heuristic strategies based on the adjacency graph and the diagonal values of the matrices for finding independent sets are introduced. Analytical bounds for the factorization and preconditioned errors are obtained for the case of a twolevel analysis. These bounds provide useful information in designing robust ILUM preconditioners. Extensive numerical experiments are conducted in order to compare robustness and efficiency of various heuristic strategies.
An Algebraic Multilevel Multigraph Algorithm
 SIAM J. on Scientific Computing
"... . We describe an algebraic multilevel multigraph algorithm. Many of the multilevel components are generalizations of algorithms originally applied to general sparse Gaussian elimination. Indeed, general sparse Gaussian elimination with minimum degree ordering is a limiting case of our algorithm. Our ..."
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Cited by 12 (1 self)
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. We describe an algebraic multilevel multigraph algorithm. Many of the multilevel components are generalizations of algorithms originally applied to general sparse Gaussian elimination. Indeed, general sparse Gaussian elimination with minimum degree ordering is a limiting case of our algorithm. Our goal is to develop a procedure which has the robustness and simplicity of use of sparse direct methods, yet oers the opportunity to obtain the optimal or nearoptimal complexity typical of classical multigrid methods. Key words. algebraic multigrid, incomplete LU factorization, multigraph methods. AMS subject classications. 65M55, 65N55 1. Introduction. In this work, we develop a multilevelmultigraph algorithm. Algebraic multigrid methods are currently a topic of intense research interest [17, 18, 20, 46, 12, 48, 38, 11, 44, 3, 4, 1, 2, 5, 16, 7, 29, 28, 27, 42, 41, 21]. An excellent recent survey is given in Wagner [49]. In many \real world" calculations, direct methods are still wid...
Parallel Adaptive Subspace Correction Schemes with Applications to Elasticity
 Comput. Methods Appl. Mech. Engrg
, 1999
"... : In this paper, we give a survey on the three main aspects of the efficient treatment of PDEs, i.e. adaptive discretization, multilevel solution and parallelization. We emphasize the abstract approach of subspace correction schemes and summarize its convergence theory. Then, we give the main featur ..."
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Cited by 8 (4 self)
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: In this paper, we give a survey on the three main aspects of the efficient treatment of PDEs, i.e. adaptive discretization, multilevel solution and parallelization. We emphasize the abstract approach of subspace correction schemes and summarize its convergence theory. Then, we give the main features of each of the three distinct topics and treat the historical background and modern developments. Furthermore, we demonstrate how all three ingredients can be put together to give an adaptive and parallel multilevel approach for the solution of elliptic PDEs and especially of linear elasticity problems. We report on numerical experiments for the adaptive parallel multilevel solution of some test problems, namely the Poisson equation and Lam'e's equation. Here, we emphasize the parallel efficiency of the adaptive code even for simple test problems with little work to distribute, which is achieved through hash storage techniques and spacefilling curves. Keywords: subspace correction, iter...
Hierarchical Basis for the ConvectionDiffusion Equation on Unstructured Meshes
 in Ninth International Symposium on Domain Decomposition Methods for Partial Differential Equations
, 1997
"... Introduction The Hierarchical Basis Multigrid Method was originally developed for sequences of refined meshes. Hierarchical basis functions can be constructed in a straightforward fashion on such sequences of nested meshes. The HBMG iteration itself is just a block symmetric GauSeidel iteration ap ..."
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Cited by 6 (2 self)
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Introduction The Hierarchical Basis Multigrid Method was originally developed for sequences of refined meshes. Hierarchical basis functions can be constructed in a straightforward fashion on such sequences of nested meshes. The HBMG iteration itself is just a block symmetric GauSeidel iteration applied to the stiffness matrix represented in the hierarchical basis. Because the stiffness matrix is less sparse than when the standard nodal basis functions are used, the iteration is carried out by forming the hierarchical basis stiffness matrix only implicitly. The resulting algorithm is strongly connected to the classical multigrid Vcycle, except that only a subset of the unknowns on each level is smoothed during the relaxation steps [BDY88]. In recent years, we have generalized such bases to completely unstructured meshes, not just those arising from some refinement process. This is done by recognizing the strong connection between the Hierarchical Basis Multigrid Method and an
Variable Preconditioning Procedures for Elliptic Problems
, 1996
"... For solving systems of grid equations approximating elliptic boundary value problems a method of constructing variable preconditioning procedures is presented. The main purpose is to discuss how an efficient preconditioning iterative procedure can be constructed in the case of elliptic problems with ..."
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Cited by 6 (0 self)
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For solving systems of grid equations approximating elliptic boundary value problems a method of constructing variable preconditioning procedures is presented. The main purpose is to discuss how an efficient preconditioning iterative procedure can be constructed in the case of elliptic problems with disproportional coefficients, e.g. equations with a large coefficient in the reaction term (or a small diffusion coefficient) . The optimality of the suggested technique is based on fictitious space and multilevel decomposition methods. Using an additive form of the preconditioners, we introduce factors into the preconditioners to optimize the corresponding convergence rate. The optimization with respect to these factors is used at each step of the iterative process. The application of this technique to twolevel phierarchical preconditioners and domain decomposition methods is considered too. Key words: Iterative methods, preconditioning operators, conjugate gradient methods, additive Sc...
Multilevel block factorizations in generalized hierarchical bases
 NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
, 2002
"... This paper studies the use of a generalized hierarchical basis transformation at each level of a multilevel block factorization. The factorization may be used as a preconditioner to the conjugate gradient method, or the structure it sets up may be used to define a multigrid method. The basis transfo ..."
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Cited by 6 (0 self)
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This paper studies the use of a generalized hierarchical basis transformation at each level of a multilevel block factorization. The factorization may be used as a preconditioner to the conjugate gradient method, or the structure it sets up may be used to define a multigrid method. The basis transformation is performed with an averaged piecewise constant interpolant and is applicable to unstructured elliptic problems. The results show greatly improved convergence rate when the transformation is applied for solving sample diffusion and elasticity problems. The cost of the method, however, grows and can get very high with the number of of nonzeros per row