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47
Algebraic Multigrid Based On Element Interpolation (AMGe)
, 1998
"... We introduce AMGe, an algebraic multigrid method for solving the discrete equations that arise in Ritztype finite element methods for partial differential equations. Assuming access to the element stiffness matrices, AMGe is based on the use of two local measures, which are derived from global meas ..."
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Cited by 104 (16 self)
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We introduce AMGe, an algebraic multigrid method for solving the discrete equations that arise in Ritztype finite element methods for partial differential equations. Assuming access to the element stiffness matrices, AMGe is based on the use of two local measures, which are derived from global measures that appear in existing multigrid theory. These new measures are used to determine local representations of algebraically "smooth" error components that provide the basis for constructing effective interpolation and, hence, the coarsening process for AMG. Here, we focus on the interpolation process; choice of the coarse "grids" based on these measures is the subject of current research. We develop a theoretical foundation for AMGe and present numerical results that demonstrate the efficacy of the method.
General Highly Accurate Algebraic Coarsening
 Electronic Trans. Num. Anal
, 2000
"... General purely algebraic approaches for repeated coarsening of deterministic or statistical field equations are presented, including a universal way to gauge and control the quality of the coarselevel set of variables, and generic procedures for deriving the coarselevel set of equations. They appl ..."
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Cited by 72 (8 self)
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General purely algebraic approaches for repeated coarsening of deterministic or statistical field equations are presented, including a universal way to gauge and control the quality of the coarselevel set of variables, and generic procedures for deriving the coarselevel set of equations. They apply to the equations arising from variational as well as nonvariational discretizations of general, elliptic as well as nonelliptic, partial differential systems, on structured or unstructured grids. They apply to many types of disordered systems, such as those arising in composite materials, inhomogeneous ground flows, "twisted geometry" discretizations and Dirac equations in disordered gauge fields, and also to nonPDE systems. The coarsening can be inexpensive with low accuracy, as needed for multigrid solvers, or more expensive and highly accurate, as needed for other applications (e.g., onceforall derivation of macroscopic equations). Extensions to nonlocal and highly indefinite (w...
Domain decomposition for multiscale PDEs
 Numer. Math
"... We consider additive Schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic PDEs with highly variable coefficients. In contrast to standard analyses, we do not assume that the coefficients can be resolved by a coarse mesh. This situation arises of ..."
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Cited by 48 (16 self)
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We consider additive Schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic PDEs with highly variable coefficients. In contrast to standard analyses, we do not assume that the coefficients can be resolved by a coarse mesh. This situation arises often in practice, for example in the computation of flows in heterogeneous porous media, in both the deterministic and (MonteCarlo simulated) stochastic cases. We consider preconditioners which combine local solves on general overlapping subdomains together with a global solve on a general coarse space of functions on a coarse grid. We perform a new analysis of the preconditioned matrix, which shows rather explicitly how its condition number depends on the variable coefficient in the PDE as well as on the coarse mesh and overlap parameters. The classical estimates for this preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse grid elements. By contrast, our new results show that, with a good choice of subdomains and coarse space basis functions, the preconditioner can still be robust even for large coefficient variation inside domains,
Multilevel Solvers For Unstructured Surface Meshes
 SIAM J. Sci. Comput
"... Parameterization of unstructured surface meshes is of fundamental importance in many applications of Digital Geometry Processing. Such parameterization approaches give rise to large and exceedingly illconditioned systems which are difficult or impossible to solve without the use of sophisticated mu ..."
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Cited by 38 (3 self)
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Parameterization of unstructured surface meshes is of fundamental importance in many applications of Digital Geometry Processing. Such parameterization approaches give rise to large and exceedingly illconditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner.
Sparse Approximate Inverse Smoother for Multigrid
 SIAM J. Matrix Anal. Appl
, 1999
"... Various forms of sparse approximate inverses (SAI) have been shown to be useful for preconditioning. Their potential usefulness in a parallel environment has motivated much interest in recent years. However, the capability of an approximate inverse in eliminating the local error has not yet been ful ..."
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Cited by 32 (2 self)
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Various forms of sparse approximate inverses (SAI) have been shown to be useful for preconditioning. Their potential usefulness in a parallel environment has motivated much interest in recent years. However, the capability of an approximate inverse in eliminating the local error has not yet been fully exploited in multigrid algorithms. A careful examination of the iteration matrices of these approximate inverses indicates their superiority in smoothing the high frequency error in addition to their inherent parallelism. We propose a new class of sparse approximate inverse smoothers in this paper and present their analytic smoothing factors for constant coecient PDEs. Several distinctive features that make this technique special are: By adjusting the quality of the approximate inverse, the smoothing factor can be improved accordingly. For hard problems, this is useful.
Energy Optimization of Algebraic Multigrid Bases
, 1998
"... . We propose a fast iterative method to optimize coarse basis functions in algebraic multigrid by minimizing the sum of their energies, subject to the condition that linear combinations of the basis functions equal to given zero energy modes, and subject to restrictions on the supports of the coarse ..."
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Cited by 30 (2 self)
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. We propose a fast iterative method to optimize coarse basis functions in algebraic multigrid by minimizing the sum of their energies, subject to the condition that linear combinations of the basis functions equal to given zero energy modes, and subject to restrictions on the supports of the coarse basis functions. The convergence rate of the minimization algorithm is bounded independently of the meshsize under usual assumptions on finite elements. The first iteration gives exactly the same basis functions as our earlier method using smoothed aggregation. The construction is presented for scalar problems as well as for linear elasticity. Computational results on difficult industrial problems demonstrate that the use of energy minimal basis functions improves algebraic multigrid performance and yields a more robust multigrid algorithm than smoothed aggregation. 1. Introduction. This paper is concerned with aspects of the design of Algebraic Multigrid Methods (AMG) for the solution of ...
Metric based upscaling
 Communications on Pure and Applied Mathematics
, 2007
"... We consider divergence form elliptic operators in dimension n ≥ 2. Although solutions of these operators are only Hölder continuous, we show that they are differentiable (C 1,α) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the med ..."
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Cited by 23 (2 self)
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We consider divergence form elliptic operators in dimension n ≥ 2. Although solutions of these operators are only Hölder continuous, we show that they are differentiable (C 1,α) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales by transferring a new metric in addition to traditional averaged (homogenized) quantities from subgrid scales into computational scales and error bounds can be given. This numerical homogenization method can also be used as a compression tool for differential operators. 1 Introduction and main results Let Ω be a bounded and convex domain of class C2. We consider the following benchmark PDE
Robust Domain Decomposition Algorithms for Multiscale PDEs
, 2006
"... In this paper we describe a new class of domain deomposition preconditioners suitable for solving elliptic PDEs in highly fractured or heterogeneous media, such as arise in groundwater flow or oil recovery applications. Our methods employ novel coarsening operators which are adapted to the heterogen ..."
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Cited by 22 (10 self)
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In this paper we describe a new class of domain deomposition preconditioners suitable for solving elliptic PDEs in highly fractured or heterogeneous media, such as arise in groundwater flow or oil recovery applications. Our methods employ novel coarsening operators which are adapted to the heterogeneity of the media. In contrast to standard methods (based on piecewise polynomial coarsening), the new methods can achieve robustness with respect to coefficient discontinuities even when these are not resolved by a coarse mesh. This situation arises often in practical flow computation, in both the deterministic and (MonteCarlo simulated) stochastic cases. An example of a suitable coarsener is provided by multiscale finite elements. In this paper we explore the linear algebraic aspects of the multiscale algorithm, showing that it involves a blend of both classical overlapping Schwarz methods and nonoverlapping Schur methods. We also extend the algorithm and the theory from its additive variant to obtain new hybrid and deflation variants. Finally we give extensive numerical experiments on a range of heterogeneous media problems illustrating the properties of the methods.
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 14 (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...
Refining an approximate inverse
 J. Comput. Appl. Math
, 1999
"... Direct methods have made remarkable progress in the computational efficiency of factorization algorithms during the last three decades. The advances in graph theoretic algorithms have not received enough attention from the iterative methods community. For example, we demonstrate how symbolic factori ..."
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Cited by 14 (1 self)
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Direct methods have made remarkable progress in the computational efficiency of factorization algorithms during the last three decades. The advances in graph theoretic algorithms have not received enough attention from the iterative methods community. For example, we demonstrate how symbolic factorization algorithms from direct methods can accelerate the computation of a factored approximate inverse preconditioner. For very sparse preconditioners, however, a reformulation of the algorithm with outer products can exploit even more zeros to good advantage. We also explore the possibilities of improving cache efficiency in the application of the preconditioner through reorderings. The article finishes by proposing a block version of the algorithm for further gains in efficiency and robustness. Key words. factored approximate inverse, implementation, symbolic factorization, ordering, block algorithms. 1