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338
Preconditioning techniques for large linear systems: A survey
 J. COMPUT. PHYS
, 2002
"... This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization i ..."
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Cited by 189 (5 self)
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This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper.
BoomerAMG: a Parallel Algebraic Multigrid Solver and Preconditioner
 Applied Numerical Mathematics
, 2000
"... Driven by the need to solve linear sytems arising from problems posed on extremely large, unstructured grids, there has been a recent resurgence of interest in algebraic multigrid (AMG). AMG is attractive in that it holds out the possibility of multigridlike performance on unstructured grids. The sh ..."
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Cited by 124 (9 self)
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Driven by the need to solve linear sytems arising from problems posed on extremely large, unstructured grids, there has been a recent resurgence of interest in algebraic multigrid (AMG). AMG is attractive in that it holds out the possibility of multigridlike performance on unstructured grids. The sheer size of many modern physics and simulation problems has led to the development of massively parallel computers, and has sparked much research into developing algorithms for them. Parallelizing AMG is a difficult task, however. While much of the AMG method parallelizes readily, the process of coarsegrid selection, in particular, is fundamentally sequential in nature. We have previously introduced a parallel algorithm [7] for the selection of coarsegrid points, based on modifications of certain parallel independent set algorithms and the application of heuristics designed to insure the quality of the coarse grids, and shown results from a prototype serial version of the algorithm. In this pa...
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 102 (15 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.
hypre: a Library of High Performance Preconditioners
 Preconditioners,” Lecture Notes in Computer Science
, 2002
"... hypre is a software library for the solution of large, sparse linear systems on massively parallel computers. Its emphasis is on modern powerful and scalable preconditioners. hypre provides various conceptual interfaces to enable application users to access the library in the way they naturally ..."
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Cited by 84 (4 self)
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hypre is a software library for the solution of large, sparse linear systems on massively parallel computers. Its emphasis is on modern powerful and scalable preconditioners. hypre provides various conceptual interfaces to enable application users to access the library in the way they naturally think about their problems. This paper presents the conceptual interfaces in hypre. An overview of the preconditioners that are available in hypre is given, including some numerical results that show the eciency of the library.
Texture segmentation by multiscale aggregation of filter responses and shape elements
 IN ICCV
, 2003
"... Texture segmentation is a difficult problem, as is apparent from camouflage pictures. A Textured region can contain texture elements of various sizes, each of which can itself be textured. We approach this problem using a bottomup aggregation framework that combines structural characteristics of te ..."
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Cited by 65 (9 self)
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Texture segmentation is a difficult problem, as is apparent from camouflage pictures. A Textured region can contain texture elements of various sizes, each of which can itself be textured. We approach this problem using a bottomup aggregation framework that combines structural characteristics of texture elements with filter responses. Our process adaptively identifies the shape of texture elements and characterize them by their size, aspect ratio, orientation, brightness, etc., and then uses various statistics of these properties to distinguish between different textures. At the same time our process uses the statistics of filter responses to characterize textures. In our process the shape measures and the filter responses crosstalk extensively. In addition, a topdown cleaning process is applied to avoid mixing the statistics of neighboring segments. We tested our algorithm on real images and demonstrate that it can accurately segment regions that contain challenging textures.
ADAPTIVE ALGEBRAIC MULTIGRID
, 2006
"... Efficient numerical simulation of physical processes is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the lim ..."
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Cited by 58 (15 self)
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Efficient numerical simulation of physical processes is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple geometries and differential equations, algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix. While this allows for efficient blackbox solution of the linear systems associated with discretizations of many elliptic differential equations, it also results in a lack of robustness due to unsatisfied assumptions made on the near null spaces of these matrices. This paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process. Emphasis is on the principles that guide the adaptivity and their application to algebraic multigrid solution of certain symmetric positivedefinite linear systems.
Multiscale scientific computation: Review 2001
 Multiscale and Multiresolution Methods
, 2001
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ARMS: An Algebraic Recursive Multilevel Solver for general sparse linear systems
 Numer. Linear Alg. Appl
, 1999
"... This paper presents a general preconditioning method based on a multilevel partial solution approach. The basic step in constructing the preconditioner is to separate the initial points into two subsets. The first subset which can be termed "coarse" is obtained by using "block" ..."
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Cited by 53 (24 self)
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This paper presents a general preconditioning method based on a multilevel partial solution approach. The basic step in constructing the preconditioner is to separate the initial points into two subsets. The first subset which can be termed "coarse" is obtained by using "block" independent sets, or "aggregates". Two aggregates have no coupling between them, but nodes in the same aggregate may be coupled. The nodes not in the coarse set are part of what might be called the "Fringe" set. The idea of the methods is to form the Schur complement related to the fringe set. This leads to a natural block LU factorization which can be used as a preconditioner for the system. This system is then solver recursively using as preconditioner the factorization that could be obtained from the next level. Unlike other multilevel preconditioners available, iterations between levels are allowed. One interesting aspect of the method is that it provides a common framework for many other technique...
An EnergyMinimizing Interpolation For Robust Multigrid Methods
 SIAM J. SCI. COMPUT
, 1998
"... We propose a robust interpolation for multigrid based on the concepts of energy minimization and approximation. The formulation is general; it can be applied to any dimensions. The analysis for one dimension proves that the convergence rate of the resulting multigrid method is independent of the coe ..."
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Cited by 47 (6 self)
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We propose a robust interpolation for multigrid based on the concepts of energy minimization and approximation. The formulation is general; it can be applied to any dimensions. The analysis for one dimension proves that the convergence rate of the resulting multigrid method is independent of the coefficient of the underlying PDE, in addition to being independent of the mesh size. We demonstrate numerically the effectiveness of the multigrid method in two dimensions by applying it to a discontinuous coefficient problem and an oscillatory coefficient problem. We also show using a onedimensional Helmholtz problem that the energy minimization principle can be applied to solving elliptic problems that are not positive definite.
A fast solver for the Stokes equations with distributed forces in complex geometries
 J. Comput. Phys
"... We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a blackbox fashion; (2) it is second order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the Embedded ..."
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Cited by 40 (10 self)
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We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a blackbox fashion; (2) it is second order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the Embedded Boundary Integral method, is based on Anita Mayo’s work for the Poisson’s equation: “The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions”, SIAM Journal on Numerical Analysis, 21 (1984), pp. 285–299. We embed the domain in a rectangular domain, for which fast solvers are available, and we impose the boundary conditions as interface (jump) conditions on the velocities and tractions. We use an indirect boundary integral formulation for the homogeneous Stokes equations to compute the jumps. The resulting equations are discretized by Nyström’s method. The rectangular domain problem is discretized by finite elements for a velocitypressure formulation with equal order interpolation bilinear elements (£¥ ¤£¥ ¤). Stabilization is used to circumvent the ¦¨§�©������� � condition for the pressure space. For the integral equations, fast matrix vector multiplications are achieved via an ���¨���� � algorithm based on a block representation of the discrete integral operator, combined with (kernel independent) singular value decomposition to sparsify lowrank blocks. The regular grid solver is a Krylov method (Conjugate Residuals) combined with an optimal twolevel Schwartzpreconditioner. For the integral equation we use GMRES. We have tested our algorithm on several numerical examples and we have observed optimal convergence rates. Key Words: Stokes equations, fast solvers, integral equations, doublelayer potential, fast multipole methods, embedded domain methods, immersed interface methods, fictitious