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A Comparative Study on Dynamic and Static Sparsity Patterns in Parallel Sparse Approximate Inverse Preconditioning
 J. Math. Model. Algor
, 2002
"... Sparse approximate inverse (SAI) techniques have recently emerged as a new class of parallel preconditioning techniques for solving large sparse linear systems on high performance computers. The choice of the sparsity pattern of the SAI matrix is probably the most important step in constructing a ..."
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Sparse approximate inverse (SAI) techniques have recently emerged as a new class of parallel preconditioning techniques for solving large sparse linear systems on high performance computers. The choice of the sparsity pattern of the SAI matrix is probably the most important step in constructing an SAI preconditioner. Both dynamic and static sparsity pattern selection approaches have been proposed by researchers.
Parallel Multilevel Sparse Approximate Inverse Preconditioners in Large Sparse Matrix Computations
 In proceedings of Supercomputing 2003: Igniting Innovation. November 15  21, 2003
"... Abstract. We investigate the use of the multistep successive preconditioning strategies (MSP) to construct a class of parallel multilevel sparse approximate inverse (SAI) preconditioners. We do not use independent set ordering, but a diagonal dominance based matrix permutation to build a multilevel ..."
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Abstract. We investigate the use of the multistep successive preconditioning strategies (MSP) to construct a class of parallel multilevel sparse approximate inverse (SAI) preconditioners. We do not use independent set ordering, but a diagonal dominance based matrix permutation to build a multilevel structure. The purpose of introducing multilevel structure into SAI is to enhance the robustness of SAI for solving difficult problems. Forward and backward preconditioning iteration and two Schur complement preconditioning strategies are proposed to improve the performance and to reduce the storage cost of the multilevel preconditioners. One version of the parallel multilevel SAI preconditioner based on the MSP strategy is implemented. Numerical experiments for solving a few sparse matrices on a distributed memory parallel computer are reported. Key words. Sparse matrices, parallel preconditioning, sparse approximate inverse, multilevel preconditioning, multistep successive preconditioning. 1. Introduction. Large
Fourth Order Compact Difference Scheme for 3D Convection Diffusion Equation with Boundary Layers on Nonuniform Grids
, 2000
"... We present a fourth order compact finite difference scheme for a general three dimensional convection diffusion equation with variable coefficients on a uniform cubic grid. This high order compact difference scheme is used to solve convection diffusion equation with boundary layers on a three dimens ..."
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We present a fourth order compact finite difference scheme for a general three dimensional convection diffusion equation with variable coefficients on a uniform cubic grid. This high order compact difference scheme is used to solve convection diffusion equation with boundary layers on a three dimensional nonuniform grid. We compare the computed accuracy and computational efficiency of the fourth order compact difference scheme with that of the standard central difference scheme and the first order upwind difference scheme. Several convection diffusion problems are solved numerically to validate the proposed fourth order compact scheme. Key words: convection diffusion equation, boundary layer, grid stretching, fourth order Technical Report 29800, Department of Computer Science, University of Kentucky, Lexington, KY, 2000. y Email: jzhang@cs.uky.edu, URL: http://www.cs.uky.edu/jzhang. The research of this author was supported in part by the U.S. National Science Foundation under g...
Multiresolution 3D Nonrigid Registration via Optimal Mass Transport on the GPU
, 2007
"... In this paper we present computationally efficient implementation of the minimizing flow approach for optimal mass transport (OMT) with applications to nonrigid 3D image registration. Our implementation solves the OMT problem via multiresolution, multigrid, and parallel methodologies on a consume ..."
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In this paper we present computationally efficient implementation of the minimizing flow approach for optimal mass transport (OMT) with applications to nonrigid 3D image registration. Our implementation solves the OMT problem via multiresolution, multigrid, and parallel methodologies on a consumer graphics processing unit (GPU). Although computing the optimal map has shown to be computationally expensive in the past, we show that our approach is almost two orders magnitude faster than previous work and is capable of finding transport maps with optimality measures (mean curl) previously unattainable by other works (which directly influences the accuracy of registration). We give results where the algorithm was used to compute nonrigid registrations of 3D synthetic data as well as intrapatient preoperative and postoperative 3D brain MRI datasets.
A Two Colorable Fourth Order Compact Difference Scheme and Parallel Iterative Solution of the 3D Convection Diffusion Equation
, 2000
"... A new fourth order compact difference scheme for the three dimensional convection diffusion equation with variable coefficients is presented. The novelty of this new difference scheme is that it only requires 15 grid points and that it can be decoupled with two colors. The entire computational grid ..."
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A new fourth order compact difference scheme for the three dimensional convection diffusion equation with variable coefficients is presented. The novelty of this new difference scheme is that it only requires 15 grid points and that it can be decoupled with two colors. The entire computational grid can be updated in two parallel subsweeps with a GaussSeidel type iterative method. This is compared with the known 19 point fourth order compact difference scheme which requires four colors to decouple the computational grid. Numerical results, with multigrid methods implemented on a shared memory parallel computer, are presented to compare the 15 point and 19 point fourth order compact schemes.
High Accuracy and Scalable Multiscale Multigrid Computation for 3D Convection Diffusion Equation
"... We present a sixth order explicit compact finite difference scheme to solve the three dimensional (3D) convection diffusion equation. We first use multiscale multigrid method to solve the linear systems arising from a 19point fourth order discretization scheme to compute the fourth order solutions ..."
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We present a sixth order explicit compact finite difference scheme to solve the three dimensional (3D) convection diffusion equation. We first use multiscale multigrid method to solve the linear systems arising from a 19point fourth order discretization scheme to compute the fourth order solutions on both the coarse grid and the fine grid. Then an operator based interpolation scheme combined with an extrapolation technique is used to approximate the sixth order accurate solution on the fine grid. Since the multigrid method using a standard point relaxation smoother may fail to achieve the optimal grid independent convergence rate for solving convection diffusion equation with a high Reynolds number, we also implement the plane relaxation smoother in the multigrid solver to achieve better grid independency. Supporting numerical results are presented to demonstrate the efficiency and accuracy of the sixth order compact scheme
High Order Compact Difference Scheme and Multigrid Method for 2D Elliptic Problems with Variable Coefficients and Interior/Boundary Layers on Nonuniform Grids
, 2015
"... In this paper, a high order compact difference scheme and a multigrid method are proposed for solving twodimensional (2D) elliptic problems with variable coefficients and interior/boundary layers on nonuniform grids. Firstly, the original equation is transformed from the physical domain (with a non ..."
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In this paper, a high order compact difference scheme and a multigrid method are proposed for solving twodimensional (2D) elliptic problems with variable coefficients and interior/boundary layers on nonuniform grids. Firstly, the original equation is transformed from the physical domain (with a nonuniform mesh) to the computational domain (with a uniform mesh) by using a coordinate transformation. Then, a fourth order compact difference scheme is proposed to solve the transformed elliptic equation on uniform girds. After that, a multigrid method is employed to solve the linear algebraic system arising from the difference equation. At last, the numerical experiments on some elliptic problems with interior/boundary layers are conducted to show high accuracy and high efficiency of the present method.
Jules Kouatchoux NASA Goddard Space Flight Center Code 931
, 2000
"... Abstract A new fourth order compact difference scheme for the three dimensional convection diffusion equation with variable coefficients is presented. The novelty of this new difference scheme is that it only requires 15 grid points and that it can be decoupled with two colors. The entire computatio ..."
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Abstract A new fourth order compact difference scheme for the three dimensional convection diffusion equation with variable coefficients is presented. The novelty of this new difference scheme is that it only requires 15 grid points and that it can be decoupled with two colors. The entire computational grid can be updated in two parallel subsweeps with a GaussSeidel type iterative method. This is compared with the known 19 point fourth order compact difference scheme which requires four colors to decouple the computational grid. Numerical results, with multigrid methods implemented on a shared memory parallel computer, are presented to compare the 15 point and 19 point fourth order compact schemes. Key words: 3D convection diffusion equation, fourth order compact difference schemes, multigrid method, parallel computation.