Algebraic multigrid (AMG) for saddle point systems from meshfree discretizations

Leem, Koung Hee; Oliveira, Suely; Stewart, David E.

doi:10.1002/nla.383

Cited by 17 publications

(15 citation statements)

References 15 publications

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“…In the experiments we compare the performance of four preconditioners in GMRES: JOR [16], AMG [2], the H-matrix method of [10] and the H-LU method of this paper. The results are plotted in log-log scale.…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

ℋ︁‐matrix preconditioners for symmetric saddle‐point systems from meshfree discretization

Borne¹,

Oliveira²,

Fang³

2008

Numerical Linear Algebra App

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Meshfree methods are suitable for solving problems on irregular domains, avoiding the use of a mesh. To deal with the boundary conditions, we can use Lagrange multipliers and obtain a sparse, symmetric and indefinite system of saddle-point type. Many methods have been developed to solve the indefinite system. Previously, we presented an algebraic method to construct an LU-based preconditioner for the saddle-point system obtained by meshfree methods, which combines the multilevel clustering method with the H-matrix arithmetic. The corresponding preconditioner has both H-matrix and sparse matrix subblocks. In this paper we refine the above method and propose a way to construct a pure H-matrix preconditioner. We compare the new method with the old method, JOR and smoothed algebraic multigrid methods. The numerical results show that the new preconditioner outperforms the preconditioners based on the other methods.In [2], a scheme based on smoothed algebraic multigrid (AMG) is proposed to solve the saddlepoint systems from meshfree methods. In this paper we present an H-matrix-based approach to solve the saddle-point system. Hierarchical matrices (H-matrices) were introduced in [3] and since then, much work has been done on the theory and applications of H-matrices [4][5][6][7][8]. Hmatrices provide a cheap but approximate way of carrying out matrix computations involving dense matrices.The basic idea of the H-matrix representation is that instead of representing a matrix exactly, approximations are used to represent a matrix based on a hierarchical block cluster tree. The leaves of the tree represent the matrix blocks that are not partitioned further, which are represented by low-rank matrices (Rk-matrix format) or by full matrices (full matrix format). An H-matrix arithmetic is developed by adapting conventional matrix operations to the H-matrix format. The required storage for a matrix and the computational complexity of operations, such as matrix-vector multiplication, matrix-matrix multiplication, matrix addition, inversion and LU factorization, are reduced to almost linear complexity (O(n log n)) [4,8].H-matrices have been combined with finite element methods to solve partial differential equations. Classic H-matrix construction [8] relies on the underlying geometric structure of the problem. Recently, a black-box H-matrix construction method was developed for sparse matrices, which is based on the matrix graph of the underlying sparse matrix [9].In [10], we present a pure algebraic method for the H-matrix construction, which is based on multilevel clustering methods. The experimental results show that it is comparable with the domaindecomposition-based H-matrix construction methods for sparse matrices. Since the algebraic method for the H-matrix construction does not depend on the geometric information underlying the problem, it can be applied to systems from meshfree methods. In [10], we present a way to build preconditioners for solving saddle-point systems, which have both H-matrix subblocks and sparse matri...

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Section: Resultsmentioning

confidence: 99%

“…In order to overcome these problems, we use an independently generated set of basis functions on the boundary [2]. These can also be generated as meshfree functions, but using a different family of kernel functions i on the boundary * : i (x) = a (x− x i ) where x ∈ * , and the points x i are chosen appropriately from * .…”

Section: Introductionmentioning

confidence: 99%

ℋ︁‐matrix preconditioners for symmetric saddle‐point systems from meshfree discretization

Borne¹,

Oliveira²,

Fang³

2008

Numerical Linear Algebra App

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show abstract

“…This kind of system of linear equations arises in a variety of scientific and engineering applications, such as computational fluid dynamics, constrained optimization, optimal control, weighted least-squares problems, electronic networks, computer graphic, the constrained least squares problems and generalized least squares problems etc; see [2,9,18,23,28,29] and the references therein. In addition, we can also obtain saddle point linear systems from the meshfree discretization of some partial differential equations [12,20] or the mixed hybrid finite element discretization of second order elliptic problems [11]. A comprehensive summary about various applications leading to saddle point matrices and a general framework of preconditioning methods and their theoretical analyses were given in [8].…”

Section: Introductionmentioning

confidence: 99%

On semi-convergence of a class of relaxation methods for singular saddle point problems

Fan

Zhu

Zheng

2015

Applied Mathematics and Computation

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“…This kind of system of linear equations arises in a variety of scientific and engineering applications, such as computational fluid dynamics, constrained optimization, optimal control, weighted least-squares problems, electronic networks, computer graphic etc; see [1][2][3][4] and the references therein. In addition, we can also obtain saddle-point linear systems from the mixed or hybrid finite element discretization of secondorder elliptic problems [5] or the meshfree discretization of some partial differential equations [6,7]. When matrix B is column rank-deficient, i.e., rankðBÞ < m 6 n, matrix M is singular.…”

Section: Introductionmentioning

confidence: 99%

On semi-convergence of the Uzawa–HSS method for singular saddle-point problems

Yang

2015

Applied Mathematics and Computation

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Algebraic multigrid (AMG) for saddle point systems from meshfree discretizations

Cited by 17 publications

References 15 publications

ℋ︁‐matrix preconditioners for symmetric saddle‐point systems from meshfree discretization

ℋ︁‐matrix preconditioners for symmetric saddle‐point systems from meshfree discretization

On semi-convergence of a class of relaxation methods for singular saddle point problems

On semi-convergence of the Uzawa–HSS method for singular saddle-point problems

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