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- Marian Brezina, Andrew J. Cleary, +5 authors John W. Ruge
- SIAM J. Scientific Computing
- 2001

We introduce AMGe, an algebraic multigrid method for solving the discrete equations that arise in Ritz-type nite element methods for partial diierential equations. Assuming access to the element stiiness 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… (More)

- Petr Vanek, Jan Mandel, Marian Brezina
- Computing
- 1996

An algebraic multigrid algorithm for symmetric, positive definite linear systems is developed based on the concept of prolongation by smoothed aggregation. Coarse levels are generated automatically. We present a set of requirements motivated heuristically by a convergence theory. The algorithm then attempts to satisfy the requirements. Input to the method… (More)

- Jan Mandel, Marian Brezina, Petr Vanek
- Computing
- 1999

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. For a particular selection of the supports,… (More)

- Petr Vanek, Marian Brezina, Jan Mandel
- Numerische Mathematik
- 2001

We prove a convergence estimate for the Algebraic Multigrid Method with prolongations deened by aggregation using zero energy modes, followed by a smoothing. The method input is the problem matrix and a matrix of the zero energy modes. The estimate depends only polylogarithmically on the mesh size, and requires only a weak approximation property for the… (More)

Gauss-Seidel method is often the smoother of choice within multigrid applications. In the context of unstructured meshes, however, maintaining good parallel efficiency is difficult with multiplicative iterative methods such as Gauss-Seidel. This leads us to consider alternative smoothers. We discuss the computational advantages of polynomial smoothers… (More)

- James Brannick, Marian Brezina, +7 authors Ludmil Zikatanov
- 2005

The linear systems arising in lattice QCD pose significant challenges for traditional iterative solvers. For physically interesting values of the so-called quark mass, these systems are nearly singular, indicating the need for efficient preconditioners. However, multi-level preconditioners cannot easily be constructed because the Dirac operator associated… (More)

- Marian Brezina, Petr Vanek
- Computing
- 1999

We propose a black-box parallel iterative method suitable for solving both elliptic and certain non-elliptic problems discretized on unstructured meshes. The method is analyzed in the case of the second order elliptic problems discretized on quasiuniform P1 and Q1 finite element meshes. The numerical experiments confirm the validity of the proved convegence… (More)

- Marian Brezina, Thomas A. Manteuffel, Stephen F. McCormick, John W. Ruge, Geoffrey Sanders
- SIAM J. Scientific Computing
- 2010

- Jan Mandel, Marian Brezina
- Math. Comput.
- 1996

The Balancing Domain Decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent. The abstract theory introduced recently by the first-named author is used to develop condition… (More)

- James J. Brannick, Marian Brezina, Scott P. MacLachlan, Thomas A. Manteuffel, Stephen F. McCormick, John W. Ruge
- Numerical Lin. Alg. with Applic.
- 2006

SUMMARY Algebraic multigrid (AMG) is an iterative method that is often optimal for solving the matrix equations that arise in a wide variety of applications, including discretized partial differential equations. It automatically constructs a sequence of increasingly smaller matrix problems that hopefully enables efficient resolution of all scales present in… (More)