Analysis of two-level domain decomposition preconditioners based on aggregation

Sala, Marzio

doi:10.1051/m2an:2004038

Cited by 12 publications

(6 citation statements)

References 17 publications

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“…The simplest aggregation algorithm produces a piecewise constant coarse space. If [ H /h] = O(1), then this preconditioner applied to the Poisson problem has condition number bounded independent of h [12,13].…”

Section: Related Workmentioning

confidence: 99%

“…Another alternative is smoothed aggregation, which smooths the basis functions, thus reducing the steep jumps at subdomain boundaries. For the Poisson problem, this transforms an H /h term in the condition number bound into an H /d term, where d is the smoothing diameter [12]. This keeps the size of the coarse problem the same as basic aggregation, but requires additional work to smooth the basis, and can increase the number of nonzeros in the coarse matrix.…”

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

The Discretely-Discontinuous Galerkin Coarse Grid for Domain Decomposition

Edwards,

Bridson

2015

Preprint

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We present an algebraic method for constructing a highly effective coarse grid correction to accelerate domain decomposition. The coarse problem is constructed from the original matrix and a small set of input vectors that span a low-degree polynomial space, but no further knowledge of meshes or continuous functionals is used. We construct a coarse basis by partitioning the problem into subdomains and using the restriction of each input vector to each subdomain as its own basis function. This basis resembles a Discontinuous Galerkin basis on subdomain-sized elements. Constructing the coarse problem by Galerkin projection, we prove a highorder convergent error bound for the coarse solutions. Used in a twolevel symmetric multiplicative overlapping Schwarz preconditioner, the resulting conjugate gradient solver shows optimal scaling. Convergence requires a constant number of iterations, independent of fine problem size, on a range of scalar and vector-valued second-order and fourth-order PDEs.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

The Discretely-Discontinuous Galerkin Coarse Grid for Domain Decomposition

Edwards,

Bridson

2015

Preprint

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“…Using this module, the user can easily create black-box two-level and multilevel preconditioners based on smoothed aggregation procedures (see Sala [2004b] and Brezina [1997] and the references therein). Parameters are specified using a Teuchos ParameterList or a Python dictionary, as in the Amesos module of Section 4.5; the list of accepted parameter names is given in .…”

Section: The Aztecoo Ifpack and ML Modulesmentioning

confidence: 99%

PyTrilinos

Sala

Spotz

Heroux

2008

ACM Trans. Math. Softw.

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PyTrilinos is a collection of Python modules that are useful for serial and parallel scientific computing. This collection contains modules that cover serial and parallel dense linear algebra, serial and parallel sparse linear algebra, direct and iterative linear solution techniques, domain decomposition and multilevel preconditioners, nonlinear solvers, and continuation algorithms. Also included are a variety of related utility functions and classes, including distributed I/O, coloring algorithms, and matrix generation. PyTrilinos vector objects are integrated with the popular NumPy Python module, gathering together a variety of high-level distributed computing operations with serial vector operations.PyTrilinos is a set of interfaces to existing, compiled libraries. This hybrid framework uses Python as front-end, and efficient precompiled libraries for all computationally expensive tasks. Thus, we take advantage of both the flexibility and ease of use of Python, and the efficiency of the underlying C++, C, and FORTRAN numerical kernels. Out numerical results show that, for many important problem classes, the overhead required by the Python interpreter is negligible.To run in parallel, PyTrilinos simply requires a standard Python interpreter. The fundamental MPI calls are encapsulated under an abstract layer that manages all interprocessor communications. This makes serial and parallel scripts using PyTrilinos virtually identical.

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“…Nonetheless, this drawback is somehow compensated for by the ease of generating nonsmoothed aggregation. For further results on this subject, see [17,25,26].…”

Section: Solution Of the Original Systemmentioning

confidence: 99%

An Interface-Strip Domain Decomposition Preconditioner

Quarteroni¹,

Sala²,

Valli³

2006

SIAM J. Sci. Comput.

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In this paper we consider domain decomposition preconditioners based on a vertexoriented (VO) decomposition of the computational domain. In element-oriented (EO) decompositions, each element of the grid belongs to a different domain, while in VO decompositions each vertex belongs to a different subdomain. Based on VO decompositions, we present some preconditioners for the solution of the original system, as well as for that of the Schur complement system. Theoretical properties are investigated for a finite element approximation of an elliptic problem. Numerical results and comparison with state-of-the-art preconditioners are also reported. The numerical results presented here show the effectiveness of the proposed preconditioners and their good scalability properties.

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Analysis of two-level domain decomposition preconditioners based on aggregation

Cited by 12 publications

References 17 publications

The Discretely-Discontinuous Galerkin Coarse Grid for Domain Decomposition

The Discretely-Discontinuous Galerkin Coarse Grid for Domain Decomposition

PyTrilinos

An Interface-Strip Domain Decomposition Preconditioner

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