5,252 research outputs found
Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods
Use of the stochastic Galerkin finite element methods leads to large systems
of linear equations obtained by the discretization of tensor product solution
spaces along their spatial and stochastic dimensions. These systems are
typically solved iteratively by a Krylov subspace method. We propose a
preconditioner which takes an advantage of the recursive hierarchy in the
structure of the global matrices. In particular, the matrices posses a
recursive hierarchical two-by-two structure, with one of the submatrices block
diagonal. Each one of the diagonal blocks in this submatrix is closely related
to the deterministic mean-value problem, and the action of its inverse is in
the implementation approximated by inner loops of Krylov iterations. Thus our
hierarchical Schur complement preconditioner combines, on each level in the
approximation of the hierarchical structure of the global matrix, the idea of
Schur complement with loops for a number of mutually independent inner Krylov
iterations, and several matrix-vector multiplications for the off-diagonal
blocks. Neither the global matrix, nor the matrix of the preconditioner need to
be formed explicitly. The ingredients include only the number of stiffness
matrices from the truncated Karhunen-Lo\`{e}ve expansion and a good
preconditioned for the mean-value deterministic problem. We provide a condition
number bound for a model elliptic problem and the performance of the method is
illustrated by numerical experiments.Comment: 15 pages, 2 figures, 9 tables, (updated numerical experiments
Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, Part I: Geometrically Conforming Meshes
This paper is concerned with the design, analysis and implementation of
preconditioning concepts for spectral Discontinuous Galerkin discretizations of
elliptic boundary value problems. While presently known techniques realize a
growth of the condition numbers that is logarithmic in the polynomial degrees
when all degrees are equal and quadratic otherwise, our main objective is to
realize full robustness with respect to arbitrarily large locally varying
polynomial degrees degrees, i.e., under mild grading constraints condition
numbers stay uniformly bounded with respect to the mesh size and variable
degrees. The conceptual foundation of the envisaged preconditioners is the
auxiliary space method. The main conceptual ingredients that will be shown in
this framework to yield "optimal" preconditioners in the above sense are
Legendre-Gauss-Lobatto grids in connection with certain associated anisotropic
nested dyadic grids as well as specially adapted wavelet preconditioners for
the resulting low order auxiliary problems. Moreover, the preconditioners have
a modular form that facilitates somewhat simplified partial realizations. One
of the components can, for instance, be conveniently combined with domain
decomposition, at the expense though of a logarithmic growth of condition
numbers. Our analysis is complemented by quantitative experimental studies of
the main components.Comment: 41 pages, 11 figures; Major revision: rearrangement of the contents
for better readability, part on wavelet preconditioner adde
Sparse approximate inverse preconditioners on high performance GPU platforms
Simulation with models based on partial differential equations often requires the solution of (sequences of) large and sparse algebraic linear systems. In multidimensional domains, preconditioned Krylov iterative solvers are often appropriate for these duties. Therefore, the search for efficient preconditioners for Krylov subspace methods is a crucial theme. Recent developments, especially in computing hardware, have renewed the interest in approximate inverse preconditioners in factorized form, because their application during the solution process can be more efficient. We present here some experiences focused on the approximate inverse preconditioners proposed by Benzi and Tůma from 1996 and the sparsification and inversion proposed by van Duin in 1999. Computational costs, reorderings and implementation issues are considered both on conventional and innovative computing architectures like Graphics Programming Units (GPUs)
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