399 research outputs found
Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin approximations of Hamilton--Jacobi--Bellman equations
We analyse a class of nonoverlapping domain decomposition preconditioners for
nonsymmetric linear systems arising from discontinuous Galerkin finite element
approximation of fully nonlinear Hamilton--Jacobi--Bellman (HJB) partial
differential equations. These nonsymmetric linear systems are uniformly bounded
and coercive with respect to a related symmetric bilinear form, that is
associated to a matrix . In this work, we construct a
nonoverlapping domain decomposition preconditioner , that is based
on , and we then show that the effectiveness of the preconditioner
for solving the} nonsymmetric problems can be studied in terms of the condition
number . In particular, we establish the
bound , where
and are respectively the coarse and fine mesh sizes, and and
are respectively the coarse and fine mesh polynomial degrees. This represents
the first such result for this class of methods that explicitly accounts for
the dependence of the condition number on ; our analysis is founded upon an
original optimal order approximation result between fine and coarse
discontinuous finite element spaces. Numerical experiments demonstrate the
sharpness of this bound. Although the preconditioners are not robust with
respect to the polynomial degree, our bounds quantify the effect of the coarse
and fine space polynomial degrees. Furthermore, we show computationally that
these methods are effective in practical applications to nonsymmetric, fully
nonlinear HJB equations under -refinement for moderate polynomial degrees
A multi-level preconditioned Krylov method for the efficient solution of algebraic tomographic reconstruction problems
Classical iterative methods for tomographic reconstruction include the class
of Algebraic Reconstruction Techniques (ART). Convergence of these stationary
linear iterative methods is however notably slow. In this paper we propose the
use of Krylov solvers for tomographic linear inversion problems. These advanced
iterative methods feature fast convergence at the expense of a higher
computational cost per iteration, causing them to be generally uncompetitive
without the inclusion of a suitable preconditioner. Combining elements from
standard multigrid (MG) solvers and the theory of wavelets, a novel
wavelet-based multi-level (WMG) preconditioner is introduced, which is shown to
significantly speed-up Krylov convergence. The performance of the
WMG-preconditioned Krylov method is analyzed through a spectral analysis, and
the approach is compared to existing methods like the classical Simultaneous
Iterative Reconstruction Technique (SIRT) and unpreconditioned Krylov methods
on a 2D tomographic benchmark problem. Numerical experiments are promising,
showing the method to be competitive with the classical Algebraic
Reconstruction Techniques in terms of convergence speed and overall performance
(CPU time) as well as precision of the reconstruction.Comment: Journal of Computational and Applied Mathematics (2014), 26 pages, 13
figures, 3 table
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