78 research outputs found
Preconditioning for Sparse Linear Systems at the Dawn of the 21st Century: History, Current Developments, and Future Perspectives
Iterative methods are currently the solvers of choice for large sparse linear systems of equations. However, it is well known that the key factor for accelerating, or even allowing for, convergence is the preconditioner. The research on preconditioning techniques has characterized the last two decades. Nowadays, there are a number of different options to be considered when choosing the most appropriate preconditioner for the specific problem at hand. The present work provides an overview of the most popular algorithms available today, emphasizing the respective merits and limitations. The overview is restricted to algebraic preconditioners, that is, general-purpose algorithms requiring the knowledge of the system matrix only, independently of the specific problem it arises from. Along with the traditional distinction between incomplete factorizations and approximate inverses, the most recent developments are considered, including the scalable multigrid and parallel approaches which represent the current frontier of research. A separate section devoted to saddle-point problems, which arise in many different applications, closes the paper
Why diffusion-based preconditioning of Richards equation works: spectral analysis and computational experiments at very large scale
We consider here a cell-centered finite difference approximation of the
Richards equation in three dimensions, averaging for interface values the
hydraulic conductivity , a highly nonlinear function, by arithmetic,
upstream, and harmonic means. The nonlinearities in the equation can lead to
changes in soil conductivity over several orders of magnitude and
discretizations with respect to space variables often produce stiff systems of
differential equations. A fully implicit time discretization is provided by
\emph{backward Euler} one-step formula; the resulting nonlinear algebraic
system is solved by an inexact Newton Armijo-Goldstein algorithm, requiring the
solution of a sequence of linear systems involving Jacobian matrices. We prove
some new results concerning the distribution of the Jacobians eigenvalues and
the explicit expression of their entries. Moreover, we explore some connections
between the saturation of the soil and the ill-conditioning of the Jacobians.
The information on eigenvalues justifies the effectiveness of some
preconditioner approaches which are widely used in the solution of Richards
equation. We also propose a new software framework to experiment with scalable
and robust preconditioners suitable for efficient parallel simulations at very
large scales. Performance results on a literature test case show that our
framework is very promising in the advance towards realistic simulations at
extreme scale
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
Efficient solvers for hybridized three-field mixed finite element coupled poromechanics
We consider a mixed hybrid finite element formulation for coupled
poromechanics. A stabilization strategy based on a macro-element approach is
advanced to eliminate the spurious pressure modes appearing in
undrained/incompressible conditions. The efficient solution of the stabilized
mixed hybrid block system is addressed by developing a class of block
triangular preconditioners based on a Schur-complement approximation strategy.
Robustness, computational efficiency and scalability of the proposed approach
are theoretically discussed and tested using challenging benchmark problems on
massively parallel architectures
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