618 research outputs found
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)
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
Aggregation-based Multilevel Methods for Lattice QCD
In Lattice QCD computations a substantial amount of work is spent in solving
the Dirac equation. In the recent past it has been observed that conventional
Krylov solvers tend to critically slow down for large lattices and small quark
masses. We present a Schwarz alternating procedure (SAP) multilevel method as a
solver for the Clover improved Wilson discretization of the Dirac equation.
This approach combines two components (SAP and algebraic multigrid) that have
separately been used in lattice QCD before. In combination with a bootstrap
setup procedure we show that considerable speed-up over conventional Krylov
subspace methods for realistic configurations can be achieved.Comment: Talk presented at the XXIX International Symposium on Lattice Field
Theory, July 10-16, 2011, Lake Tahoe, Californi
Generalizing Reduction-Based Algebraic Multigrid
Algebraic Multigrid (AMG) methods are often robust and effective solvers for
solving the large and sparse linear systems that arise from discretized PDEs
and other problems, relying on heuristic graph algorithms to achieve their
performance. Reduction-based AMG (AMGr) algorithms attempt to formalize these
heuristics by providing two-level convergence bounds that depend concretely on
properties of the partitioning of the given matrix into its fine- and
coarse-grid degrees of freedom. MacLachlan and Saad (SISC 2007) proved that the
AMGr method yields provably robust two-level convergence for symmetric and
positive-definite matrices that are diagonally dominant, with a convergence
factor bounded as a function of a coarsening parameter. However, when applying
AMGr algorithms to matrices that are not diagonally dominant, not only do the
convergence factor bounds not hold, but measured performance is notably
degraded. Here, we present modifications to the classical AMGr algorithm that
improve its performance on matrices that are not diagonally dominant, making
use of strength of connection, sparse approximate inverse (SPAI) techniques,
and interpolation truncation and rescaling, to improve robustness while
maintaining control of the algorithmic costs. We present numerical results
demonstrating the robustness of this approach for both classical isotropic
diffusion problems and for non-diagonally dominant systems coming from
anisotropic diffusion
Analysis of dielectric photonic-crystal problems with MLFMA and Schur-complement preconditioners
Cataloged from PDF version of article.We present rigorous solutions of electromagnetics problems involving 3-D dielectric photonic crystals (PhCs). Problems are formulated with recently developed surface integral equations and solved iteratively using the multilevel fast multipole algorithm (MLFMA). For efficient solutions, iterations are accelerated via robust Schur-complement preconditioners. We show that complicated PhC structures can be analyzed with unprecedented efficiency and accuracy by an effective solver based on the combined tangential formulation, MLFMA, and Schur-complement preconditioners. © 2006 IEEE
Multilevel quasiseparable matrices in PDE-constrained optimization
Optimization problems with constraints in the form of a partial differential
equation arise frequently in the process of engineering design. The
discretization of PDE-constrained optimization problems results in large-scale
linear systems of saddle-point type. In this paper we propose and develop a
novel approach to solving such systems by exploiting so-called quasiseparable
matrices. One may think of a usual quasiseparable matrix as of a discrete
analog of the Green's function of a one-dimensional differential operator. Nice
feature of such matrices is that almost every algorithm which employs them has
linear complexity. We extend the application of quasiseparable matrices to
problems in higher dimensions. Namely, we construct a class of preconditioners
which can be computed and applied at a linear computational cost. Their use
with appropriate Krylov methods leads to algorithms of nearly linear
complexity
A hybrid recursive multilevel incomplete factorization preconditioner for solving general linear systems
In this paper we introduce an algebraic recursive multilevel incomplete factorization preconditioner, based on a distributed Schur complement formulation, for solving general linear systems. The novelty of the proposed method is to combine factorization techniques of both implicit and explicit type, recursive combinatorial algorithms, multilevel mechanisms and overlapping strategies to maximize sparsity in the inverse factors and consequently reduce the factorization costs. Numerical experiments demonstrate the good potential of the proposed solver to precondition effectively general linear systems, also against other state-of-the-art iterative solvers of both implicit and explicit form
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