183 research outputs found
On large-scale diagonalization techniques for the Anderson model of localization
We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model
of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the Jacobi–Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete
LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization
by several orders of magnitude
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)
A two-level ILU preconditioner for electromagnetic applications
[EN] Computational electromagnetics based on the solution of the integral form of Maxwell s
equations with boundary element methods require the solution of large and dense linear
systems. For large-scale problems the solution is obtained by using iterative Krylov-type
methods provided that a fast method for performing matrix vector products is available.
In addition, for ill-conditioned problems some kind of preconditioning technique must
be applied to the linear system in order to accelerate the convergence of the iterative
method and improve its performance. For many applications it has been reported that
incomplete factorizations often suffer from numerical instability due to the indefiniteness
of the coefficient matrix. In this context, approximate inverse preconditioners based on
Frobenius-norm minimization have emerged as a robust and highly parallel alternative.
In this work we propose a two-level ILU preconditioner for the preconditioned GMRES
method. The computation and application of the preconditioner is based on graph
partitioning techniques. Numerical experiments are presented for different problems and
show that with this technique it is possible to obtain robust ILU preconditioners that
perform competitively compared with Frobenius-norm minimization preconditioners.This work was supported by the Spanish Ministerio de Economía y Competitividad under grant MTM2014-58159-P and MTM2015-68805-REDT.Cerdán Soriano, JM.; Marín Mateos-Aparicio, J.; Mas Marí, J. (2017). A two-level ILU preconditioner for electromagnetic applications. Journal of Computational and Applied Mathematics. 309:371-382. https://doi.org/10.1016/j.cam.2016.03.012S37138230
A domain decomposing parallel sparse linear system solver
The solution of large sparse linear systems is often the most time-consuming
part of many science and engineering applications. Computational fluid
dynamics, circuit simulation, power network analysis, and material science are
just a few examples of the application areas in which large sparse linear
systems need to be solved effectively. In this paper we introduce a new
parallel hybrid sparse linear system solver for distributed memory
architectures that contains both direct and iterative components. We show that
by using our solver one can alleviate the drawbacks of direct and iterative
solvers, achieving better scalability than with direct solvers and more
robustness than with classical preconditioned iterative solvers. Comparisons to
well-known direct and iterative solvers on a parallel architecture are
provided.Comment: To appear in Journal of Computational and Applied Mathematic
Efficient approximation of functions of some large matrices by partial fraction expansions
Some important applicative problems require the evaluation of functions
of large and sparse and/or \emph{localized} matrices . Popular and
interesting techniques for computing and , where
is a vector, are based on partial fraction expansions. However,
some of these techniques require solving several linear systems whose matrices
differ from by a complex multiple of the identity matrix for computing
or require inverting sequences of matrices with the same
characteristics for computing . Here we study the use and the
convergence of a recent technique for generating sequences of incomplete
factorizations of matrices in order to face with both these issues. The
solution of the sequences of linear systems and approximate matrix inversions
above can be computed efficiently provided that shows certain decay
properties. These strategies have good parallel potentialities. Our claims are
confirmed by numerical tests
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Using Jacobi iterations and blocking for solving sparse triangular systems in incomplete factorization preconditioning
When using incomplete factorization preconditioners with an iterative method to solve large sparse linear systems, each application of the preconditioner involves solving two sparse triangular systems. These triangular systems are challenging to solve efficiently on computers with high levels of concurrency. On such computers, it has recently been proposed to use Jacobi iterations, which are highly parallel, to approximately solve the triangular systems from incomplete factorizations. The effectiveness of this approach, however, is problem-dependent: the Jacobi iterations may not always converge quickly enough for all problems. Thus, as a necessary and important step to evaluate this approach, we experimentally test the approach on a large number of realistic symmetric positive definite problems. We also show that by using block Jacobi iterations, we can extend the range of problems for which such an approach can be effective. For block Jacobi iterations, it is essential for the blocking to be cognizant of the matrix structure
Approximate and Incomplete Factorizations
In this chapter, we give a brief overview of a particular class of preconditioners known as incomplete factorizations. They can be thought of as approximating the exact LU factorization of a given matrix A (e.g. computed via Gaussian elimination) by disallowing certain ll-ins. As opposed to other PDE-based preconditioners such as multigrid and domain decomposition, this class of preconditioners are primarily algebraic in nature and can in principle be applied to any sparse matrices. When applied to PDE problems, they are usually not optimal in the sense that the condition number of the preconditioned system will grow as the mesh size h is reduced, although usually at a slower rate than for the unpreconditioned system. On the other hand, they are often quite robust with respect to other more algebraic features of the problem such as rough and anisotropic coecients and strong convection terms.
We will describe the basic ILU and (modied) MILU preconditioners. Then we will review brie
y several variants: more lls, relaxed ILU, shifted ILU, ILQ, as well as block and multilevel variants. We will also touch on a related class of approximate factorization methods which arise more directly from approximating a partial dierential operator by a product of simpler operators.
Finally, we will discuss parallelization aspects, including re-ordering, series expansion and domain decomposition techniques. Generally, this class of preconditioner does not possess a high degree of parallelism in its original form. Re-ordering and approximations by truncating certain series expansion will increase the parallelism, but usually with a deterioration in convergence rate. Domain decomposition oers a compromise
Combinatorial problems in solving linear systems
42 pages, available as LIP research report RR-2009-15Numerical linear algebra and combinatorial optimization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise. Sparse matrices therefore form the basis of the interaction of these two seemingly disparate subjects. As the core of many of today's numerical linear algebra computations consists of the solution of sparse linear system by direct or iterative methods, we survey some combinatorial problems, ideas, and algorithms relating to these computations. On the direct methods side, we discuss issues such as matrix ordering; bipartite matching and matrix scaling for better pivoting; task assignment and scheduling for parallel multifrontal solvers. On the iterative method side, we discuss preconditioning techniques including incomplete factorization preconditioners, support graph preconditioners, and algebraic multigrid. In a separate part, we discuss the block triangular form of sparse matrices
Implicit solvers for unstructured meshes
Implicit methods were developed and tested for unstructured mesh computations. The approximate system which arises from the Newton linearization of the nonlinear evolution operator is solved by using the preconditioned GMRES (Generalized Minimum Residual) technique. Three different preconditioners were studied, namely, the incomplete LU factorization (ILU), block diagonal factorization, and the symmetric successive over relaxation (SSOR). The preconditioners were optimized to have good vectorization properties. SSOR and ILU were also studied as iterative schemes. The various methods are compared over a wide range of problems. Ordering of the unknowns, which affects the convergence of these sparse matrix iterative methods, is also studied. Results are presented for inviscid and turbulent viscous calculations on single and multielement airfoil configurations using globally and adaptively generated meshes
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