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Substructure-by-subspace preconditioners for structured linear systems

# Substructure-by-subspace preconditioners for structured linear systems

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Published by Rutherford Appleton Laboratory in Chilton .
Written in English

Edition Notes

 ID Numbers Statement M.J. Dayde et al .... Series Rutherford Appleton Laboratory Technical Report -- RAL-TR-98-005 Contributions Dayde, M. J., Rutherford Appleton Laboratory., Council For The Central Laboratory of The Research Councils. Open Library OL17267949M

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### Substructure-by-subspace preconditioners for structured linear systems Download PDF EPUB FB2

KEY WORDS Sparse structured linear systems, iterative methods, preconditioned conjugate gradient, element-by-element preconditioners 1. Introduction We consider the solution of n by n real linear systems of equations Ax D b () where A is symmetric positive-definite and has the form A D Xe iD1 AiA T i () Here Ai is an n by ni real matrix.

CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): We consider the iterative solution of symmetric positive-definite linear systems whose coefficient matrix may be expressed as the outer-product of low-rank terms.

We derive suitable preconditioners for such systems, and demonstrate their effectiveness on a number of test examples. Iterative Methods and Preconditioning for Large and Sparse Linear Systems with Applications book.

Preconditioners for some structured linear systems. By Daniele Bertaccini, Fabio Durastante. The use of circulant preconditioners is therefore good from the point of view of computational costs. In the real symmetric case, a sufficient. We proposed a substructure preconditioner for a class of structured linear system of equations.

We show that a preconditioner with half of the constraint terms is able to make the preconditioned. We have performed some tests on artificially constructed structure linear systems of equations of the always use the restarted GMRES method as the iterative solver, and the maximum subspace dimension is set to be The initial guess is the zero vector and the stopping criterion is ‖ b − A x k ‖ 2 ‖ b ‖ ≤ 1 × 1 0 − 6, where x k is the k-th approximate by: 2.

Keywords: Krylov subspace approximations, linear systems, iterative methods, preconditioners, finite precision arithmetic, multigrid methods, domain decomposition methods - Hide Description Much recent research has concentrated on the efficient solution of large sparse or structured linear systems using iterative methods.

() Substructure preconditioners for a class of structured linear systems of equations. Mathematical and Computer Modelling() The spectral properties of the preconditioned matrix for nonsymmetric saddle point problems.

Much recent research has concentrated on the efficient solution of large sparse or structured linear systems using iterative methods. A language loaded with acronyms for a thousand different algorithms has developed, and it is often difficult even for specialists to identify the basic principles involved.

Here is a book that focuses on the analysis of iterative methods. () Block preconditioners with circulant blocks for general linear systems. Computers & Mathematics with Applications() A modified T. Chan’s preconditioner for Toeplitz systems. () An MHSS-like iteration method for two-by-two linear systems with application to FDE optimization problems.

Journal of Computational and Applied Mathematics() Regularizing properties of a class of matrices including the optimal and the superoptimal preconditioners.

Linear systems with tensor product structure can be regarded as linear matrix equations for d = 2 and appear to be their most natural extension for d > 2.

this book introduces Sinc methods to. Recently, Bai proposed a block-counter-diagonal and a block-counter-triangular preconditioning matrices to precondition the GMRES method for solving the structured system of linear equations.

In this paper, the minimization problem with a half-quadratic (HQ) regularization for image restoration is studied. For the structured linear system arising at each step of the Newton method for solving the minimization problem, we propose improved block preconditioners based on approximate inversion of the Schur complement and matrix decomposition of the Hessian matrix.

For the iterative solution of such systems, we use Krylov type iterative meth-ods with preconditioners, which exploit the natural block decomposition pro-vided by the physics and saddle point structure of the matrices.

The saddle point structure is provided by both multiphysics and mixed nite element dis. This book describes, in a basic way, the most useful and effective iterative solvers and appropriate preconditioning techniques for some of the most important classes of large and sparse linear systems. The solution of large and sparse linear systems is the most time-consuming part for most of the scientific computing simulations.

Indeed, mathematical models become more and more accurate by. The BDCS preconditioners are higher order extensions of the DCS preconditioner constructed in for solving the discrete linear systems with respect to the 1D spatial fractional diffusion equations in.

They can be effectively executed by the fast Fourier transform, and can tightly cluster the eigenvalues of the corresponding preconditioned.

Abstract. For the solution of the SID (Symmetric InDefinite) linear systems, the use of the GLS (Generalized Least-Squares) polynomial preconditioner can improve the execution efficiency of solvers, particularly for some specially structured systems. Block preconditioner with circulant blocks (BPCB) has been used for solving linear systems with block Toeplitz structure since [R.

Chan, X. Jin, A family of block preconditioners for block systems, SIAM J. Sci. Statist. Comput. (13) () –]. In this new paper, we use BPCBs to general linear systems (with no block structure. () New preconditioners for systems of linear equations with Toeplitz structure. Calcolo() Augmented Lagrangian with Variable Splitting for Faster Non-Cartesian ${\rm L}_{1}$-SPIRiT MR Image Reconstruction.

This report gives some insight into OpenFOAM's structure of linear solvers, i.e. iterative solvers for linear sets of equations Ax = b.

Also matrix preconditioners and smoothers will be presented. In a tutorial section we will use the icoFoam application solver on the cavity test case. A comparison. In either case, each processor will end up with a set of equations (rows of the linear system) and a vector of the variables associated with these rows.

This natural way of distributing a sparse linear system has been adopted by most developers of software for distributed sparse linear systems (see, e.g.,3.,  This book describes, in a basic way, the most useful and effective iterative solvers and appropriate preconditioning techniques for some of the most important classes of large and sparse linear systems.

The solution of large and sparse linear systems is the most time-consuming part for most of the scientific computing simulations. Indeed, mathematical models become more and more accurate. Zhu Y., Sameh A.H. () How to Generate Effective Block Jacobi Preconditioners for Solving Large Sparse Linear Systems.

In: Bazilevs Y., Takizawa K. (eds) Advances in Computational Fluid-Structure Interaction and Flow Simulation. Modeling and Simulation in Science, Engineering and Technology.

These results are elementary in constructing high-quality structured preconditioners for the systems of linear equations arising from the sinc discretizations of ordinary and partial differential.

linear operation (i.e. a computational procedure which applies a linear op-eration to a vector) is what is required. This article is a personal perspective on the subject of preconditioning. It addresses preconditioning only in the most common context of the so-lution of linear systems of equations.

Much research on this problem has. In this paper preconditioners for linear systems arising in interior-point methods for the solution of distributed control problems are derived and analyzed. The matrices K in these systems have a block structure with blocks obtained from the discretization of the objective function and the governing differential equation.

Block preconditioner with circulant blocks (BPCB) has been used for solving linear systems with block Toeplitz structure since [R. Chan, X. Jin, A family of block preconditioners for block. Abstract. A key ingredient in the solution of a large, sparse system of linear equations by an iterative method like conjugate gradients is a preconditioner, which is in a sense an approximation to the matrix of y, the iterative method converges much faster on the preconditioned system at the extra cost of one solve against the preconditioner per iteration.

In this note, the authors describe a new Krylov-subspace iteration for solving symmetric indefinite linear systems that can be combined with arbitrary symmetric preconditioners.

The algorithm can be interpreted as a special case of the quasi-minimal residual method for general non-Hermitian linear systems, and like the latter, it produces. This paper is concerned with block preconditioners for linear KKT systems that arise in optimization problems governed by partial differential equations.

The preconditioners exhibit, like the KKT system, a specific block structure and are composed of preconditioners for submatrices in the general case. Implementation issues are discussed. This book describes, in a basic way, the most useful and effective iterative solvers and appropriate preconditioning techniques for some of the most important classes of large and sparse linear systems.

The solution of large and sparse linear systems is the most time-consuming part for most of the scientific computing simulations.

Block preconditioner with circulant blocks (BPCB) has been used for solving linear systems with block Toeplitz structure since [R. Chan, X. Jin, A family of block preconditioners for block systems, SIAM J.

Sci. Statist. Comput. (13) () ]. In this new paper, we use BPCBs to general linear systems (with no block structure usually). The resulting discretized sparse linear systems can be highly indefinite, nonsymmetric and extremely ill-conditioned.

For such problems, factorization based algorithms are often the most robust algorithmic choices among many alternatives, either being used as direct solvers, or as coarse-grid solvers in multigrid, or as preconditioners for.

Much recent research has concentrated on the efficient solution of large sparse or structured linear systems using iterative methods. A language loaded with acronyms for a thousand different algorithms has developed, and it is often difficult even for specialists to identify the basic principles Rating: % positive.

This book gathers selected contributions presented at the INdAM Meeting Structured Matrices in Numerical Linear Algebra: Analysis, Algorithms and Applications, held in Cortona, Italy on SeptemberHighlights cutting-edge research on Structured Matrix Analysis, it covers theoretical issues, computational aspects, and applications alike.

Roughly speaking they can be grouped in two classes: implicit preconditioners and explicit preconditioners. Preconditioners of the first class typically compute incomplete factorizations of A, such as incomplete LU, and therefore the preconditioning step is done by solving two triangular linear systems; see for example [18, 19, 22, 23].

We are interested in fast and stable iterative regularization methods for image deblurring problems with space invariant blur. The associated coefficient matrix has a Block Toeplitz Toeplitz Blocks (BTTB) like structure plus a small rank correction depending on the boundary conditions imposed on the imaging model.

In the literature, several strategies have been proposed in the attempt to. Much recent research has concentrated on the efficient solution of large sparse or structured linear systems using iterative methods.

A language loaded with acronyms for a thousand different algorithms has developed, and it is often difficult even for specialists to identify the basic principles s: 2.

In this paper, we propose an improved preconditioned algorithm for the conjugate gradient squared method (improved PCGS) for the solution of linear equations.

Further, the logical structures underlying the formation of this preconditioned algorithm are demonstrated via a number of theorems. This improved PCGS algorithm retains some mathematical properties that are associated with the CGS.

The book is written in a systematic way and generally easy to follow. The ideas are presented in a systematic and coherent manner.

the monograph is suited for researchers, practitioners and graduate students, in particular from an (systems) engineering community. It provides an excellent reference book for realization theory and linear. Saddle point linear systems of the form () appear in many applications; see [1] for a comprehensive survey.

Frequently they are large and sparse, and iterative solvers must be applied. In recent years, a lot of research has focused on seeking effective preconditioners.The construction of preconditioners is a large research area. History. Probably the first iterative method for solving a linear system appeared in a letter of Gauss to a student of his.

He proposed solving a 4-by-4 system of equations by repeatedly solving the component in which the residual was the largest.Since the first edition of this book was published intremendous progress has been made in the scientific and engineering disciplines regarding the use of iterative methods for linear systems.

The size and complexity of the new generation of linear and nonlinear systems arising in .