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Incomplete Factorization Preconditioning For Linear Least Squares Problems
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Book Synopsis Incomplete Factorization Preconditioning for Linear Least Squares Problems by : Xiaoge Wang
Download or read book Incomplete Factorization Preconditioning for Linear Least Squares Problems written by Xiaoge Wang and published by . This book was released on 1993 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: When certain conditions are imposed on the sparsity pattern of the preconditioning matrix, incomplete Cholesky (IC) factorization is identical to IMGS. Therefore, a sufficient condition on the sparsity pattern for a stable IC factorization can be derived. Based on this condition, three algorithms for modifying a sparsity pattern for which IC may not succeed have been designed. These methods differ from previously proposed methods to guarantee the existence and improve the stability of the IC factorization since they do not require numerical information. Numerical experiments illustrating the capabilities of the preconditioners are also presented."
Book Synopsis Robust Preconditioners for Large Sparse Linear Systems and Least-squares Problems by : Na Li
Download or read book Robust Preconditioners for Large Sparse Linear Systems and Least-squares Problems written by Na Li and published by . This book was released on 2006 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Numerical Methods for Least Squares Problems, Second Edition by : Åke Björck
Download or read book Numerical Methods for Least Squares Problems, Second Edition written by Åke Björck and published by SIAM. This book was released on 2024-07-05 with total page 509 pages. Available in PDF, EPUB and Kindle. Book excerpt: The method of least squares, discovered by Gauss in 1795, is a principal tool for reducing the influence of errors when fitting a mathematical model to given observations. Applications arise in many areas of science and engineering. The increased use of automatic data capturing frequently leads to large-scale least squares problems. Such problems can be solved by using recent developments in preconditioned iterative methods and in sparse QR factorization. The first edition of Numerical Methods for Least Squares Problems was the leading reference on the topic for many years. The updated second edition stands out compared to other books on this subject because it provides an in-depth and up-to-date treatment of direct and iterative methods for solving different types of least squares problems and for computing the singular value decomposition. It also is unique because it covers generalized, constrained, and nonlinear least squares problems as well as partial least squares and regularization methods for discrete ill-posed problems. The bibliography of over 1,100 historical and recent references provides a comprehensive survey of past and present research in the field. This book will be of interest to graduate students and researchers in applied mathematics and to researchers working with numerical linear algebra applications.
Book Synopsis Numerical Methods for Least Squares Problems by : Ake Bjorck
Download or read book Numerical Methods for Least Squares Problems written by Ake Bjorck and published by SIAM. This book was released on 1996-01-01 with total page 425 pages. Available in PDF, EPUB and Kindle. Book excerpt: The method of least squares was discovered by Gauss in 1795. It has since become the principal tool to reduce the influence of errors when fitting models to given observations. Today, applications of least squares arise in a great number of scientific areas, such as statistics, geodetics, signal processing, and control. In the last 20 years there has been a great increase in the capacity for automatic data capturing and computing. Least squares problems of large size are now routinely solved. Tremendous progress has been made in numerical methods for least squares problems, in particular for generalized and modified least squares problems and direct and iterative methods for sparse problems. Until now there has not been a monograph that covers the full spectrum of relevant problems and methods in least squares. This volume gives an in-depth treatment of topics such as methods for sparse least squares problems, iterative methods, modified least squares, weighted problems, and constrained and regularized problems. The more than 800 references provide a comprehensive survey of the available literature on the subject.
Book Synopsis Robust Preconditioning for Sparse Linear Systems by : Edmond Ten-Fu Chow
Download or read book Robust Preconditioning for Sparse Linear Systems written by Edmond Ten-Fu Chow and published by . This book was released on 1997 with total page 372 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Preconditioning Techniques for Sparse Linear Systems by : Shaked Regev
Download or read book Preconditioning Techniques for Sparse Linear Systems written by Shaked Regev and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We present two novel methods, SSAI and HyKKT, for sparse linear systems. The methods differ in that SSAI is meant to be an out of the box solver that is robust on many different types of Hermitian positive definite (HPD) linear systems and has a variant that can be used to solve general systems (including rectangular ones), but is not necessarily the best for a given linear system. This is due to the fact that SSAI in an effort to be general, ignores domain knowledge of the problem. In contrast, HyKKT is a specialized solver for very sparse symmetric indefinite linear systems with a Karush-Kuhn-Tucker (KKT) structure. These two methods represent the main parts of the thesis. Chapter 2 introduces SSAI, a method for solving a Hermitian positive definite linear system Ax=b, where A is an explicit sparse matrix (real or complex). A sparse approximate right inverse is computed and replaced by its symmetrization M, which is used as a left-right preconditioner in a modified version of the preconditioned conjugate-gradient method (PCG), where M is modified occasionally, if necessary, to make it more positive definite. Before symmetrization, M is formed column by column and can therefore be computed in parallel with no communication except at the beginning and end. PCG requires only matrix-vector multiplications with A and M (not solving a linear system with M), and so too can be carried out in parallel. We compare it with incomplete Cholesky factorization (the gold standard for PCG) and with a direct Cholesky factorization and solve on sparse matrices from various applications and show it is robust. For least-squares problems, we implement an analogous form of preconditioned Conjugate Gradient Least-Squares (PCGLS) and show it is also robust. The contributions of the work in Chapter 2 are summarized in Section 2.2. Chapter 3 introduces HyKKT, a solution strategy for the large indefinite linear systems arising in interior methods for nonlinear optimization. The method is suitable for implementation on hardware accelerators such as graphical processing units (GPUs). The current gold standard for sparse indefinite systems is the LBL factorization, where L is a lower triangular matrix and B is 1X1 or 2X2 block diagonal. However, this requires pivoting, which substantially increases communication cost and degrades performance on GPUs. Our approach solves a large indefinite system by solving multiple smaller positive definite systems, using an iterative solver on the Schur complement and an inner direct solve (via Cholesky factorization) within each iteration. Cholesky is stable without pivoting, thereby reducing communication and allowing reuse of the symbolic factorization. We demonstrate the practicality of our approach on large optimal power flow problems and show that it can efficiently utilize GPUs and outperform LBL factorization of the full system. The contributions of the work in Chapter 3 are summarized in Section 3.3.
Book Synopsis Iterative Incomplete Factorization Methods by : Valeri? Pavlovich Il?in
Download or read book Iterative Incomplete Factorization Methods written by Valeri? Pavlovich Il?in and published by World Scientific. This book was released on 1992 with total page 212 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to numerical methods for solving sparse linear algebra systems of very large dimension which arise in the implementation of the mesh approximations of the partial differential equations. Incomplete factorization is the basis of the wide class of preconditioning interative processes with acceleration by conjugate gradients or the Chebyshev technique. Different kinds of explicit and implicit algorithms are considered. Theoretical grounds of correctness and estimates of the convergence velocity of iterations are presented. Together with the results of experimental investigations for the typical examples, this book is the first on systematic studying of the incomplete factorization methods.
Book Synopsis Direct Methods for Sparse Matrices by : I. S. Duff
Download or read book Direct Methods for Sparse Matrices written by I. S. Duff and published by Oxford University Press. This book was released on 2017-02-10 with total page 539 pages. Available in PDF, EPUB and Kindle. Book excerpt: The subject of sparse matrices has its root in such diverse fields as management science, power systems analysis, surveying, circuit theory, and structural analysis. Efficient use of sparsity is a key to solving large problems in many fields. This second edition is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all examples in the first edition were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer architectures are now much more complex, requiring new ways of adapting algorithms to parallel environments with memory hierarchies. Because the area is such an important one to all of computational science and engineering, a huge amount of research has been done in the last 30 years, some of it by the authors themselves. This new research is integrated into the text with a clear explanation of the underlying mathematics and algorithms. New research that is described includes new techniques for scaling and error control, new orderings, new combinatorial techniques for partitioning both symmetric and unsymmetric problems, and a detailed description of the multifrontal approach to solving systems that was pioneered by the research of the authors and colleagues. This includes a discussion of techniques for exploiting parallel architectures and new work for indefinite and unsymmetric systems.
Book Synopsis Iterative Methods for Sparse Linear Systems by : Yousef Saad
Download or read book Iterative Methods for Sparse Linear Systems written by Yousef Saad and published by SIAM. This book was released on 2003-04-01 with total page 537 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Computing -- General.
Book Synopsis A Survey of Preconditioned Iterative Methods by : Are Magnus Bruaset
Download or read book A Survey of Preconditioned Iterative Methods written by Are Magnus Bruaset and published by Routledge. This book was released on 2018-12-13 with total page 175 pages. Available in PDF, EPUB and Kindle. Book excerpt: The problem of solving large, sparse, linear systems of algebraic equations is vital in scientific computing, even for applications originating from quite different fields. A Survey of Preconditioned Iterative Methods presents an up to date overview of iterative methods for numerical solution of such systems. Typically, the methods considered are w
Book Synopsis A Survey of Preconditioned Iterative Methods by : Are Magnus Bruaset
Download or read book A Survey of Preconditioned Iterative Methods written by Are Magnus Bruaset and published by Routledge. This book was released on 2018-12-13 with total page 140 pages. Available in PDF, EPUB and Kindle. Book excerpt: The problem of solving large, sparse, linear systems of algebraic equations is vital in scientific computing, even for applications originating from quite different fields. A Survey of Preconditioned Iterative Methods presents an up to date overview of iterative methods for numerical solution of such systems. Typically, the methods considered are w
Book Synopsis Iterative Methods and Preconditioning for Large and Sparse Linear Systems with Applications by : Daniele Bertaccini
Download or read book Iterative Methods and Preconditioning for Large and Sparse Linear Systems with Applications written by Daniele Bertaccini and published by CRC Press. This book was released on 2018-02-19 with total page 321 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 including a greater volume of data, but this requires the solution of larger and harder algebraic systems. In recent years, research has focused on the efficient solution of large sparse and/or structured systems generated by the discretization of numerical models by using iterative solvers.
Book Synopsis Numerical Analysis and Its Applications by : Lubin Vulkov
Download or read book Numerical Analysis and Its Applications written by Lubin Vulkov and published by Springer Science & Business Media. This book was released on 1997-02-26 with total page 628 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the First International Workshop on Numerical Analysis and Its Applications, WNAA'96, held in Rousse, Bulgaria, in June 1996. The 57 revised full papers presented were carefully selected and reviewed for inclusion in the volume; also included are 14 invited presentations. All in all, the book offers a wealth of new results and methods of numerical analysis applicable in computational science, particularly in computational physics and chemistry. The volume reflects that the cooperation of computer scientists, mathematicians and scientists provides new numerical tools for computational scientists and, at the same time, stimulates numerical analysis.
Book Synopsis Matrix Computations by : Gene H. Golub
Download or read book Matrix Computations written by Gene H. Golub and published by JHU Press. This book was released on 1996-10-15 with total page 734 pages. Available in PDF, EPUB and Kindle. Book excerpt: Revised and updated, the third edition of Golub and Van Loan's classic text in computer science provides essential information about the mathematical background and algorithmic skills required for the production of numerical software. This new edition includes thoroughly revised chapters on matrix multiplication problems and parallel matrix computations, expanded treatment of CS decomposition, an updated overview of floating point arithmetic, a more accurate rendition of the modified Gram-Schmidt process, and new material devoted to GMRES, QMR, and other methods designed to handle the sparse unsymmetric linear system problem.
Book Synopsis Threshold Incomplete Factorization as a Preconditioner of an Iterative Solution for Nonsymmetric Systems of Linear Equations by : Christian J. Corley
Download or read book Threshold Incomplete Factorization as a Preconditioner of an Iterative Solution for Nonsymmetric Systems of Linear Equations written by Christian J. Corley and published by . This book was released on 1988 with total page 66 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Iterative Solution of Symmetric Quasi-definite Linear Systems by : Dominique Orban
Download or read book Iterative Solution of Symmetric Quasi-definite Linear Systems written by Dominique Orban and published by SIAM. This book was released on 2017-04-07 with total page 101 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerous applications, including computational optimization and fluid dynamics, give rise to block linear systems of equations said to have the quasi-definite structure. In practical situations, the size or density of those systems can preclude a factorization approach, leaving only iterative methods as the solution technique. Known iterative methods, however, are not specifically designed to take advantage of the quasi-definite structure.? This book discusses the connection between quasi-definite systems and linear least-squares problems, the most common and best understood problems in applied mathematics, and explains how quasi-definite systems can be solved using tailored iterative methods for linear least squares (with half as much work!). To encourage researchers and students to use the software, it is provided in MATLAB, Python, and Julia.? The authors provide a concise account of the most well-known methods for symmetric systems and least-squares problems, research-level advances in the solution of problems with specific illustrations in optimization and fluid dynamics, and a website that hosts software in three languages.?
Book Synopsis Parallel Processing and Applied Mathematics by : Roman Wyrzykowski
Download or read book Parallel Processing and Applied Mathematics written by Roman Wyrzykowski and published by Springer. This book was released on 2016-04-05 with total page 622 pages. Available in PDF, EPUB and Kindle. Book excerpt: This two-volume set LNCS 9573 and LNCS 9574 constitutes the refereed proceedings of the 11th International Conference of Parallel Processing and Applied Mathematics, PPAM 2015, held in Krakow, Poland, in September 2015.The 111 revised full papers presented in both volumes were carefully reviewed and selected from 196 submissions. The focus of PPAM 2015 was on models, algorithms, and software tools which facilitate efficient and convenient utilization of modern parallel and distributed computing architectures, as well as on large-scale applications, including big data problems.
