Download or read book A Framework of Conjugate Direction Methods for Symmetric Linear Systems in Optimization written by Giovanni Fasano and published by . This book was released on 2014 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper we introduce a parameter dependent class of Krylov-based methods, namely CD, for the solution of symmetric linear systems. We give evidence that in our proposal we generate sequences of conjugate directions, extending some properties of the standard Conjugate Gradient (CG) method, in order to preserve the conjugacy. For specific values of the parameters in our framework we obtain schemes equivalent to both the CG and the scaled-CG. We also prove the finite convergence of the algorithms in CD, and we provide some error analysis. Finally, preconditioning is introduced for CD, and we show that standard error bounds for the preconditioned CG also hold for the preconditioned CD.
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.
Download or read book Conjugate Direction Methods in Optimization written by M.R. Hestenes and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 334 pages. Available in PDF, EPUB and Kindle. Book excerpt: Shortly after the end of World War II high-speed digital computing machines were being developed. It was clear that the mathematical aspects of com putation needed to be reexamined in order to make efficient use of high-speed digital computers for mathematical computations. Accordingly, under the leadership of Min a Rees, John Curtiss, and others, an Institute for Numerical Analysis was set up at the University of California at Los Angeles under the sponsorship of the National Bureau of Standards. A similar institute was formed at the National Bureau of Standards in Washington, D. C. In 1949 J. Barkeley Rosser became Director of the group at UCLA for a period of two years. During this period we organized a seminar on the study of solu tions of simultaneous linear equations and on the determination of eigen values. G. Forsythe, W. Karush, C. Lanczos, T. Motzkin, L. J. Paige, and others attended this seminar. We discovered, for example, that even Gaus sian elimination was not well understood from a machine point of view and that no effective machine oriented elimination algorithm had been developed. During this period Lanczos developed his three-term relationship and I had the good fortune of suggesting the method of conjugate gradients. We dis covered afterward that the basic ideas underlying the two procedures are essentially the same. The concept of conjugacy was not new to me. In a joint paper with G. D.
Download or read book On the Solution of Symmetric Linear Systems by the Preconditioned Conjugate Gradient Method written by Anna Psimarni and published by . This book was released on 1993 with total page 46 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Numerical Infinities and Infinitesimals in Optimization written by Yaroslav D. Sergeyev and published by Springer Nature. This book was released on 2022-07-05 with total page 372 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a friendly introduction to the paradigm and proposes a broad panorama of killing applications of the Infinity Computer in optimization: radically new numerical algorithms, great theoretical insights, efficient software implementations, and interesting practical case studies. This is the first book presenting to the readers interested in optimization the advantages of a recently introduced supercomputing paradigm that allows to numerically work with different infinities and infinitesimals on the Infinity Computer patented in several countries. One of the editors of the book is the creator of the Infinity Computer, and another editor was the first who has started to use it in optimization. Their results were awarded by numerous scientific prizes. This engaging book opens new horizons for researchers, engineers, professors, and students with interests in supercomputing paradigms, optimization, decision making, game theory, and foundations of mathematics and computer science. “Mathematicians have never been comfortable handling infinities... But an entirely new type of mathematics looks set to by-pass the problem... Today, Yaroslav Sergeyev, a mathematician at the University of Calabria in Italy solves this problem... ” MIT Technology Review “These ideas and future hardware prototypes may be productive in all fields of science where infinite and infinitesimal numbers (derivatives, integrals, series, fractals) are used.” A. Adamatzky, Editor-in-Chief of the International Journal of Unconventional Computing. “I am sure that the new approach ... will have a very deep impact both on Mathematics and Computer Science.” D. Trigiante, Computational Management Science. “Within the grossone framework, it becomes feasible to deal computationally with infinite quantities, in a way that is both new (in the sense that previously intractable problems become amenable to computation) and natural”. R. Gangle, G. Caterina, F. Tohmé, Soft Computing. “The computational features offered by the Infinity Computer allow us to dynamically change the accuracy of representation and floating-point operations during the flow of a computation. When suitably implemented, this possibility turns out to be particularly advantageous when solving ill-conditioned problems. In fact, compared with a standard multi-precision arithmetic, here the accuracy is improved only when needed, thus not affecting that much the overall computational effort.” P. Amodio, L. Brugnano, F. Iavernaro & F. Mazzia, Soft Computing
Download or read book Iterative Solution of Large Linear Systems written by David M. Young and published by Elsevier. This book was released on 2014-05-10 with total page 599 pages. Available in PDF, EPUB and Kindle. Book excerpt: Iterative Solution of Large Linear Systems describes the systematic development of a substantial portion of the theory of iterative methods for solving large linear systems, with emphasis on practical techniques. The focal point of the book is an analysis of the convergence properties of the successive overrelaxation (SOR) method as applied to a linear system where the matrix is "consistently ordered". Comprised of 18 chapters, this volume begins by showing how the solution of a certain partial differential equation by finite difference methods leads to a large linear system with a sparse matrix. The next chapter reviews matrix theory and the properties of matrices, as well as several theorems of matrix theory without proof. A number of iterative methods, including the SOR method, are then considered. Convergence theorems are also given for various iterative methods under certain assumptions on the matrix A of the system. Subsequent chapters deal with the eigenvalues of the SOR method for consistently ordered matrices; the optimum relaxation factor; nonstationary linear iterative methods; and semi-iterative methods. This book will be of interest to students and practitioners in the fields of computer science and applied mathematics.
Download or read book The Newton Cauchy Framework written by John L. Nazareth and published by Springer Science & Business Media. This book was released on 1994-02-28 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational unconstrained nonlinear optimization comes to life from a study of the interplay between the metric-based (Cauchy) and model-based (Newton) points of view. The motivating problem is that of minimizing a convex quadratic function. This research monograph reveals for the first time the essential unity of the subject. It explores the relationships between the main methods, develops the Newton-Cauchy framework and points out its rich wealth of algorithmic implications and basic conceptual methods. The monograph also makes a valueable contribution to unifying the notation and terminology of the subject. It is addressed topractitioners, researchers, instructors, and students and provides a useful and refreshing new perspective on computational nonlinear optimization.
Download or read book Optimization and Control with Applications written by Liqun Qi and published by Springer Science & Business Media. This book was released on 2006-03-30 with total page 587 pages. Available in PDF, EPUB and Kindle. Book excerpt: A collection of 28 refereed papers grouped according to four broad topics: duality and optimality conditions, optimization algorithms, optimal control, and variational inequality and equilibrium problems. Suitable for researchers, practitioners and postgrads.
Download or read book Numerical Optimization written by Jorge Nocedal and published by Springer Science & Business Media. This book was released on 2006-12-11 with total page 686 pages. Available in PDF, EPUB and Kindle. Book excerpt: Optimization is an important tool used in decision science and for the analysis of physical systems used in engineering. One can trace its roots to the Calculus of Variations and the work of Euler and Lagrange. This natural and reasonable approach to mathematical programming covers numerical methods for finite-dimensional optimization problems. It begins with very simple ideas progressing through more complicated concepts, concentrating on methods for both unconstrained and constrained optimization.
Download or read book Templates for the Solution of Linear Systems written by Richard Barrett and published by SIAM. This book was released on 1994-01-01 with total page 141 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, which focuses on the use of iterative methods for solving large sparse systems of linear equations, templates are introduced to meet the needs of both the traditional user and the high-performance specialist. Templates, a description of a general algorithm rather than the executable object or source code more commonly found in a conventional software library, offer whatever degree of customization the user may desire. Templates offer three distinct advantages: they are general and reusable; they are not language specific; and they exploit the expertise of both the numerical analyst, who creates a template reflecting in-depth knowledge of a specific numerical technique, and the computational scientist, who then provides "value-added" capability to the general template description, customizing it for specific needs. For each template that is presented, the authors provide: a mathematical description of the flow of algorithm; discussion of convergence and stopping criteria to use in the iteration; suggestions for applying a method to special matrix types; advice for tuning the template; tips on parallel implementations; and hints as to when and why a method is useful.
Download or read book Parameter Free Iterative Linear Solvers written by Rüdiger Weiss and published by Wiley-VCH. This book was released on 1996-08-15 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a unifying theoretical framework for iterative solvers for linear systems. The focus is on parameter-free methods, i. e. methods that can be applied without a priori knowledge of inner properties of the systems. Despite the generality of the approach convergence estimates and geometrical properties can be derived. State-of-the-art solvers are surveyed and classified. Many new relations between these methods are furnished leading to a deeper insight. Existing gaps in the theory are pointed out and some visions for the design of more efficient and robust solvers are given. A valuation of established methods based on the theoretical results and on numerical tests is given. Recipes for the application of particular techniques are derived. The criteria are efficiency, robustness and the possibility of parallelization. This book has a triple aim: to simplify the analysis by a unified theory, to survey and valuate state-of-the-art solvers and to give an introduction to iterative solvers for non-experts in the field.
Download or read book Encyclopedia of Optimization written by Christodoulos A. Floudas and published by Springer Science & Business Media. This book was released on 2008-09-04 with total page 4646 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of the Encyclopedia of Optimization is to introduce the reader to a complete set of topics that show the spectrum of research, the richness of ideas, and the breadth of applications that has come from this field. The second edition builds on the success of the former edition with more than 150 completely new entries, designed to ensure that the reference addresses recent areas where optimization theories and techniques have advanced. Particularly heavy attention resulted in health science and transportation, with entries such as "Algorithms for Genomics", "Optimization and Radiotherapy Treatment Design", and "Crew Scheduling".
Download or read book Iterative Methods for Solving Linear Systems written by Anne Greenbaum and published by SIAM. This book was released on 1997-01-01 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Computing -- Numerical Analysis.
Download or read book Conjugate gradient methods for partial differential equations written by Rati Chandra and published by . This book was released on 1978 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt: The method of conjugate gradients is known as an efficient iterative method for solving large sparse symmetric positive definite systems of linear equations. Such systems arise frequently in scientific computing, as in the solution of self-adjoint elliptic partial differential equations using finite difference and finite element techniques.
Download or read book The Lanczos and Conjugate Gradient Algorithms written by Gerard Meurant and published by SIAM. This book was released on 2006-01-01 with total page 380 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Lanczos and conjugate gradient (CG) algorithms are fascinating numerical algorithms. This book presents the most comprehensive discussion to date of the use of these methods for computing eigenvalues and solving linear systems in both exact and floating point arithmetic. The author synthesizes the research done over the past 30 years, describing and explaining the "average" behavior of these methods and providing new insight into their properties in finite precision. Many examples are given that show significant results obtained by researchers in the field. The author emphasizes how both algorithms can be used efficiently in finite precision arithmetic, regardless of the growth of rounding errors that occurs. He details the mathematical properties of both algorithms and demonstrates how the CG algorithm is derived from the Lanczos algorithm. Loss of orthogonality involved with using the Lanczos algorithm, ways to improve the maximum attainable accuracy of CG computations, and what modifications need to be made when the CG method is used with a preconditioner are addressed.
Download or read book Distributed Optimization and Statistical Learning Via the Alternating Direction Method of Multipliers written by Stephen Boyd and published by Now Publishers Inc. This book was released on 2011 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: Surveys the theory and history of the alternating direction method of multipliers, and discusses its applications to a wide variety of statistical and machine learning problems of recent interest, including the lasso, sparse logistic regression, basis pursuit, covariance selection, support vector machines, and many others.
Download or read book Iterative Methods for Large Linear Systems written by David Ronald Kincaid and published by . This book was released on 1990 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: Very Good,No Highlights or Markup,all pages are intact.
