Download or read book Numerical Solution of Saddle Point Problems by Projection written by Gul Karaduman and published by . This book was released on 2020 with total page 67 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we work on iterative solutions of large linear systems of saddle point problems of the form where A ∈ Rn×n, B1, B2 ∈ Rm×n, f ∈ Rn, and n ≥ m. Many applications in computational sciences and engineering give rise to saddle point problems such as finite element approximations to Stokes problems, image reconstruction, tomography, genetics, statistics and model order reduction for dynamical systems. Such problems are typically large and sparse. We develop new techniques to solve the saddle point problems depending on the rank of B2. First, we deal with the case when B2 has full row rank, i.e., rank (B2) = m. The key idea is to construct a projection matrix and transform the original problem to a least squares problem then solve the least squares problem by using one of the iterative methods such as LSMR. In most applications B2 has full rank, but not always. Next, we turn to the saddle point systems with the rank-deficient matrix B2. Similarly we construct a new projection matrix by using only maximal linearly independent rows of B2. By using this projection matrix, the original problem can still be transformed into a least squares problem. Again, the new system can be solved by using one of the iterative techniques for least squares problems. Numerical experiments show that the new iterative solution techniques work very well for large sparse saddle point systems with both full rank and rank-deficient matrix B2.

Download or read book Saddle Point Problems and Their Iterative Solution written by Miroslav Rozložník and published by Springer. This book was released on 2018-11-19 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides essential lecture notes on solving large linear saddle-point systems, which arise in a wide range of applications and often pose computational challenges in science and engineering. The focus is on discussing the particular properties of such linear systems, and a large selection of algebraic methods for solving them, with an emphasis on iterative methods and preconditioning. The theoretical results presented here are complemented by a case study on potential fluid flow problem in a real world-application. This book is mainly intended for students of applied mathematics and scientific computing, but also of interest for researchers and engineers working on various applications. It is assumed that the reader has completed a basic course on linear algebra and numerical mathematics.

Download or read book Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations written by Tarek Mathew and published by Springer Science & Business Media. This book was released on 2008-06-25 with total page 775 pages. Available in PDF, EPUB and Kindle. Book excerpt: Domain decomposition methods are divide and conquer computational methods for the parallel solution of partial differential equations of elliptic or parabolic type. The methodology includes iterative algorithms, and techniques for non-matching grid discretizations and heterogeneous approximations. This book serves as a matrix oriented introduction to domain decomposition methodology. A wide range of topics are discussed include hybrid formulations, Schwarz, and many more.

Download or read book Numerical Methods for Nonsmooth Dynamical Systems written by Vincent Acary and published by Springer Science & Business Media. This book was released on 2008-01-30 with total page 529 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book concerns the numerical simulation of dynamical systems whose trajec- ries may not be differentiable everywhere. They are named nonsmooth dynamical systems. They make an important class of systems, rst because of the many app- cations in which nonsmooth models are useful, secondly because they give rise to new problems in various elds of science. Usually nonsmooth dynamical systems are represented as differential inclusions, complementarity systems, evolution va- ational inequalities, each of these classes itself being split into several subclasses. The book is divided into four parts, the rst three parts being sketched in Fig. 0. 1. The aim of the rst part is to present the main tools from mechanics and applied mathematics which are necessary to understand how nonsmooth dynamical systems may be numerically simulated in a reliable way. Many examples illustrate the th- retical results, and an emphasis is put on mechanical systems, as well as on electrical circuits (the so-called Filippov’s systems are also examined in some detail, due to their importance in control applications). The second and third parts are dedicated to a detailed presentation of the numerical schemes. A fourth part is devoted to the presentation of the software platform Siconos. This book is not a textbook on - merical analysis of nonsmooth systems, in the sense that despite the main results of numerical analysis (convergence, order of consistency, etc. ) being presented, their proofs are not provided.

Download or read book Variational Methods for the Numerical Solution of Nonlinear Elliptic Problem written by Roland Glowinski and published by SIAM. This book was released on 2015-11-04 with total page 473 pages. Available in PDF, EPUB and Kindle. Book excerpt: Variational Methods for the Numerical Solution of Nonlinear Elliptic Problems addresses computational methods that have proven efficient for the solution of a large variety of nonlinear elliptic problems. These methods can be applied to many problems in science and engineering, but this book focuses on their application to problems in continuum mechanics and physics. This book differs from others on the topic by presenting examples of the power and versatility of operator-splitting methods; providing a detailed introduction to alternating direction methods of multipliers and their applicability to the solution of nonlinear (possibly nonsmooth) problems from science and engineering; and showing that nonlinear least-squares methods, combined with operator-splitting and conjugate gradient algorithms, provide efficient tools for the solution of highly nonlinear problems. The book provides useful insights suitable for advanced graduate students, faculty, and researchers in applied and computational mathematics as well as research engineers, mathematical physicists, and systems engineers.

Download or read book Numerical Solution of Partial Differential Equations Theory Algorithms and Their Applications written by Oleg P. Iliev and published by Springer Science & Business Media. This book was released on 2013-06-04 with total page 334 pages. Available in PDF, EPUB and Kindle. Book excerpt: One of the current main challenges in the area of scientific computing​ is the design and implementation of accurate numerical models for complex physical systems which are described by time dependent coupled systems of nonlinear PDEs. This volume integrates the works of experts in computational mathematics and its applications, with a focus on modern algorithms which are at the heart of accurate modeling: adaptive finite element methods, conservative finite difference methods and finite volume methods, and multilevel solution techniques. Fundamental theoretical results are revisited in survey articles and new techniques in numerical analysis are introduced. Applications showcasing the efficiency, reliability and robustness of the algorithms in porous media, structural mechanics and electromagnetism are presented. Researchers and graduate students in numerical analysis and numerical solutions of PDEs and their scientific computing applications will find this book useful.

Download or read book Efficient Iterative Methods for Saddle Point Problems written by Vivek Sarin and published by . This book was released on 1997 with total page 86 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "This thesis investigates efficient iterative methods for a type of saddle-point problem, namely the generalized Stokes problem, which arises frequently in the simulation of time-dependent Navier-Stokes equations for incompressible fluid flow. These systems are indefinite due to a set of linear constraints on the velocity, causing difficulty for most preconditioners and iterative methods. A multilevel algorithm is proposed for the solution of such systems, which uses a novel technique for the construction of a basis for the space satisfying the constraints. The proposed algorithm achieves faster convergence on account of implicit preconditioning of the linear system, and can be implemented efficiently on parallel processors. Along with a scalable parallel implementation described in the thesis, the multilevel algorithm yields a competitive parallel preconditioned iterative method for the solution of these problems."

Download or read book Numerical Analysis of Some Saddle Point Formulation with X FEM Type Approximation on Cracked Or Fictitious Domains written by Saber Amdouni and published by . This book was released on 2013 with total page 123 pages. Available in PDF, EPUB and Kindle. Book excerpt: This Ph.D. thesis was done in collaboration with "La Manufacture Française des Pneumatiques Michelin". It concerns the mathematical and numerical analysis of convergence and stability of mixed or hybrid formulation of constrained optimization problem with Lagrange multiplier method in the framework of the eXtended Finite Element Method (XFEM). First we try to prove the stability of the X-FEM discretization for incompressible elastostatic problem by ensured a LBB condition. The second axis, which present the main content of the thesis, is dedicated to the use of some stabilized Lagrange multiplier methods. The particularity of these stabilized methods is that the stability of the multiplier is provided by adding supplementary terms in the weak formulation. In this context, we study the Barbosa-Hughes stabilization technique applied to the frictionless unilateral contact problem with XFEM-cut-off. Then we present a new consistent method based on local projections for the stabilization of a Dirichlet condition in the framework of extended finite element method with a fictitious domain approach. Moreover we make comparative study between the local projection stabilization and the Barbosa-Hughes stabilization. Finally we use the local projection stabilization to approximate the two-dimensional linear elastostatics unilateral contact problem with Tresca frictional in the framework of the eXtended Finite Element Method X-FEM.

Download or read book Handbook of Numerical Analysis written by Philippe G. Ciarlet and published by Gulf Professional Publishing. This book was released on 1990 with total page 1187 pages. Available in PDF, EPUB and Kindle. Book excerpt: Includes following subjects: Solution of equations in Rn, Finite difference methods, Finite element methods, Techniques of scientific computing, Optimization theory and systems science, Numerical methods for fluids, Numerical methods for solids, Specific applications

Download or read book Numerical Methods in Turbulence Simulation written by Robert Moser and published by Elsevier. This book was released on 2022-11-30 with total page 568 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical Methods in Turbulence Simulation provides detailed specifications of the numerical methods needed to solve important problems in turbulence simulation. Numerical simulation of turbulent fluid flows is challenging because of the range of space and time scales that must be represented. This book provides explanations of the numerical error and stability characteristics of numerical techniques, along with treatments of the additional numerical challenges that arise in large eddy simulations. Chapters are written as tutorials by experts in the field, covering specific both contexts and applications. Three classes of turbulent flow are addressed, including incompressible, compressible and reactive, with a wide range of the best numerical practices covered. A thorough introduction to the numerical methods is provided for those without a background in turbulence, as is everything needed for a thorough understanding of the fundamental equations. The small scales that must be resolved are generally not localized around some distinct small-scale feature, but instead are distributed throughout a volume. These characteristics put particular strain on the numerical methods used to simulate turbulent flows. Includes a detailed review of the numerical approximation issues that impact the simulation of turbulence Provides a range of examples of large eddy simulation techniques Discusses the challenges posed by boundary conditions in turbulence simulation and provides approaches to addressing them

Download or read book Domain Decomposition Methods 10 written by Jan Mandel and published by American Mathematical Soc.. This book was released on 1998 with total page 569 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the Tenth International Conference on Domain Decomposition Methods, which focused on the latest developments in realistic applications in structural mechanics, structural dynamics, computational fluid dynamics, and heat transfer. The proceedings of these conferences have become standard references in the field and contain seminal papers as well as the latest theoretical results and reports on practical applications.

Download or read book Milestones in Matrix Computation written by Gene Howard Golub and published by Oxford University Press. This book was released on 2007-02-22 with total page 581 pages. Available in PDF, EPUB and Kindle. Book excerpt: The text presents and discusses some of the most influential papers in Matrix Computation authored by Gene H. Golub, one of the founding fathers of the field. Including commentaries by leading experts and a brief biography, this text will be of great interest to students and researchers in numerical analysis and scientific computation.

Download or read book Milestones in Matrix Computation The selected works of Gene H Golub with commentaries written by Raymond Chan and published by OUP Oxford. This book was released on 2007-02-22 with total page 584 pages. Available in PDF, EPUB and Kindle. Book excerpt: The text presents and discusses some of the most influential papers in Matrix Computation authored by Gene H. Golub, one of the founding fathers of the field. The collection of 21 papers is divided into five main areas: iterative methods for linear systems, solution of least squares problems, matrix factorizations and applications, orthogonal polynomials and quadrature, and eigenvalue problems. Commentaries for each area are provided by leading experts: Anne Greenbaum, Ake Bjorck, Nicholas Higham, Walter Gautschi, and G. W. (Pete) Stewart. Comments on each paper are also included by the original authors, providing the reader with historical information on how the paper came to be written and under what circumstances the collaboration was undertaken. Including a brief biography and facsimiles of the original papers, this text will be of great interest to students and researchers in numerical analysis and scientific computation.

Download or read book Mathematical Foundations of Finite Elements and Iterative Solvers written by SCI085000 and published by SIAM. This book was released on 2022-06-27 with total page 186 pages. Available in PDF, EPUB and Kindle. Book excerpt: “This book combines an updated look, at an advanced level, of the mathematical theory of the finite element method (including some important recent developments), and a presentation of many of the standard iterative methods for the numerical solution of the linear system of equations that results from finite element discretization, including saddle point problems arising from mixed finite element approximation. For the reader with some prior background in the subject, this text clarifies the importance of the essential ideas and provides a deeper understanding of how the basic concepts fit together.” — Richard S. Falk, Rutgers University “Students of applied mathematics, engineering, and science will welcome this insightful and carefully crafted introduction to the mathematics of finite elements and to algorithms for iterative solvers. Concise, descriptive, and entertaining, the text covers all of the key mathematical ideas and concepts dealing with finite element approximations of problems in mechanics and physics governed by partial differential equations while interweaving basic concepts on Sobolev spaces and basic theorems of functional analysis presented in an effective tutorial style.” — J. Tinsley Oden, The University of Texas at Austin This textbook describes the mathematical principles of the finite element method, a technique that turns a (linear) partial differential equation into a discrete linear system, often amenable to fast linear algebra. Reflecting the author’s decade of experience in the field, Mathematical Foundations of Finite Elements and Iterative Solvers examines the crucial interplay between analysis, discretization, and computations in modern numerical analysis; furthermore, it recounts historical developments leading to current state-of-the-art techniques. While self-contained, this textbook provides a clear and in-depth discussion of several topics, including elliptic problems, continuous Galerkin methods, iterative solvers, advection-diffusion problems, and saddle point problems. Accessible to readers with a beginning background in functional analysis and linear algebra, this text can be used in graduate-level courses on advanced numerical analysis, data science, numerical optimization, and approximation theory. Professionals in numerical analysis and finite element methods will also find the book of interest.

Download or read book Acta Numerica 2005 Volume 14 written by Arieh Iserles and published by Cambridge University Press. This book was released on 2005-06-30 with total page 584 pages. Available in PDF, EPUB and Kindle. Book excerpt: A high-impact factor, prestigious annual publication containing invited surveys by subject leaders: essential reading for all practitioners and researchers.

Download or read book Optimization on Solution Sets of Common Fixed Point Problems written by Alexander J. Zaslavski and published by Springer Nature. This book was released on 2021-08-09 with total page 434 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to a detailed study of the subgradient projection method and its variants for convex optimization problems over the solution sets of common fixed point problems and convex feasibility problems. These optimization problems are investigated to determine good solutions obtained by different versions of the subgradient projection algorithm in the presence of sufficiently small computational errors. The use of selected algorithms is highlighted including the Cimmino type subgradient, the iterative subgradient, and the dynamic string-averaging subgradient. All results presented are new. Optimization problems where the underlying constraints are the solution sets of other problems, frequently occur in applied mathematics. The reader should not miss the section in Chapter 1 which considers some examples arising in the real world applications. The problems discussed have an important impact in optimization theory as well. The book will be useful for researches interested in the optimization theory and its applications.

Download or read book Finite Elements and Fast Iterative Solvers written by Howard Elman and published by OUP Oxford. This book was released on 2014-06-19 with total page 495 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a description of why and how to do Scientific Computing for fundamental models of fluid flow. It contains introduction, motivation, analysis, and algorithms and is closely tied to freely available MATLAB codes that implement the methods described. The focus is on finite element approximation methods and fast iterative solution methods for the consequent linear(ized) systems arising in important problems that model incompressible fluid flow. The problems addressed are the Poisson equation, Convection-Diffusion problem, Stokes problem and Navier-Stokes problem, including new material on time-dependent problems and models of multi-physics. The corresponding iterative algebra based on preconditioned Krylov subspace and multigrid techniques is for symmetric and positive definite, nonsymmetric positive definite, symmetric indefinite and nonsymmetric indefinite matrix systems respectively. For each problem and associated solvers there is a description of how to compute together with theoretical analysis that guides the choice of approaches and describes what happens in practice in the many illustrative numerical results throughout the book (computed with the freely downloadable IFISS software). All of the numerical results should be reproducible by readers who have access to MATLAB and there is considerable scope for experimentation in the "computational laboratory " provided by the software. Developments in the field since the first edition was published have been represented in three new chapters covering optimization with PDE constraints (Chapter 5); solution of unsteady Navier-Stokes equations (Chapter 10); solution of models of buoyancy-driven flow (Chapter 11). Each chapter has many theoretical problems and practical computer exercises that involve the use of the IFISS software. This book is suitable as an introduction to iterative linear solvers or more generally as a model of Scientific Computing at an advanced undergraduate or beginning graduate level.