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Newton Methods for Nonlinear Problems

Affine Invariance and Adaptive Algorithms

  • Textbook
  • © 2011

Overview

Authors:
  1. Peter Deuflhard

    1. Zuse-Institut Berlin (ZIB), Berlin, Germany

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Part of the book series: Springer Series in Computational Mathematics (SSCM, volume 35)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    • Peter Deuflhard
    Pages 7-41

  3. Algebraic Equations

    1. Front Matter

      Pages 43-43
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  4. ALGEBRAIC EQUATIONS

    1. Systems of Equations: Local Newton Methods

      • Peter Deuflhard
      Pages 45-107

    2. Systems of Equations: Global Newton Methods

      • Peter Deuflhard
      Pages 109-172

    3. Least Squares Problems: Gauss-Newton Methods

      • Peter Deuflhard
      Pages 173-231

    4. Parameter Dependent Systems: Continuation Methods

      • Peter Deuflhard
      Pages 233-282

  5. Differential Equations

    1. Front Matter

      Pages 283-283
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  6. DIFFERENTIAL EQUATIONS

    1. Stiff ODE Initial Value Problems

      • Peter Deuflhard
      Pages 285-314

    2. ODE Boundary Value Problems

      • Peter Deuflhard
      Pages 315-368

    3. PDE Boundary Value Problems

      • Peter Deuflhard
      Pages 369-404

  7. Back Matter

    Pages 405-424
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Keywords

About this book

This book deals with the efficient numerical solution of challenging nonlinear problems in science and engineering, both in finite dimension (algebraic systems) and in infinite dimension (ordinary and partial differential equations). Its focus is on local and global Newton methods for direct problems or Gauss-Newton methods for inverse problems. The term 'affine invariance' means that the presented algorithms and their convergence analysis are invariant under one out of four subclasses of affine transformations of the problem to be solved. Compared to traditional textbooks, the distinguishing affine invariance approach leads to shorter theorems and proofs and permits the construction of fully adaptive algorithms. Lots of numerical illustrations, comparison tables, and exercises make the text useful in computational mathematics classes. At the same time, the book opens many directions for possible future research.

Reviews

From the reviews:

“This monograph covers a multitude of Newton methods and presents the algorithms and their convergence analysis from the perspective of affine invariance, which has been the subject of research by the author since 1970. … The book is intended for graduate students of mathematics and computational science and also for researchers in the area of numerical analysis and scientific computing. … As a research monograph, the book not only assembles the current state of the art, but also points to future research prospects.” (Gudula Runger, ACM Computing Reviews, June, 2012)

Authors and Affiliations

  • Zuse-Institut Berlin (ZIB), Berlin, Germany

    Peter Deuflhard

About the author

Peter Deuflhard is founder and head of the internationally renowned Zuse Institute Berlin (ZIB) and full professor of Numerical Analysis and Scientific Computing at the Free University of Berlin. He is a regular invited speaker at international conferences and universities as well as industry places all over the world.

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