Dynamical systems : method and applications : theoretical developments and numerical examples
Demonstrates the application of DSM to solve a broad range of operator equations The dynamical systems method (DSM) is a powerful computational method for solving operator equations. With this book as their guide, readers will master the application of DSM to solve a variety of linear and nonlinear...
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| Main Authors: | , |
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| Format: | eBook Book |
| Language: | English |
| Published: |
Hoboken, N.J
Wiley
2012
John Wiley & Sons, Incorporated Wiley-Blackwell |
| Edition: | 1 |
| Subjects: | |
| ISBN: | 1118024281, 9781118024287, 9781118199619, 1118199618 |
| Online Access: | Get full text |
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Table of Contents:
- Intro -- Dynamical Systems Method and Applications: Theoretical Developments and Numerical Examples -- CONTENTS -- List of Figures -- List of Tables -- Preface -- Acknowledgments -- PART I -- 1 Introduction -- 1.1 What this book is about -- 1.2 What the DSM (Dynamical Systems Method) is -- 1.3 The scope of the DSM -- 1.4 A discussion of DSM -- 1.5 Motivations -- 2 III-posed problems -- 2.1 Basic definitions. Examples -- 2.2 Variational regularization -- 2.3 Quasi-solutions -- 2.4 Iterative regularization -- 2.5 Quasi-inversion -- 2.6 Dynamical systems method (DSM) -- 2.7 Variational regularization for nonlinear equations -- 3 DSM for well-posed problems -- 3.1 Every solvable well-posed problem can be solved by DSM -- 3.2 DSM and Newton-type methods -- 3.3 DSM and the modified Newton's method -- 3.4 DSM and Gauss-Newton-type methods -- 3.5 DSM and the gradient method -- 3.6 DSM and the simple iterations method -- 3.7 DSM and minimization methods -- 3.8 Ulm's method -- 4 DSM and linear ill-posed problems -- 4.1 Equations with bounded operators -- 4.2 Another approach -- 4.3 Equations with unbounded operators -- 4.4 Iterative methods -- 4.5 Stable calculation of values of unbounded operators -- 5 Some inequalities -- 5.1 Basic nonlinear differential inequality -- 5.2 An operator inequality -- 5.3 A nonlinear inequality -- 5.4 The Gronwall-type inequalities -- 5.5 Another operator inequality -- 5.6 A generalized version of the basic nonlinear inequality -- 5.6.1 Formulations and results -- 5.6.2 Applications -- 5.7 Some nonlinear inequalities and applications -- 5.7.1 Formulations and results -- 5.7.2 Applications -- 6 DSM for monotone operators -- 6.1 Auxiliary results -- 6.2 Formulation of the results and proofs -- 6.3 The case of noisy data -- 7 DSM for general nonlinear operator equations -- 7.1 Formulation of the problem. The results and proofs
- 24.2.1 Numerical experiments for computing second derivative -- 24.3 Numerical experiments with an image restoration problem -- 24.4 Numerical experiments with Volterra integral equations of the first kind -- 24.4.1 Numerical experiments with an inverse problem for the heat equation -- 24.5 Numerical experiments with numerical differentiation -- 24.5.1 The first approach -- 24.5.2 The second approach -- 25 Stable solutions of Hammerstein-type integral equations -- 25.1 DSM of Newton type -- 25.1.1 An experiment with an operator defined on H = L2[0, 1] -- 25.1.2 An experiment with an operator defined on a dense subset of H = L2[0, 1] -- 25.2 DSM of gradient type -- 25.3 DSM of simple iteration type -- 26 Inversion of the Laplace transform from the real axis using an adaptive iterative method -- 26.1 Introduction -- 26.2 Description of the method -- 26.2.1 Noisy data -- 26.2.2 Stopping rule -- 26.2.3 The algorithm -- 26.3 Numerical experiments -- 26.3.1 The parameters k, a0, d -- 26.3.2 Experiments -- 26.4 Conclusion -- Appendix A: Auxiliary results from analysis -- A.l Contraction mapping principle -- A.2 Existence and uniqueness of the local solution to the Cauchy problem -- A.3 Derivatives of nonlinear mappings -- A.4 Implicit function theorem -- A.5 An existence theorem -- A.6 Continuity of solutions to operator equations with respect to a parameter -- A.7 Monotone operators in Banach spaces -- A.8 Existence of solutions to operator equations -- A.9 Compactness of embeddings -- Appendix B: Bibliographical notes -- References -- Index
- 7.2 Noisy data -- 7.3 Iterative solution -- 7.4 Stability of the iterative solution -- 8 DSM for operators satisfying a spectral assumption -- 8.1 Spectral assumption -- 8.2 Existence of a solution to a nonlinear equation -- 9 DSM in Banach spaces -- 9.1 Well-posed problems -- 9.2 Ill-posed problems -- 9.3 Singular perturbation problem -- 10 DSM and Newton-type methods without inversion of the derivative -- 10.1 Well-posed problems -- 10.2 Ill-posed problems -- 11 DSM and unbounded operators -- 11.1 Statement of the problem -- 11.2 Ill-posed problems -- 12 DSM and nonsmooth operators -- 12.1 Formulation of the results -- 12.2 Proofs -- 13 DSM as a theoretical tool -- 13.1 Surjectivity of nonlinear maps -- 13.2 When is a local homeomorphism a global one? -- 14 DSM and iterative methods -- 14.1 Introduction -- 14.2 Iterative solution of well-posed problems -- 14.3 Iterative solution of ill-posed equations with monotone operator -- 14.4 Iterative methods for solving nonlinear equations -- 14.5 Ill-posed problems -- 15 Numerical problems arising in applications -- 15.1 Stable numerical differentiation -- 15.2 Stable differentiation of piecewise-smooth functions -- 15.3 Simultaneous approximation of a function and its derivative by interpolation polynomials -- 15.4 Other methods of stable differentiation -- 15.5 DSM and stable differentiation -- 15.6 Stable calculating singular integrals -- PART II -- 16 Solving linear operator equations by a Newton-type DSM -- 16.1 An iterative scheme for solving linear operator equations -- 16.2 DSM with fast decaying regularizing function -- 17 DSM of gradient type for solving linear operator equations -- 17.1 Formulations and Results -- 17.1.1 Exact data -- 17.1.2 Noisy data fδ -- 17.1.3 Discrepancy principle -- 17.2 Implementation of the Discrepancy Principle -- 17.2.1 Systems with known spectral decomposition
- 17.2.2 On the choice of t0 -- 18 DSM for solving linear equations with finite-rank operators -- 18.1 Formulation and results -- 18.1.1 Exact data -- 18.1.2 Noisy data fδ -- 18.1.3 Discrepancy principle -- 18.1.4 An iterative scheme -- 18.1.5 An iterative scheme with a stopping rule based on a discrepancy principle -- 18.1.6 Computing uδ(tδ) -- 19 A discrepancy principle for equations with monotone continuous operators -- 19.1 Auxiliary results -- 19.2 A discrepancy principle -- 19.3 Applications -- 20 DSM of Newton-type for solving operator equations with minimal smoothness assumptions -- 20.1 DSM of Newton-type -- 20.1.1 Inverse function theorem -- 20.1.2 Convergence of the DSM -- 20.1.3 The Newton method -- 20.2 A justification of the DSM for global homeomorphisms -- 20.3 DSM of Newton-type for solving nonlinear equations with monotone operators -- 20.3.1 Existence of solution and a justification of the DSM for exact data -- 20.3.2 Solving equations with monotone operators when the data are noisy -- 20.4 Implicit Function Theorem and the DSM -- 20.4.1 Example -- 21 DSM of gradient type -- 21.1 Auxiliary results -- 21.2 DSM gradient method -- 21.3 An iterative scheme -- 22 DSM of simple iteration type -- 22.1 DSM of simple iteration type -- 22.1.1 Auxiliary results -- 22.1.2 Main results -- 22.2 An iterative scheme for solving equations with σ-inverse monotone operators -- 22.2.1 Auxiliary results -- 22.2.2 Main results -- 23 DSM for solving nonlinear operator equations in Banach spaces -- 23.1 Proofs -- 23.2 The case of continuous F'(u) -- PART III -- 24 Solving linear operator equations by the DSM -- 24.1 Numerical experiments with ill-conditioned linear algebraic systems -- 24.1.1 Numerical experiments with Hilbert matrix -- 24.2 Numerical experiments with Fredholm integral equations of the first kind