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Dynamical Systems Method and Applications : Theoretical Developments and Numerical Examples.
Demonstrates the application of DSM to solve a broad range of operator equationsThe 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 p...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Hoboken :
John Wiley & Sons,
2012.
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| Temas: | |
| Acceso en línea: | Texto completo |
MARC
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| 100 | 1 | |a Ramm, A. G. |q (Alexander G.) |1 https://id.oclc.org/worldcat/entity/E39PBJht6c7MRHvFtQDQJ4MVmd | |
| 245 | 1 | 0 | |a Dynamical Systems Method and Applications : |b Theoretical Developments and Numerical Examples. |
| 260 | |a Hoboken : |b John Wiley & Sons, |c 2012. | ||
| 300 | |a 1 online resource (572 pages) | ||
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| 505 | 0 | |a 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. | |
| 505 | 8 | |a 3 DSM for well-posed problems3.1 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. | |
| 505 | 8 | |a 5.1 Basic nonlinear differential inequality5.2 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. | |
| 505 | 8 | |a 7.1 Formulation of the problem. The results and proofs7.2 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. | |
| 505 | 8 | |a 12 DSM and nonsmooth operators12.1 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. | |
| 500 | |a 15.3 Simultaneous approximation of a function and its derivative by interpolation polynomials. | ||
| 520 | |a Demonstrates the application of DSM to solve a broad range of operator equationsThe 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 problems as well as ill-posed and well-posed problems. The authors offer a clear, step-by-step, systematic development of DSM that enables readers to grasp the method's underlying logic and its numerous applications. Dynamical Systems Method and Applications begins with a general introduction and the. | ||
| 588 | 0 | |a Print version record. | |
| 504 | |a Includes bibliographical references and index. | ||
| 546 | |a English. | ||
| 590 | |a ProQuest Ebook Central |b Ebook Central Academic Complete | ||
| 650 | 0 | |a Differentiable dynamical systems. | |
| 650 | 6 | |a Dynamique différentiable. | |
| 650 | 7 | |a SCIENCE |x Mechanics |x Dynamics. |2 bisacsh | |
| 650 | 7 | |a Differentiable dynamical systems |2 fast | |
| 700 | 1 | |a Hoang, Nguyen S. | |
| 758 | |i has work: |a Dynamical systems method and applications (Text) |1 https://id.oclc.org/worldcat/entity/E39PCFCWxjQwqFgXwPkw3vfCkP |4 https://id.oclc.org/worldcat/ontology/hasWork | ||
| 776 | 0 | 8 | |i Print version: |a Ramm, Alexander G. |t Dynamical Systems Method and Applications : Theoretical Developments and Numerical Examples. |d Hoboken : John Wiley & Sons, ©2012 |z 9781118024287 |
| 856 | 4 | 0 | |u https://ebookcentral.uam.elogim.com/lib/uam-ebooks/detail.action?docID=817767 |z Texto completo |
| 880 | 8 | |6 505-00/(S |a 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. | |
| 880 | 8 | |6 505-00/(S |a 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. | |
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