# Dynamical Systems Method and Applications : Theoretical Developments and Numerical Examples.

Publisher: New York : John Wiley & Sons, Incorporated, 2011Copyright date: ©2012Edition: 1st edDescription: 1 online resource (572 pages)Content type:- text

- computer

- online resource

- 9781118199596

- 515/.35

- QA614.8 -- .R35 2012eb

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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.

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.

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.

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 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 then sets forth the scope of DSM in Part One. Part Two introduces the discrepancy principle, and Part Three offers examples of numerical applications of DSM to solve a broad range of problems in science and engineering. Additional featured topics include: General nonlinear operator equations Operators satisfying a spectral assumption Newton-type methods without inversion of the derivative Numerical problems arising in applications Stable numerical differentiation Stable solution to ill-conditioned linear algebraic systems Throughout the chapters, the authors employ the use of figures and tables to help readers grasp and apply new concepts. Numerical examples offer original theoretical results based on the solution of practical problems involving ill-conditioned linear algebraic systems, and stable differentiation of noisy data. Written by internationally recognized authorities on the topic, Dynamical Systems Method and Applications is an excellent book for courses on numerical analysis, dynamical systems, operator theory, and applied mathematics at the graduate level. The book also serves as a valuable resource for professionals in the fields of mathematics, physics, and engineering.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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