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1st ed 
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1 online resource (572 pages) 

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computer c rdamedia 

online resource cr rdacarrier 
附註 
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 IIIposed problems  2.1 Basic definitions. Examples  2.2 Variational regularization  2.3 Quasisolutions  2.4 Iterative regularization  2.5 Quasiinversion  2.6 Dynamical systems method (DSM)  2.7 Variational regularization for nonlinear equations  3 DSM for wellposed problems  3.1 Every solvable wellposed problem can be solved by DSM  3.2 DSM and Newtontype methods  3.3 DSM and the modified Newton's method  3.4 DSM and GaussNewtontype 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 illposed 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 Gronwalltype 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 Wellposed problems  9.2 Illposed problems  9.3 Singular perturbation problem  10 DSM and Newtontype methods without inversion of the derivative  10.1 Wellposed problems  10.2 Illposed problems  11 DSM and unbounded operators  11.1 Statement of the problem  11.2 Illposed 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 wellposed problems  14.3 Iterative solution of illposed equations with monotone operator  14.4 Iterative methods for solving nonlinear equations  14.5 Illposed problems  15 Numerical problems arising in applications  15.1 Stable numerical differentiation  15.2 Stable differentiation of piecewisesmooth 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 Newtontype 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 finiterank 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 Newtontype for solving operator equations with minimal smoothness assumptions  20.1 DSM of Newtontype  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 Newtontype 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 σinverse 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 illconditioned 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 Hammersteintype 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 illposed and wellposed problems. The authors offer a clear, stepbystep, 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 Newtontype methods without inversion of the derivative Numerical problems arising in applications Stable numerical differentiation Stable solution to illconditioned 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 illconditioned 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 

Description based on publisher supplied metadata and other sources 

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries 
鏈接 
Print version: Ramm, Alexander G. Dynamical Systems Method and Applications : Theoretical Developments and Numerical Examples
New York : John Wiley & Sons, Incorporated,c2011 9781118024287

主題 
Differentiable dynamical systems


Electronic books

Alt Author 
Hoang, Nguyen S

