Dynamical Systems Method and Applications

Dynamical Systems Method (DSM) is a powerful general method for solving operator equations. These equations can be linear or nonlinear or well-posed or ill-posed. The book presents a systematic development of the DSM, and theoretical results are illustrated by a number of numerical examples, which are of independent interest. These include: stable differentiation of noisy data, stable solution of ill-conditioned linear algebraic systems, stable solution of Fredholm and Volterra integral equations of the first kind, stable inversion of the Laplace transform from the real axis, solution of nonlinear integral equations, and other examples. Chapter coverage includes ill-posed problems; well-posed problems; linear ill-posed problems; inequalities; monotone operators; general nonlinear operator equations; operators satisfying a spectral assumption; Banach spaces; Newton-type methods without inversion of the derivative; unbound operators; nonsmooth operators; DSM as a theoretical tool; iterative methods; numerical problems arising in applications; auxiliary results from analysis; a discrepancy principle for solving equations with monotone operators; solving linear equations; stable numerical differentiation; deconvolution problems; numerical implementation; and stable solution to ill-conditioned linear algebraic systems.

Alexander G. Ramm, PhD, is Professor in the Department of Mathematics at Kansas State University. Dr. Ramm serves as associate editor for several journals.

Nguyen S. Hoang, PhD, is Visiting Assistant Professor in the Department of Mathematics at the University of Oklahoma. He has published numerous journal articles in the areas of numerical analysis, operator theory, ordinary and partial differential equations, optimization, and inverse and ill-posed problems.

PART I

1 Introduction 3

2 Ill-posed problems 11

3 DSM for well-posed problems 57

4 DSM and linear ill-posed problems 71

5 Some inequalities 93

6 DSM for monotone operators 133

7 DSM for general nonlinear operator equations 145

8 DSM for operators satisfying a spectral assumption 155

9 DSM in Banach spaces 161

10 DSM and Newton-type methods without inversion of the derivative 169

11 DSM and unbounded operators 177

12 DSM and nonsmooth operators 181

13 DSM as a theoretical tool 195

14 DSM and iterative methods 201

15 Numerical problems arising in applications 213

PART II

16 Solving linear operator equations by a Newton-type DSM 255

17 DSM of gradient type for solving linear operator equations 269

18 DSM for solving linear equations with finite-rank operators 281

19 A discrepancy principle for equations with monotone continuous operators 295

20 DSM of Newton-type for solving operator equations with minimal smoothness assumptions 307

21 DSM of gradient type 347

22 DSM of simple iteration type 373

23 DSM for solving nonlinear operator equations in Banach spaces 409

PART III

24 Solving linear operator equations by the DSM 423

25 Stable solutions of Hammerstein-type integral equations 441

26 Inversion of the Laplace transform from the real axis using an adaptive iterative method 455

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Dynamical Systems Method and Applications Theoretical Developments and Numerical Examples

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