Dynamical Systems Method and Applications
Theoretical Developments and Numerical Examples

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.

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

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.