Dynamical Systems Method and Applications (eBook)

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

"The book is well organized and presents the DSM method to
solve a broad range of operator equations. Suitable for senior
under graduate and under graduate students as well as practical
engineers and researchers interested in dynamical systems methods
and application for operator equations".
(Zentralblatt MATH, 1 December 2012)