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.
Tabella dei contenuti
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
Circa l’autore
Alexander G. Ramm, Ph D, is Professor in the Department of
Mathematics at Kansas State University. Dr. Ramm serves as
associate editor for several journals.
Nguyen S. Hoang, Ph D, 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.
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