|  | Ramm, Alexander G. / Hoang, Nguyen S.
Dynamical Systems Method and Applications
Theoretical Developments and Numerical Examples
 1. Edition January 2012
119.- Euro
2012. 576 Pages, Hardcover
ISBN 978-1-118-02428-7 - John Wiley & Sons
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| Short description
Dynamical Systems Method (DSM) is a powerful general method for solving operator equations. These equations can be linear or nonlinear, 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.
From the contents
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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