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Inverse and Ill-Posed Problems Series 54
Managing Editor
Sergey I. Kabanikhin, Novosibirsk, Russia /Almaty, Kazakhstan

Anatoly B. Bakushinsky
Mihail Yu. Kokurin
Alexandra Smirnova
Iterative Methods for
Ill-Posed Problems
An Introduction
De Gruyter

Mathematics Subject Classification 2010: Primary: 47A52; Secondary: 65J20.
ISBN 978-3-11-025064-0
e-ISBN 978-3-11-025065-7
ISSN 1381-4524
Library of Congress Cataloging-in-Publication Data
Bakushinskii, A. B. (Anatolii Borisovich).
[Iterativnye metody resheniia nekorrektnykh zadach. English]
Iterative methods for ill-posed problems : an introduction / by
Anatoly Bakushinsky, Mikhail Kokurin, Alexandra Smirnova.
p. cm. ⫺ (Inverse and ill-posed problems series ; 54)
Includes bibliographical references and index.
ISBN 978-3-11-025064-0 (alk. paper)
1. Differential equations, Partial ⫺ Improperly posed problems.
2. Iterative methods (Mathematics) I. Kokurin, M. IU. (Mikhail
IUr’evich) II. Smirnova, A. B. (Aleksandra Borisovna) III. Title.
QA377.B25513 2011
5151.353⫺dc22
2010038154
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.
” 2011 Walter de Gruyter GmbH & Co. KG, Berlin/New York
Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de
Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen
앪
앝 Printed on acid-free paper
Printed in Germany
www.degruyter.com

Preface
A variety of processes in science and engineering is commonly modeled by alge-
braic, differential, integral and other equations. In a more difficult case, it can be
systems of equations combined with the associated initial and boundary conditions.
Frequently, the study of applied optimization problems is also reduced to solving the
corresponding equations. Typical examples include Euler’s equation in calculus of
variations and boundary value problems for Pontrjagin’s maximal principle in con-
trol theory. All such equations, encountered both in theoretical and applied areas,
may naturally be classified as operator equations. These equations connect the un-
known parameters of the model with some given quantities describing the model.
The above quantities, which can be either measured or calculated at the preliminary
stage, form the so-called input data. Generally, the input data as well as the unknown
parameters are the elements of certain metric spaces, in particular, Banach or Hilbert
spaces, with the operator of the model acting from the solution space to the data
space. The current textbook will focus on iterative methods for operator equations in
Hilbert spaces.
Iterative methods in their simplest form are first introduced in an undergraduate
numerical analysis course, among which Newton’s method for approximating a root
of a differentiable function in one variable is probably the best known. This is a typ-
ical iterative process widely used in applications. It can be generalized to the case of
finite systems of nonlinear equations with a finite number of unknowns, and also to
the case of operator equations in infinite dimensional spaces. It should, however, be
noted that direct generalization of this kind is only possible for regular operator equa-
tions and systems of equations. The regularity condition generalizes the requirement
on the derivative to be different from zero in a neighborhood of the root. This require-
ment is used for the convergence analysis of Newton’s scheme in a one-dimensional
case. Without the regularity condition, Newton’s iterations are not necessarily well-
defined. The lack of regularity is a major obstacle when it comes to applicability of
not only the Newton method, but all classical iterative methods, gradient-type meth-
ods for example, although often these methods are formally executable for irregular
problems as well. Still, a lot of important mathematical models give rise to either
irregular operator equations or to operator equations whose regularity is extremely

vi Preface
difficult to investigate, for instance numerous nonlinear inverse problems in PDEs.
Thus, the question is whether or not it is possible to construct iterative methods for
nonlinear operator equations without the regularity condition.
In the last few years the authors have been developing a unified approach to
the construction of such methods for irregular equations. The approach under devel-
opment is closely related to modern theory of ill-posed problems. The goal of our
textbook is to give a brief account of this approach. There are 16 chapters (lectures)
in the manuscript, which is based on the lecture notes prepared by the authors for
graduate students at Moscow Institute of Physics and Technology and Mari State
University, Russia, and Georgia State University, USA. A set of exercises appears at
the end of each chapter. These range from routine tests of comprehension to more
challenging problems helping to get a working understanding of the material. The
book does not require any prior knowledge of classical iterative methods for nonlin-
ear operator equations. The first three chapters investigate the basic iterative meth-
ods, the Newton, the Gauss–Newton and the gradient ones, in great detail. They also
give an overview of some relevant functional analysis and infinite dimensional op-
timization theory. Further chapters gradually take the reader to the area of iterative
methods for irregular operator equations. The last three chapters contain a number
of realistic nonlinear test problems reduced to finite systems of nonlinear equations
with a finite number of unknowns, integral equations of the first kind, and parameter
identification problems in PDEs. The test problems are specially selected in order to
emphasize numerical implementation of various iteratively regularized procedures
addressed in this book, and to enable the reader to conduct his/her own computa-
tional experiments.
As it follows from the title, this textbook is meant to illuminate only the primary
approaches to the construction and investigation of iterative methods for solving ill-
posed operator equations. These methods are being constantly perfected and aug-
mented with new algorithms. Applied inverse problems are the main sources of this
development: to solve them, the successful implementation of well-known theoreti-
cal procedures is often impossible without a deep analysis of the nature of a problem
and a successful resolution of the difficulties related to the choice of control parame-
ters, which sometimes necessitates modification of the original iterative schemes. At
times, by analyzing the structure of particular applied problems, researchers develop
new procedures (iterative algorithms, for instance), aimed at these problems exclu-
sively. The new ‘problem-oriented’ procedures may turn out to be more effective than
those designed for general operator equations. Examples of such procedures include,
but are not limited to, the method of quasi-reversibility (Lattes and Lions, 1967)
for solving unstable initial value problems (IVPs) for the diffusion equation with
reversed time, iteratively regularized schemes for solving unstable boundary value
problems (BVPs), which reduce the original BVP to a sequence of auxiliary BVPs
for the same differential equation with ‘regularized’ boundary conditions (Kozlov
and Mazya, 1990), and various procedures for solving inverse scattering problems.
For applied problems of shape design and shape recovery, the level set method is
widely used (Osher and Sethian, 1988). The reader may consult [59, 27, 63, 69, 40]
for a detailed theoretical and numerical analysis of these algorithms.

Preface vii
The formulas within the text are doubly numbered, with the ﬁrst number being
the number of the chapter and the second number being the number of the formula
within the chapter. The problems are doubly numbered as well. A few references
are given to the extensive bibliography at the end of the book; they are indicated by
initials in square brackets. Standard notations are used throughout the book; R is
the set of real numbers, N is the set of natural numbers. All other notations are
introduced as they appear.
The authors hope that the textbook will be useful to graduate students pursuing
their degrees in computational and applied mathematics, as well as to researchers
and engineers who may encounter numerical methods for nonlinear models in their
work.
Anatoly Bakushinsky
Mikhail Kokurin
Alexandra Smirnova

Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
1 The regularity condition. Newton’s method . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Linearization procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 The Gauss–Newton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Convergence rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 The gradient method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1 The gradient method for regular problems . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Ill-posed case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 Tikhonov’s scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1 The Tikhonov functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Properties of a minimizing sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 Other types of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 Equations with noisy data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5 Tikhonov’s scheme for linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.1 The main convergence result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.2 Elements of spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3 Minimizing sequences for linear equations . . . . . . . . . . . . . . . . . . . . . 35
5.4 A priori agreement between the regularization parameter and the
error for equations with perturbed right-hand sides . . . . . . . . . . . . . . . 37

x Contents
5.5 The discrepancy principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.6 Approximation of a quasi-solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6 The gradient scheme for linear equations . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.1 The technique of spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.2 A priori stopping rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.3 A posteriori stopping rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7 Convergence rates for the approximation methods in the case
of linear irregular equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7.1 The source-type condition (STC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7.2 STC for the gradient method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.3 The saturation phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7.4 Approximations in case of a perturbed STC. . . . . . . . . . . . . . . . . . . . . 61
7.5 Accuracy of the estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
8 Equations with a convex discrepancy functional by Tikhonov’s
method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
8.1 Some difficulties associated with Tikhonov’s method in case
of a convex discrepancy functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
8.2 An illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
9 Iterative regularization principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
9.1 The idea of iterative regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
9.2 The iteratively regularized gradient method . . . . . . . . . . . . . . . . . . . . . 70
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
10 The iteratively regularized Gauss–Newton method . . . . . . . . . . . . . . . . . 76
10.1 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
10.2 Further properties of IRGN iterations . . . . . . . . . . . . . . . . . . . . . . . . . . 79
10.3 A unified approach to the construction of iterative methods
for irregular equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
10.4 The reverse connection control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
11 The stable gradient method for irregular nonlinear equations . . . . . . . 90
11.1 Solving an auxiliary finite dimensional problem by the gradient
descent method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
11.2 Investigation of a difference inequality . . . . . . . . . . . . . . . . . . . . . . . . . 94
11.3 The case of noisy data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Contents xi
12 Relative computational efﬁciency of iteratively regularized methods . 98
12.1 Generalized Gauss–Newton methods . . . . . . . . . . . . . . . . . . . . . . . . . . 98
12.2 A more restrictive source condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
12.3 Comparison to iteratively regularized gradient scheme. . . . . . . . . . . . 101
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
13 Numerical investigation of two-dimensional inverse gravimetry
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
13.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
13.2 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
13.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
14 Iteratively regularized methods for inverse problem in optical
tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
14.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
14.2 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
14.3 Forward simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
14.4 The inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
14.5 Numerical results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
15 Feigenbaum’s universality equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
15.1 The universal constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
15.2 Ill-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
15.3 Numerical algorithm for 2 z 12 . . . . . . . . . . . . . . . . . . . . . . . . . . 125
15.4 Regularized method for z 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
16 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

1
The regularity condition. Newton’s method
1.1 Preliminary results
In this textbook we primarily focus on the operator equation in the form
F.x/ D 0; (1.1)
where F W H1 ! H2 is a nonlinear operator acting on a pair of real Hilbert spaces
.H1; H2/. Some results can also be generalized to the case of a complex Hilbert
space. Let x
be the solution of interest, and let L.H1; H2/ be the normed space
of all linear continuous operators from H1 into H2. Suppose that F is deﬁned and
Fréchet differentiable everywhere in H1. Recall that F 0
.x0/ 2 L.H1; H2/ is said to
be the Fréchet derivative of F at a point x0 if for any x in H1
F.x/ F.x0/ D F 0
.x0/.x x0/ C !.x0; x/;
k!.x0; x/k D o.kx x0k/ as x ! x0. Below by k kH and . ; /H we denote the
norm and the scalar product in a Hilbert space H. Throughout the book it is assumed
that the following conditions on F 0
hold:
kF 0
.x/kL.H1;H2/ N1 8x 2 H1I (1.2)
kF 0
.x/ F 0
.y/kL.H1;H2/ N2kx ykH1
8x; y 2 H1: (1.3)
For some iterative processes one may weaken conditions (1.2) and (1.3) by replacing
8x; y 2 H1 with 8x; y 2 , where is some bounded subset of H1. One can take
D B.0; R/, a ball of a sufﬁciently large radius, for example. In general,
B.x; r/ D ¹y 2 H1 W ky xkH1
rº:
The replacement is usually possible when the methods under investigation are of
a special nature, and it is known a priori that all iterations they generate are contained
in a bounded subset of H1. One can easily see that if we use 8x; y 2 instead of
8x; y 2 H1 in (1.2) and (1.3), then condition (1.2) with
N1 D kF 0
. N
x/kL.H1;H2/ C N2 sup
x2
kx N
xkH1

2 1 The regularity condition. Newton’s method
is a consequence of (1.3) for any N
x 2 . Therefore, inequality (1.3) alone needs to
be verified in that case, and we can require the differentiability of F on only. We’ll
keep that remark in mind, while using conditions (1.2) and (1.3) in order to simplify
our presentation. Let F .N1; N2/ be the class of operators F satisfying these two
conditions.
It follows from (1.3) that F 0
.x/ depends continuously on x as a map from H1
to L.H1; H2/. For all operators with a continuous derivative and for operators from
the class F .N1; N2/ in particular, one has
F.x C h/ F.x/ D
Z 1
0
F 0
.x C th/ h dt; x; h 2 H1; (1.4)
as a result of the Newton–Leibniz theorem. Clearly, from (1.3) and (1.4) one can
derive the following version of Taylor’s formula
F.x C h/ D F.x/ C F 0
.x/h C G.x; h/ 8x; h 2 H1; (1.5)
which will be used later on. Here the remainder G.x; h/ satisfies the estimate
kG.x; h/kH2

1
2
N2 khk2
H1
: (1.6)
Finally, one more helpful inequality
kF.x C h/ F.x/kH2
N1khkH1
8x; h 2 H1 (1.7)
is also a consequence of (1.4).
1.2 Linearization procedure
One of the most popular approaches to the construction of iterative methods for var-
ious classes of equations with differentiable operators is linearization of these equa-
tions. Take an arbitrary element x0 2 H1. Equation (1.1) can be written in the form
F.x0 C h/ D 0; (1.8)
where h D x x0 is a new unknown. The linearization procedure for equation (1.1)
at the point x0 is as follows. Discard the last term G.x; h/ in expression (1.5) and get
the approximate identity
F.x C h/ F.x/ C F 0
.x/h:
As the result, equation (1.8) becomes linear with respect to h
F.x0/ C F 0
.x0/h D 0; h 2 H1: (1.9)
A solution to above linearized equation (1.9), if exists, defines certain element
O
h 2 H1, which is assumed to be ’an approximate solution’ to equation (1.8). Hence,

1.2 Linearization procedure 3
the point O
x D x0 C O
h may be considered as some kind of approximation to the
solution x
of initial equation (1.1).
Let us analyze this procedure in more detail. Obviously, it is based on the as-
sumption that the term G.x; h/ in the right-hand side of (1.5) is small for x D x0.
Moreover, in order to deal with linear equation (1.9), to guarantee that it is uniquely
solvable for any F.x0/ 2 H2 for example, one usually requires continuous invert-
ibility of the operator F 0
.x0/ together with some estimate on the norm of the inverse
operator F 0
.x0/1
. Recall, that an operator A 2 L.H1; H2/ is said to be continu-
ously invertible if there exists an inverse operator A1
and A1
2 L.H2; H1/. Con-
tinuous invertibility of the operator F 0
.x0/ is required due to the fact that otherwise
the inverse operator F 0
.x0/1
is either undefined, or its domain D.F 0
.x0/1
/ does
not coincide with the entire space H2. As the result, equation (1.9) may be either un-
solvable, or it can have infinitely many solutions. Also, if F 0
.x0/1
… L.H2; H1/,
then even for F.x0/ 2 D.F 0
.x0/1
/ the solution to (1.9) cannot continuously de-
pend on F.x0/. Therefore, small errors in the problem, which are not possible to
avoid in any computational process, can change the solution considerably, or even
turn (1.9) into an equation with no solutions. In other words, when F 0
.x0/ is not
continuously invertible, equation (1.9) is an ill-posed problem. We’ll talk more on
how to solve such problems below.
Let F 0
.x0/ be continuously invertible, i.e.,
F 0
.x0/1
2 L.H2; H1/:
Then F is called regular at the point x0. Equation (1.1) is called regular in a neigh-
borhood of the solution x
, if there is a neighborhood B.x
; /, such that for any
x 2 B.x
; / the operator F 0
.x/1
exists and belongs to L.H2; H1/. Otherwise
equation (1.1) is called irregular in a neighborhood of x
. Thus, irregularity of (1.1)
in a neighborhood of x
means that there exist points x, arbitrarily close to x
, where
either the operator F 0
.x/1
is undefined, or F 0
.x/1
… L.H2; H1/: Equation (1.1)
with F 0
.x/ being a compact linear operator for all x in a neighborhood of x
, is
a typical example of irregular problem when H1 is infinite dimensional (see Prob-
lem 1.12).
Assume that for some m 0
kF 0
.x0/1
kL.H2;H1/ m 1: (1.10)
Using (1.5), (1.6) (1.9), and (1.10), it is not difficult to estimate the error k O
x x
kH1
in terms of kx0 x
kH1
. Indeed, from (1.9) one has
O
h D F 0
.x0/1
F.x0/;
and therefore
O
x D x0 F 0
.x0/1
F.x0/:
Hence,
k O
x x
kH1
D kx0 x
F 0
.x0/1
.F.x0/ F.x
//kH1
: (1.11)

4 1 The regularity condition. Newton’s method
By (1.5) and (1.6),
F.x
/ F.x0/ D F 0
.x0/.x
x0/ C G.x0; x
x0/; (1.12)
where
kG.x0; x
x0/kH2

1
2
N2 kx0 x
k2
H1
:
Substitute (1.12) into (1.11). Then inequality (1.10) yields
k O
x x
kH1

1
2
m N2 kx0 x
k2
H1
: (1.13)
From (1.13) it follows that O
x, the solution of linearized equation (1.9), is much closer
to x
, the solution of original equation (1.1), than the initial element x0, provided that
x0 is not too far from x
.
The above linearization procedure can now be repeated at the point x1 D O
x.
Thus, one gets an iterative process, which is called Newton’s method. Formally, the
process is as follows. Starting at x0 2 H1 and linearizing equation (1.1) at every
step, one obtains the sequence of elements x0; x1; : : : ; where
xnC1 D xn F 0
.xn/1
F.xn/; n D 0; 1; : : : : (1.14)
1.3 Error analysis
Clearly, method (1.14) is well-defined only if the corresponding linear operator
F 0
.xn/ is continuously invertible at every step, i.e., F 0
.xn/1
2 L.H2; H1/. In order
to guarantee that, it is sufficient to assume that regularity condition (1.10) is fulfilled
for all elements xn (n D 0; 1; : : : ), defined in (1.14). However, this a posteriori as-
sumption is not convenient in practice, since it is not possible to verify before the
iterative process has begun. The following a priori assumption of uniform regularity
is more suitable: inequality (1.10) holds for all x 2 H1, i.e.,
sup
x2H1
kF 0
.x/1
kL.H2;H1/ m 1: (1.15)
For now, let us accept condition (1.15) and use it to study the behavior of iterations
(1.14). Denote
n D kxn x
kH1
:
By (1.13) the following estimate is satisfied:
nC1
1
2
m N2 2
n; n D 0; 1; : : : : (1.16)

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FOOTNOTES:
[1] Wilson (’00) describes it as Pelomyxa, but it has much closer affinities with
Amoeba. It is in fact perhaps the closest relative of Amoeba proteus. Ectoplasm
formation, and especially the formation of ectoplasmic ridges in carolinensis, is
exactly like that in proteus.
[2] This is shown by the fact that after this ameba has taken on a spherical shape
due to some disturbance in the water, the number of small ridgeless pseudopods
thrown out upon resuming movement, is about the same as in dubia; but after ridges
begin to form, the number of pseudopods decreases.
[3] That is, resemblances in nuclear division stages are not correlated with
corresponding degrees of resemblance in somatic characters. It is not generally held
that the shape or size or number of chromosomes is correlated with any external
characters. It is the presence of hypothetical factors or genes which are held to be
correlated with somatic characters and their number or arrangement in a chromosome
is not in any way related to their character.
[4] It is possible that Gruber was led to suggest a gelatinous composition for the
layer in question on the strength of assertions made by several writers that amebas
secrete mucus. It is true that amebas may be displaced by threads of mucus hanging
to glass needles which has collected on the needles while manipulating the amebas in
the culture medium, but that is not to be taken as evidence that the mucus is secreted
by the amebas. Ameba cultures are always full of gelatinous material formed by
bacteria. I have not thus far been able to convince myself that amebas actually secrete
mucus.
[5] According to Ewart (’03) the viscosity of streaming protoplasm in plant cells
lies between η = .04 and η = .2. But the velocity of streaming endoplasm in ameba is
considerably slower than that in the plant cells which formed the basis for Ewart’s
calculations. In comparison, we may estimate the viscosity of the endoplasm of

ameba as η = .1 dynes per sq. cm. The velocity of streaming endoplasm, as
ascertained by observations on Amoeba dubia (in which the endoplasm flows usually
rapidly) is 1/880 cm. per second.
Now, given a unit mass of endoplasm moving at a given instant with a 1 velocity
of 1/880 cm. per second against viscosity of η = .1 dynes per sq. how far will the unit
mass travel before coming to rest?
Force = Mass × Acceleration, and Acceleration = Velocity/Time.
η = Viscosity = F = MA = MV/T.
Now if M = 1, η = F = A = V/T.
T = V/η = 1/880/.1 = 1/88.
The space travelled over in uniformly accelerated motion equals
S = 1/2 AT² = 1/2 × 1/10 × (1/88)² = 1/154880 = .00000645 cm.
If, therefore, the force moving the central stream of endoplasm should suddenly be
discontinued, the resistance offered by the viscosity of the enveloping endoplasm
would allow it to move only .0000645 mm. before coming to rest. But the ameba as a
whole moves more slowly than the central stream of endoplasm, the average rate of
movement being about 1/300 mm. per second. The effect of the streaming endoplasm
on the forward movement of the whole ameba would therefore be correspondingly
decreased. Now if the ameba was perfectly homogeneous and perfectly symmetrical,
and free from external stimulation, and moved in a perfectly homogeneous liquid on a
perfectly plane surface, the excessively small amount of mechanical inertia would
then be sufficient, theoretically, to cause the ameba to move in a straight instead of an
irregular path. But these conditions are never realized. The ameba is unsymmetrical in
form, heterogeneous in composition and always unsymmetrically stimulated; hence it
is impossible that the excessively small amount of mechanical inertia can be
considered a factor in determining the direction of the ameba’s path.
[6] The gap between the rate of movement of a pseudopod and that of a flagellum
is however very wide. Insofar as the character of the movement is concerned,
pseudopods such as those of flagellipodia, probably resemble the flagella of the soil
ameba and of flagellates. But the very much greater speed of contraction of a
flagellum and the presence of a special organ (blepharoplast) at the base of the
flagellum, and their connection with the nucleus, indicates that a special mechanism
is necessary to cause the rapid contraction. A flagellum appears to be a pseudopod
supplied with something like nerve tissue and a ganglion capable of setting free a
rapid succession of impulses.
[7] It should be added here that since this paragraph was written I have been very
fortunate to secure numerous records of paths swam by blindfolded swimmers, which

strikingly resemble those of persons walking blindfolded as described above. Most of
the common swimming strokes were employed in these observations and
occasionally several strokes were employed in a single experiment. In a few cases the
spiral path was made up of over twenty turns, and in one case of over fifty turns. A
fuller discussion of these results does not seem pertinent here, and must be deferred to
a later date.
[8] Since this was written I have been able to examine the movement of live sperm
cells in a number of representative animals, including the jellyfish Aurelia; the
molluscs Ostrea, Solemya, Pandora; the arthropods Limulus and Anisolabia, and the
vertebrates frog, turtle, snake, cat, dog and man, with the result that all these
spermatozoa revolve on their long axes and swim in spiral paths resembling those of
flagellates. Owing to their minute size their movements are made out only with great
difficulty, but so far as could be determined all the sperms of any one species turn on
their axes in the same way, that is, either right-handed or left-handed. Recently there
has also come to my notice the very informing paper of W. D. Hoyt, 1910, in the
Botanical Gazette, in which it is stated that fern sperms of various species swim in
spiral paths.

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