Contemporary Mathematics

Volume 313, 2002

Regularization of nonlinear unstable operator equations by

secant methods with application to gravitational sounding

problem

Roman B. Alexeev and Alexandra B. Smirnova

ABSTRACT. A

novel iteratively regularized secant-type algorithm with simul-

taneous updates of the operator (F'*(xn)F'(xn)

+

E:nl)- 1 is suggested for

solving nonlinear ill-posed operator equations F(x)

=

0,

H1

---

H2,

on a

pair of Hilbert spaces

H1

and

H2.

A convergence theorem is proved. The sta-

bility of the process towards noise in the data is analyzed, and a stopping time

is chosen so that the method converges as the noise level tends to zero. The

proposed scheme is illustrated by a numerical example in which a nonlinear

inverse problem of gravitational sounding is considered.

Introduction

The theme of our paper is solving nonlinear operator equations of the form:

(1.1)

F(x)

=

0,

F:

H1--+ H2,

on a pair of Hilbert spaces H

1

and H

2

.

We assume here that equation (1.1) has a

solution

x,

not necessarily unique, and the operator F is twice Frechet differentiable

without such structural assumptions as monotonicity, invertibility of

F'(x)

etc.

In order to avoid the ill-posed inversion of the Frechet derivative operator

F'(x)

various discrete and continuous methods based on a regularization are suggested.

A principal point in the numerical implementation of regularized Newton's and

Gauss-Newton's procedures is the computation of the operators

(F'(x)

+

c:J)-

1

and

(F'*(x)F'(x)

+

c:J)-

1

respectively (see, for example,

[3]

or

[1], [2]).

This com-

putation for certain operators requires a considerable effort in many applications.

Besides it may decrease the accuracy of the approximate solution.

For finite dimensional well-posed problems, when the Jacobian

F'(:i;)

is bound-

edly invertible, several approaches are taken in order to reduce the cost associated

with the storage and inversion of

F'(x)

in Quasi-Newton schemes. Probably the

most used approach is a so called secant method: at every step of an iterative pro-

cess the Jacobian

F'(xn)

is replaced with an approximation, obtained from

F(xn+d

and

F(xn)·

In fact, multivariable generalizations of the secant method have also

been proposed. Although they require some extra calculations of

F(x),

they have

1991

Mathematics Subject Classification.

65J15, 58C15, 47H17.

Key words and phrases.

nonlinear problem, regularization, Frechet derivative, secant method.

©

2002 American Mathematical Society

http://dx.doi.org/10.1090/conm/313/05365