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
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