Solving Nonlinear Inverse Problems Based on the Regularized Modified Gauss–Newton Method

A nonlinear operator equation is investigated in the case when the Hadamard correctness conditions are violated. A two-stage method is proposed for constructing a stable method for solving the equation. It includes modified Tikhonov regularization and a modified iterative Gauss–Newton process for approximating the solution of the regularized equation. The convergence of the iterations and the strong Fejér property of the process are proved. An order optimal estimate for the error of the two-stage method is established in the class of sourcewise representable functions.


INTRODUCTION
Consider the inverse problem in the form of a nonlinear ill-posed operator equation (1) given on two Hilbert spaces U, F. Here, A is a continuously differentiable operator that has no continuous inverse and the right-hand side f is specified by its -approximation such that . To construct a regularized family of approximate solutions, we propose a two-step method in which the equation is first regularized by the modified Lavrentyev-Tikhonov method (2) and the solution of Eq. (2) is approximated by applying the modified Newton method (3) in which the derivative in the step operator is computed at the fixed point u 0 in the entire iteration process. That is why iterative process (3) is a modified Gauss-Newton method (GNM) with respect to the operator equation (1). In applications, the operator B plays the role of an information operator containing important a priori data on the solution (see, e.g., works [1,2] on thermal sounding of the atmosphere) that have to be taken into account in the algorithm. Under certain conditions on the operator and the initial approximation, for sufficiently small step sizes, it is possible to prove the linear convergence of process (3), to establish the strong Fejér property of the step operator , and to estimate the error of the two-step method. Method (2), (3) with (identity operator) and was investigated in [3], where the linear convergence of process (3) was proved and a weaker property (namely, the Fejér one) of the step operator T was established. Additionally, a three-parameter version of process (3) that is more difficult to implement was considered in [3,4]; in that method, for B = I, the operator to be inverted involved an additional parameter whose role was other than that of the parameter α. Thus, as in [3], we prove the convergence of the iterations, establish the strong Fejér property of and find an error estimate for the two-step method with respect to the residual in a more general situation (with the information operator B) on the basis of the simpler iterative scheme (3) with two control parameters . In the special case of B = I, an error estimate for the approximate solution produced by the two-step method is established in the class of source representable solutions.
(ii) B is a positive definite operator satisfying the conditions of Lemma 1.
and, for , we have the estimate (5) where is independent of α. Proof. We introduce the following notation: Then (6) Taking into account conditions (i) of Theorem 1 and the inequalities − + α ≤ κα α > . it follows from (6) that (7) Taking into account condition (iii) of Theorem 1 and the fact that , from inequalities (7) we obtain the final lower bound (8) Additionally, the following upper bound holds: The strong Fejér property of the operator T means that (10) for some , which is equivalent to the inequality (11) Combining (8) with (9) yields (12) Comparing (11) and (12), we conclude that the operator T generating iterative process (3) is strongly Fejér for .
For , the right-hand side of this inequality takes the smallest value, which, for u = u k , implies estimate (5). Consider the special case B = I. To find the error of the approximate solution produced by the two-step method, we need to estimate the error introduced by the regularization method (2). For this purpose, we impose additional conditions on the operator and assume that the solution is sourcewise representable, namely, (i) assume that and there exist a constant and an element such that, for any and ,

ERROR ESTIMATION FOR THE TWO-STEP METHOD
(ii) the solution is assumed to be sourcewise representable in the class According to Theorem 3.1 from [4] and Lemma 1 from [5], the solution of regularized equation (2) satisfies the estimate Equating the terms in parentheses in this inequality, we find the parameter value and an estimate for the regularization method (2): Combining relations (5) and (13) yields (14) Equating the terms on the right-hand side of (14), we find an expression for the number of iterations: γ = κα + α