1 INTRODUCTION

Consider the inverse problem in the form of a nonlinear ill-posed operator equation

$$A(u) = f$$
(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 \(\delta \)‑approximation \({{f}_{\delta }}\) such that \(\left\| {f - {{f}_{\delta }}} \right\| \leqslant \delta \).

To construct a regularized family of approximate solutions, we propose a two-step method in which the equation is first regularized by the modified Lavren-tyev–Tikhonov method

$$A{\kern 1pt} '({{u}^{0}}){\kern 1pt} {\text{*}}(A(u) - {{f}_{\delta }}) + \alpha B(u - {{u}^{0}}) = 0,$$
(2)

and the solution \({{u}_{\alpha }}\) of Eq. (2) is approximated by applying the modified Newton method

$$\begin{gathered} {{u}^{{k + 1}}} = {{u}^{k}} - \gamma {{[A{\kern 1pt} '({{u}^{0}}){\kern 1pt} {\text{*}}A{\kern 1pt} '({{u}^{0}}) + \alpha B]}^{{ - 1}}} \\ \, \times [A{\kern 1pt} '({{u}^{0}}){\kern 1pt} {\text{*}}(A({{u}^{k}}) - {{f}_{\delta }}) + \alpha B({{u}^{k}} - {{u}^{0}})] \equiv T({{u}^{k}}), \\ \end{gathered} $$
(3)

in which the derivative in the step operator \(T({{u}^{k}})\) is computed at the fixed point u0 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 \(T\), and to estimate the error of the two-step method.

Method (2), (3) with \(B = I\) (identity operator) and \(\gamma = 1\) 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 \(\bar {\alpha }\) 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 \(T,\) 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 \(\alpha ,\;\gamma \). 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.

2 CONVERGENCE AND ESTIMATION OF THE ITERATION ERROR

Lemma 1. Assume that Eqs. (1) and (2) are both solvable. Let A be a linear bounded operator such that

$$\left\langle {B{\kern 1pt} u,u} \right\rangle \geqslant \kappa {{\left\| u \right\|}^{2}},\quad \kappa > 0.$$

Then we have the estimate

$$\left\| {{{{(A{\kern 1pt} {\text{*}}A + \alpha B)}}^{{ - 1}}}} \right\| \leqslant 1{\text{/}}(\kappa {\kern 1pt} \alpha ),\quad \alpha > 0.$$

Now we consider the second stage of the method, i.e., analyze the convergence of process (3).

Theorem 1. Assume that Eq. (1) is solvable and Eq. (2) is solvable for any \(\alpha ,{{f}_{\delta }}\). Suppose that the following conditions hold:

(i) \(\left\| {A{\kern 1pt} '({{u}^{0}})} \right\| \leqslant {{N}_{0}}\), \(\left\| {A({{u}_{1}}) - A({{u}_{2}})} \right\| \leqslant {{N}_{1}}\left\| {{{u}_{1}} - {{u}_{2}}} \right\|\), \(\left\| {A{\kern 1pt} '({{u}_{1}}) - A{\kern 1pt} '({{u}_{2}})} \right\| \leqslant {{N}_{2}}\left\| {{{u}_{1}} - {{u}_{2}}} \right\|\)

in a ball \(S({{u}^{0}};R)\) containing \(\hat {u},\;{{u}_{\alpha }}\); here, \(\hat {u}\) and \({{u}_{\alpha }}\) are solutions of Eqs. (1) and (2), respectively.

(ii) B is a positive definite operator satisfying the conditions of Lemma 1.

(iii) \(\left\| {{{u}_{\alpha }} - {{u}^{0}}} \right\| \leqslant {{r}^{0}}\), \({{r}^{0}} = \kappa {\kern 1pt} \alpha {\text{/}}(5{\kern 1pt} {{N}_{0}}{{N}_{2}})\),

where u0 is the starting point of iterative process (3), which coincides with the trial solution of Eq. (2) and with the point at which the derivative of the operator A is computed.

Then, for \(\gamma < (\alpha \kappa ){\text{/}}({{N}_{0}}{{N}_{2}} + \alpha \left\| B \right\|)\), the operator \(T\) implementing process (3) is strongly {uα}-Fejér in the ball \(S({{u}_{\alpha }};{{r}^{0}})\) and the iterations converge, i.e.,

$$\mathop {\lim }\limits_{k \to \infty } \left\| {{{u}^{k}} - {{u}_{\alpha }}} \right\| = 0,$$
(4)

and, for \(\gamma = (\alpha \kappa ){\text{/}}2({{N}_{0}}{{N}_{2}} + \alpha \left\| B \right\|)\), we have the estimate

$$\begin{gathered} \left\| {{{u}^{k}} - {{u}_{\alpha }}} \right\| \leqslant {{q}^{k}}{{r}^{0}} \leqslant {{q}^{k}}{\kern 1pt} \bar {r}, \\ q = {{\left( {1 - {{{(\kappa \alpha )}}^{2}}{\text{/}}4{\kern 1pt} {{{\left( {{{N}_{0}}{{N}_{2}} + \alpha \left\| B \right\|} \right)}}^{2}}} \right)}^{{(1/2)}}}, \\ \end{gathered} $$
(5)

where \({{r}^{0}}{\kern 1pt} \leqslant {\kern 1pt} {\kern 1pt} \,\bar {r}{\kern 1pt} \) is independent of α.

Proof. We introduce the following notation:

$$\begin{gathered} {{D}_{0}} = A{\kern 1pt} '({{u}^{0}}){\kern 1pt} {\text{*}}A{\kern 1pt} '({{u}^{0}}) + \alpha {\kern 1pt} B, \\ F(u) = D_{0}^{{ - 1}}[A'({{u}^{0}}){\kern 1pt} {\text{*}}(A(u) - {{f}_{\delta }}) + \alpha {\kern 1pt} B(u - {{u}^{0}})]. \\ \end{gathered} $$

Then

$$\begin{gathered} \left\langle {F(u),u - {{u}_{\alpha }}} \right\rangle = \left\langle {F(u) - F({{u}_{\alpha }}),u - {{u}_{\alpha }}} \right\rangle \\ \, = \langle D_{0}^{{ - 1}}[A{\kern 1pt} '({{u}^{0}}){\kern 1pt} {\text{*}}(A(u) - A({{u}_{\alpha }})) + \alpha {\kern 1pt} B{\kern 1pt} (u - {{u}_{\alpha }})],u - {{u}_{\alpha }}\rangle \\ \end{gathered} $$
$$\begin{gathered} \, = \langle D_{0}^{{ - 1}}[A{\kern 1pt} '({{u}^{0}}){\kern 1pt} {\text{*}}(A(u) - (A(u) + A{\kern 1pt} '(u)({{u}_{\alpha }} - u) + \xi )) \\ \, + \alpha {\kern 1pt} B{\kern 1pt} (u - {{u}_{\alpha }})],u - {{u}_{\alpha }}\rangle \\ \, = \langle D_{0}^{{ - 1}}[A{\kern 1pt} '({{u}^{0}}){\kern 1pt} {\text{*}}A{\kern 1pt} '({{u}^{0}})(u - {{u}_{\alpha }}) + \alpha B(u - {{u}_{\alpha }})]{\kern 1pt} ,u - {{u}_{\alpha }}\rangle \\ \end{gathered} $$
(6)
$$\begin{gathered} \, + \langle D_{0}^{{ - 1}}[A{\kern 1pt} '({{u}^{0}}){\kern 1pt} {\text{*}}A{\kern 1pt} '(u)(u - {{u}_{\alpha }}) \\ \, - A{\kern 1pt} '({{u}^{0}}){\kern 1pt} {\text{*}}A{\kern 1pt} '({{u}^{0}})(u - {{u}_{\alpha }})],u - {{u}_{\alpha }}\rangle \\ \, - \langle D_{0}^{{ - 1}}A{\kern 1pt} '({{u}^{0}}){\kern 1pt} {\text{*}}\xi ,u - {{u}_{\alpha }}\rangle . \\ \end{gathered} $$

Taking into account conditions (i) of Theorem 1 and the inequalities

$$\left\| {D_{0}^{{ - 1}}} \right\| \leqslant 1{\text{/}}(\kappa \alpha ),\quad \left\| \xi \right\| \leqslant {{N}_{2}}{{\left\| {u - {{u}_{\alpha }}} \right\|}^{2}}{\text{/}}2,$$

it follows from (6) that

$$\begin{gathered} \left\langle {F(u),u - {{u}_{\alpha }}} \right\rangle \geqslant {{\left\| {u - {{u}_{\alpha }}} \right\|}^{2}} \\ \, - {{\frac{{\left\| {A{\kern 1pt} '({{u}^{0}})\text{*}} \right\|\left\| {A{\kern 1pt} '(u) - A{\kern 1pt} '({{u}^{0}})} \right\|\left\| {u - {{u}_{\alpha }}} \right\|}}{{\kappa \alpha }}}^{2}} \\ \end{gathered} $$
$$\, - \frac{{{{N}_{0}}{{N}_{2}}{{{\left\| {u - {{u}_{\alpha }}} \right\|}}^{3}}}}{{2\kappa \alpha }} \geqslant {{\left\| {u - {{u}_{\alpha }}} \right\|}^{2}}$$
(7)
$$\begin{gathered} \, - \left[ {\frac{{{{N}_{0}}{{N}_{2}}(\left\| {u - {{u}_{\alpha }}} \right\| + \,{\text{||}}{{u}_{\alpha }} - {{u}^{0}}{\text{||}})}}{{\kappa \alpha }}} \right. \\ \, - \left. {\frac{{{{N}_{0}}{{N}_{2}}\left\| {u - {{u}_{\alpha }}} \right\|}}{{2\kappa \alpha }}} \right]{{\left\| {u - {{u}_{\alpha }}} \right\|}^{2}}. \\ \end{gathered} $$

Taking into account condition (iii) of Theorem 1 and the fact that \(u \in S({{u}_{\alpha }};{\kern 1pt} {\kern 1pt} {{r}^{0}})\), from inequalities (7) we obtain the final lower bound

$$\left\langle {F(u),u - {{u}_{\alpha }}} \right\rangle \geqslant (1{\text{/}}2){{\left\| {u - {{u}_{\alpha }}} \right\|}^{2}}.$$
(8)

Additionally, the following upper bound holds:

$$\begin{gathered} {{\left\| {F(u)} \right\|}^{2}} = {{\left\| {F(u) - F({{u}_{\alpha }})} \right\|}^{2}} \\ \, = \left\| {D_{0}^{{ - 1}}{{{[A{\kern 1pt} '({{u}^{0}}){\kern 1pt} {\text{*}}(A(u) - A({{u}_{\alpha }})) + \alpha B(u - {{u}_{\alpha }})]}}^{2}}} \right\| \\ \, \leqslant \frac{{{{{({{N}_{0}}{{N}_{1}} + \alpha \left\| B \right\|)}}^{2}}}}{{{{{(\kappa \alpha )}}^{2}}}}{{\left\| {u - {{u}_{\alpha }}} \right\|}^{2}}. \\ \end{gathered} $$
(9)

The strong Fejér property of the operator T means that

$$\begin{gathered} {{\left\| {T(u) - {{u}_{\alpha }}} \right\|}^{2}} - {{\left\| {u - {{u}_{\alpha }}} \right\|}^{2}} + \nu {{\left\| {T(u) - u} \right\|}^{2}} \leqslant 0 \\ \forall u \in S({{u}_{\alpha }};{{r}^{0}}) \\ \end{gathered} $$
(10)

for some \(\nu > 0\), which is equivalent to the inequality

$${{\left\| {F(u)} \right\|}^{2}} \leqslant \frac{2}{{\gamma (1 + \nu )}}\left\langle {F(u),u - {{u}_{\alpha }}} \right\rangle .$$
(11)

Combining (8) with (9) yields

$${{\left\| {F(u)} \right\|}^{2}} \leqslant \frac{{2{{{({{N}_{0}}{{N}_{1}} + \alpha \left\| B \right\|)}}^{2}}}}{{{{{(\kappa \alpha )}}^{2}}}}\left\langle {F(u),u - {{u}_{\alpha }}} \right\rangle .$$
(12)

Comparing (11) and (12), we conclude that the operator T generating iterative process (3) is strongly Fejér for \(\gamma < (\kappa \alpha ){\text{/}}{{({{N}_{0}}{{N}_{1}} + \alpha \left\| B \right\|)}^{2}}\).

Setting \(u = {{u}^{k}}\) in (10), we conclude that \(\mathop {\lim }\limits_{k \to \infty } \nu \gamma {\text{||}}T({{u}^{k}}) - {{u}^{k}}{\text{|}}{{{\text{|}}}^{k}}\mathop {\lim }\limits_{k \to \infty } \nu \gamma F({{u}^{k}})\) = 0, which, combined with inequality (9), yields the convergence relation (4). It follows from (8) and (9) that

$$\begin{gathered} {{\left\| {T(u) - {{u}_{\alpha }}} \right\|}^{2}} \\ \, = {{\left\| {u - {{u}_{\alpha }}} \right\|}^{2}} - 2\gamma \left\langle {F(u),u - {{u}_{\alpha }}} \right\rangle + {{\gamma }^{2}}{{\left\| {F(u)} \right\|}^{2}} \\ {\kern 1pt} \, \leqslant \left( {1 - \gamma + {{\gamma }^{2}}\frac{{{{{({{N}_{0}}{{N}_{1}} + \alpha \left\| B \right\|)}}^{2}}}}{{{{{(\kappa \alpha )}}^{2}}}}} \right){{\left\| {u - {{u}_{\alpha }}} \right\|}^{2}}. \\ \end{gathered} $$

For \(\gamma = (\kappa \alpha ){\text{/}}2{{({{N}_{0}}{{N}_{1}} + \alpha \left\| B \right\|)}^{2}}\), the right-hand side of this inequality takes the smallest value, which, for u = uk, implies estimate (5).

3 ERROR ESTIMATION FOR THE TWO-STEP METHOD

Lemma 2. Assume that the conditions of Theorem 1 are satisfied. Suppose that Eq. (2) has a solution \({{u}_{\alpha }}\) for any \(\alpha > 0,\) \({{f}_{\delta }} \in F\) and \({{u}_{{\alpha (\delta )}}}\) belongs to a ball \(S({{u}^{0}};R),\) where \(\alpha (\delta ) \to 0\) as \(\delta \to 0.\) Then the residual of the two-step method satisfies the estimate

$$\begin{gathered} {\text{||}}A{\kern 1pt} '({{u}^{0}}){\kern 1pt} {\text{*}}(A({{u}^{k}}) - {{f}_{\delta }}){\text{||}} \\ \, \leqslant [(5{{N}_{0}}{{N}_{1}}{\text{/}}{{N}_{2}}){{q}^{k}} + {{N}_{0}}\left\| B \right\|R]\alpha (\delta ). \\ \end{gathered} $$

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 \(\hat {u},{{u}_{\alpha }} \in S({{u}^{0}},{{r}^{0}}),{\kern 1pt} \) and there exist a constant \({{k}_{0}} > 0\) and an element \(\phi (u,{{u}^{0}},w) \in U\) such that, for any\({\kern 1pt} {\kern 1pt} w \in U\) and \(u \in S({{u}^{0}};{\kern 1pt} {\kern 1pt} {{r}^{0}})\),

$$\begin{gathered} \text{[}A{\kern 1pt} '(u) - A{\kern 1pt} '({{u}^{0}})]w = A{\kern 1pt} '({{u}^{0}})\phi (u,{{u}^{0}},w), \\ {\text{||}}\phi (u,{{u}^{0}},w){\text{||}} \leqslant {{k}_{0}}{\text{||}}u - {{u}^{0}}{\text{||}}\left\| w \right\|; \\ \end{gathered} $$

(ii) the solution is assumed to be sourcewise representable in the class

$$\begin{gathered} K = \{ \hat {u}:{{u}^{0}} - \hat {u} = {{(A{\kern 1pt} '({{u}^{0}}){\kern 1pt} {\text{*}}A{\kern 1pt} '({{u}^{0}}))}^{p}}{v},\left\| v \right\| \leqslant r\} , \\ {\kern 1pt} 0 < p \leqslant 1. \\ \end{gathered} $$

According to Theorem 3.1 from [4] and Lemma 1 from [5], the solution of regularized equation (2) satisfies the estimate

$$\begin{gathered} \left\| {u - {{u}_{\alpha }}} \right\| \leqslant \frac{{\max \{ 1{\text{/}}2,r\} }}{{1 - d}}\left( {\frac{\delta }{{\sqrt \alpha }} + {{\alpha }^{p}}} \right), \\ d = {{k}_{0}}{{r}^{0}} < 1. \\ \end{gathered} $$

Equating the terms in parentheses in this inequality, we find the parameter value \(\alpha = {{\delta }^{{2/(2p + 1)}}}\) and an estimate for the regularization method (2):

$$\left\| {{{u}_{\alpha }} - \hat {u}} \right\| \leqslant \bar {c}{{\delta }^{{2p/(2p + 1)}}},\quad \bar {c} = \frac{{2\max \{ (1{\text{/}}2),r\} }}{{1 - d}}.$$
(13)

Combining relations (5) and (13) yields

$$\begin{gathered} \left\| {{{u}_{\alpha }} - \hat {u}} \right\| \leqslant {\text{||}}{{u}^{k}} - {{u}_{{\alpha (\delta )}}}{\text{||}} + \left\| {{{u}_{{\alpha (\delta )}}} - \hat {u}} \right\| \\ \, \leqslant {{q}^{k}}\bar {r} + \bar {c}{{\delta }^{{2p/(2p + 1)}}}. \\ \end{gathered} $$
(14)

Equating the terms on the right-hand side of (14), we find an expression for the number of iterations:

$$k(\delta ) = [\ln (\bar {c}{{\delta }^{{2p/(2p + 1)}}}{\text{/}}\bar {r}){\text{/}}\ln q(\alpha (\delta ))].$$

Substituting \(k(\delta )\) into (14) yields an error estimate for the two-step method:

$$\left\| {{{u}^{{k(\delta )}}} - \hat {u}} \right\| \leqslant 2\bar {c}{{\delta }^{{2p/(2p + 1)}}}.$$

Its optimality follows from [6, Lemma 4.2.3, Theo-rem 4.9.1].

Remark 1. In the classical (unmodified) GNM [7, 8], the derivative \(A{\kern 1pt} '(u)\) in the step operator is computed at every iteration, while, in the modified GNM (3), it is computed only at the initial point and is stored in the whole iteration process. As a result, the method simplifies significantly in terms of the number of operations, while preserving its optimality in the class of source representable functions.