Two-Step Modified Newton Method for Nonlinear Lavrentiev Regularization

Abstract

A two step modified Newton method is considered for obtaining an approximate solution for the nonlinear ill-posed equation 𝐹(𝑥)=𝑓 when the available data are 𝑓^𝛿 with ‖𝑓−𝑓^𝛿‖≤𝛿 and the operator 𝐹 is monotone. The derived error estimate under a general source condition on 𝑥₀−𝑥̂ is of optimal order; here 𝑥₀ is the initial guess and 𝑥̂ is the actual solution. The regularization parameter is chosen according to the adaptive method considered by Perverzev and Schock (2005). The computational results provided endorse the reliability and effectiveness of our method.

1. Introduction

Recently George and Elmahdy [1] studied iterative methods for solving the ill-posed operator equation:𝐹(𝑥)=𝑓,(1.1) where 𝐹∶𝐷(𝐹)⊆𝑋→𝑋 is a nonlinear monotone operator (i.e., ⟨𝐹(𝑥)−𝐹(𝑦),𝑥−𝑦⟩≥0,forall𝑥,𝑦∈𝐷(𝐹)) and 𝑋 is a real Hilbert space. The convergence analysis in [1] is based on suitably constructed majorizing sequences. Recall that a sequence (𝑡_𝑛) in ℝ is said to be a majorizing sequence of the sequence (𝑥_𝑛)∈𝑋 if‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤𝑡_𝑛+1−𝑡_𝑛,∀𝑛≥0.(1.2)

Throughout this paper, the inner product and the corresponding norm on the Hilbert space 𝑋 are denoted by ⟨⋅,⋅⟩ and ‖⋅‖ respectively, 𝐷(𝐹) is the domain of 𝐹 and 𝐹^′(⋅) is the Fréchet derivative of 𝐹.

In application, usually only noisy data 𝑓^𝛿 are available, such that‖‖𝑓−𝑓^𝛿‖‖≤𝛿.(1.3) Then the problem of recovery of 𝑥̂ from noisy equation 𝐹(𝑥)=𝑓^𝛿 is ill posed, in the sense that a small perturbation in the data can cause large deviation in the solution.

For monotone operators, one usually uses the Lavrentiev regularization method (see [2–5]) for solving (1.1). In this method, the regularized approximation 𝑥^𝛿𝛼 is obtained by solving the operator equation𝐹(𝑥)+𝛼𝑥−𝑥₀=𝑓^𝛿.(1.4) It is known (cf. [5], Theorem 1.1) that (1.4) has a unique solution 𝑥^𝛿𝛼∈𝐵_𝑟(𝑥̂)∶={𝑥∈𝑋∶‖𝑥−𝑥̂‖<𝑟}⊂𝐷(𝐹) for any 𝛼>0 provided that 𝑟=‖𝑥₀−𝑥̂‖+𝛿/𝛼.

The optimality of the Lavrentiev method was proved in [5] under a general source condition on 𝑥₀−𝑥̂. However the main drawback here is that, the regularized equation (1.4) remains nonlinear and one may have difficulties in solving them numerically.

Thus in the last few years, more emphasis was put on the investigation of iterative regularization methods (see [6–12]). In this paper, we considered a modified form of the method considered in [1], but we analyse the method as a two step modified Newton Lavrentiev method (TSMNLM). The proposed analysis is motivated by the two step directional Newton method (TSDNM) considered in [13], by Argyros and Hilout, for approximating a zero 𝑥^∗ of a differentiable function 𝐹 defined on a convex subset 𝒟 of a Hilbert space 𝐻 with values in ℝ. The TSMNLM for approximating the zero 𝑥^𝛿𝛼 of 𝐹(𝑥)−𝑓^𝛿+𝛼(𝑥−𝑥₀)=0 is defined as𝑦^{𝛿𝑛,𝛼}=𝑥^{𝛿𝑛,𝛼}−𝑅_𝛼𝑥₀⁻¹𝐹𝑥^{𝛿𝑛,𝛼}−𝑓^𝛿+𝛼𝑥^{𝛿𝑛,𝛼}−𝑥₀,(1.5)𝑥^{𝛿𝑛+1,𝛼}=𝑦^{𝛿𝑛,𝛼}−𝑅_𝛼𝑥₀⁻¹𝐹𝑦^{𝛿𝑛,𝛼}−𝑓^𝛿+𝛼𝑦^{𝛿𝑛,𝛼}−𝑥₀,(1.6) where 𝑥^𝛿0,𝛼∶=𝑥₀ and 𝑅_𝛼(𝑥₀)=𝐹^(𝑥₀)+𝛼𝐼. Here the regularization parameter 𝛼 is chosen from the finite set 𝐷_𝑀={𝛼_𝑖∶0<𝛼₀<𝛼₁<⋯<𝛼_𝑀}.

The organization of this paper is as follows. Convergence analysis of TSMNLM is given in Section 2, error bounds under an a priori and under an adaptive choice of the regularization parameter is given are Section 3 and Section 4 deals with implementation of the adaptive choice rule. Example and results of computational experiments are given in Section 5. Finally the paper ends with conclusion in Section 6.

2. Convergence Analysis for TSMNLM

We need the following assumptions for the convergence analysis of TSMNLM.

Assumption 2.1 (see [4]). 𝐹 possesses a locally uniformly bounded Fréchet derivative 𝐹′(⋅) at all 𝑥 in the domain 𝐷(𝐹).

Assumption 2.2 (cf. [4], Assumption 3). There exists a constant 𝑘₀>0 such that for every 𝑥,𝑢∈𝐷(𝐹) and 𝑣∈𝑋, there exists an element Φ(𝑥,𝑢,𝑣)∈𝑋 satisfying 𝐹^(𝑥)−𝐹^(𝑢)𝑣=𝐹^(𝑢)Φ(𝑥,𝑢,𝑣),‖Φ(𝑥,𝑢,𝑣)‖≤𝑘₀‖𝑣‖‖𝑥−𝑢‖(2.1) for all 𝑥,𝑢 in 𝐷(𝐹) and 𝑣∈𝑋.

Let𝛿𝑒^{𝛿𝑛,𝛼}∶=‖‖𝑦^{𝛿𝑛,𝛼}−𝑥^{𝛿𝑛,𝛼}‖‖,∀𝑛≥0,₀<3𝛼⁰4𝑘₀,𝜌<√3/2−2𝑘₀𝛿₀/𝛼₀−1𝑘₀,(2.2)𝛾_𝜌∶=𝛿⁰𝛼₀+𝑘⁰2𝜌²+𝜌.(2.3) Hereafter, for convenience, we use the notations 𝑥_𝑛,𝑦_𝑛, and 𝑒_𝑛 for 𝑥^{𝛿𝑛,𝛼},𝑦^{𝛿𝑛,𝛼}, and 𝑒^{𝛿𝑛,𝛼}, respectively.

Remark 2.3. Note that if ‖𝑥₀−𝑥̂‖≤𝜌, then by Assumption 2.2 it follows that 𝑒₀=‖‖𝑦₀−𝑥₀‖‖==‖‖‖‖𝑅𝑅_𝛼𝑥₀⁻¹𝑓^𝛿−𝐹𝑥₀_𝛼‖‖≤‖‖𝑅𝑥₀⁻¹𝑓^𝛿−𝑓+𝐹(𝑥̂)−𝐹𝑥₀−𝐹^𝑥₀𝑥̂−𝑥₀+𝐹^𝑥₀𝑥̂−𝑥₀_𝛼‖‖+‖‖𝑅_𝛼𝑥₀⁻¹𝑓^𝛿−𝑓‖‖+‖‖‖𝑅_𝛼𝑥₀⁻¹¹⁰𝐹^𝑥̂+𝑡𝑥̂−𝑥₀−𝐹^𝑥₀𝑑𝑡𝑥̂−𝑥₀‖‖‖≤𝛿𝑥₀⁻¹𝐹^𝑥₀𝑥̂−𝑥₀‖‖𝛼+𝑘2⁰𝜌²≤𝛿+𝜌₀𝛼₀+𝑘⁰2𝜌²=𝛾_𝜌+𝜌.(2.4)

Let𝑞=𝑘₀𝑟.(2.5) Then 𝛾_𝜌/(1−𝑞)<𝑟, if 𝑟∈1−√1−4𝑘₀𝛾_𝜌2𝑘₀,1+√1−4𝑘₀𝛾_𝜌2𝑘₀.(2.6)

Theorem 2.4. Let q and r be as in (2.5) and (2.6), respectively, and let 𝛾_𝜌 be as in (2.3). Let 𝑒_𝑛 be as in (2.2) and let 𝑦_𝑛 and 𝑥_𝑛 be as in (1.5) and (1.6), respectively, with 𝛿∈[0,𝛿₀) and 𝛼∈𝐷_𝑀. Then(a)‖𝑥_𝑛−𝑦_𝑛−1‖≤𝑞‖𝑦_𝑛−1−𝑥_𝑛−1‖; (b)‖𝑦_𝑛−𝑥_𝑛‖≤𝑞²‖𝑦_𝑛−1−𝑥_𝑛−1‖; (c)𝑒_𝑛≤𝑞^2𝑛𝛾_𝜌; (d)𝑥_𝑛,𝑦_𝑛∈𝐵_𝑟(𝑥₀).

Proof. Observe that if 𝑥_𝑛,𝑦_𝑛∈𝐵_𝑟(𝑥₀), then by Assumption 2.2 we have 𝑥_𝑛−𝑦_𝑛−1==𝑦𝑅_𝑛−1𝛼−𝑥_𝑛−1−𝑅_𝛼𝑥₀⁻¹𝐹𝑦_𝑛−1−𝐹𝑥_𝑛−1+𝛼𝑦_𝑛−1−𝑥_𝑛−1=𝑅_𝛼𝑥₀⁻¹𝑅_𝛼𝑥₀𝑦_𝑛−1−𝑥_𝑛−1−𝐹𝑦_𝑛−1−𝐹𝑥_𝑛−1−𝛼𝑦_𝑛−1−𝑥_𝑛−1=𝑅_𝛼𝑥₀⁻¹¹⁰𝐹^𝑥₀−𝐹^𝑥_𝑛−1+𝑡𝑦_𝑛−1−𝑥_𝑛−1×𝑦_𝑛−1−𝑥_𝑛−1𝑑𝑡𝑥₀⁻¹𝐹^𝑥₀¹⁰Φ𝑥₀,𝑥_𝑛−1+𝑡𝑦_𝑛−1−𝑥_𝑛−1,𝑦_𝑛−1−𝑥_𝑛−1𝑑𝑡,(2.7) and hence ‖‖𝑥_𝑛−𝑦_𝑛−1‖‖≤𝑘₀𝑟‖‖𝑦_𝑛−1−𝑥_𝑛−1‖‖.(2.8) Again observe that if 𝑥_𝑛,𝑦_𝑛∈𝐵_𝑟(𝑥₀), by Assumption 2.2 and (2.8), we have 𝑦_𝑛−𝑥_𝑛==𝑅𝑥_𝑛𝛼−𝑦_𝑛−1−𝑅_𝛼𝑥₀⁻¹𝐹𝑥_𝑛−𝑓^𝛿+𝛼𝑥_𝑛−𝑥₀=𝑅_𝛼𝑥₀⁻¹𝑅_𝛼𝑥₀𝑥_𝑛−𝑦_𝑛−1−𝐹𝑥_𝑛−𝐹𝑦_𝑛−1−𝛼𝑥_𝑛−𝑦_n−1=𝑅_𝛼𝑥₀⁻¹¹⁰𝐹^𝑥₀−𝐹^𝑦_𝑛−1+𝑡𝑥_𝑛−𝑦_𝑛−1𝑑𝑡𝑥_𝑛−𝑦_𝑛−1𝑥₀⁻¹𝐹^𝑥₀¹⁰Φ𝑥₀,𝑦_𝑛−1+𝑡𝑥_𝑛−𝑦_𝑛−1,𝑥_𝑛−𝑦_𝑛−1𝑑𝑡,(2.9) and hence ‖‖𝑦_𝑛−𝑥_𝑛‖‖≤𝑘₀𝑟‖‖𝑥_𝑛−𝑦_𝑛−1‖‖≤𝑞²‖‖𝑦_𝑛−1−𝑥_𝑛−1‖‖.(2.10) Thus if 𝑥_𝑛,𝑦_𝑛∈𝐵_𝑟(𝑥₀), then (a) and (b) follows from (2.8) and (2.10), respectively. Now using induction we will prove that 𝑥_𝑛,𝑦_𝑛∈𝐵_𝑟(𝑥₀). Note that 𝑥₀,𝑦₀∈𝐵_𝑟(𝑥₀) and hence by (2.8) ‖‖𝑥₁−𝑥₀‖‖≤≤(‖‖1𝑥+₁−𝑞)𝑦𝑒₀₀‖‖+‖‖𝑦₀−𝑥₀‖‖≤𝑒⁰≤1𝛾−𝑞_𝜌<𝑟1,−𝑞(2.11) that is, 𝑥₁∈𝐵_𝑟(𝑥₀); again by (2.10) ‖‖𝑦₁−𝑥₀‖‖≤≤𝑞‖‖₂𝑦₁−𝑥₁‖‖+‖‖𝑥₁−𝑥₀‖‖𝑒₀+(1+𝑞)𝑒₀≤𝑒⁰≤(1𝛾−𝑞)_𝜌<𝑟1,−𝑞(2.12) that is, 𝑦₁∈𝐵_𝑟(𝑥₀). Suppose 𝑥_𝑘,𝑦_𝑘∈𝐵_𝑟(𝑥₀) for some 𝑘>1. Then since ‖‖𝑥_𝑘+1−𝑥₀‖‖≤‖‖𝑥_𝑘+1−𝑥_𝑘‖‖+‖‖𝑥_𝑘−𝑥_𝑘−1‖‖+⋯+‖‖𝑥₁−𝑥₀‖‖,(2.13) we will first find an estimate for ‖𝑥_𝑘+1−𝑥_𝑘‖. Note that by (a) and (b) we have ‖‖𝑥_𝑘+1−𝑥_𝑘‖‖≤≤(‖‖𝑞𝑥+1_𝑘+1)−‖‖𝑦𝑦_𝑘𝑘‖‖+‖‖𝑦_𝑘−𝑥_𝑘‖‖−𝑥_𝑘‖‖≤(1+𝑞)𝑞^2𝑘𝑒₀.(2.14) Therefore by (2.13) we have ‖‖𝑥_𝑘+1−𝑥₀‖‖≤(1+𝑞)𝑞^2𝑘+𝑞^2(𝑘−1)+⋯+1𝑒₀≤(1+𝑞)1−𝑞^2𝑘+11−𝑞²𝑒₀≤𝑒⁰≤1𝛾−𝑞_𝜌<𝑟1.−𝑞(2.15) So by induction 𝑥_𝑛∈𝐵_𝑟(𝑥₀) for all 𝑛≥0. Again by (a), (b), and (2.15) we have ‖‖𝑦_𝑘+1−𝑥₀‖‖≤≤𝑞‖‖𝑦_{𝑘+12𝑘+2}−𝑥_𝑘+1‖‖+‖‖𝑥_𝑘+1−𝑥₀‖‖𝑒₀+(1+𝑞)𝑞^2𝑘+𝑞^2(𝑘−1)+⋯+1𝑒₀≤(1+𝑞)1−𝑞^2𝑘+31−𝑞²𝑒₀≤𝑒⁰≤1𝛾−𝑞_𝜌<𝑟1.−𝑞(2.16) Thus 𝑦_𝑘+1∈𝐵_𝑟(𝑥₀) and hence by induction 𝑦_𝑛∈𝐵_𝑟(𝑥₀) for all 𝑛≥0. This completes the proof of the Theorem.

The main result of this section is the following theorem.

Theorem 2.5. Let 𝑦_𝑛 and 𝑥_𝑛 be as in (1.5) and (1.6), respectively, and assumptions of Theorem 2.4 hold. Then (𝑥_𝑛) is Cauchy sequence in 𝐵_𝑟(𝑥₀) and converges to 𝑥^𝛿𝛼∈𝐵_𝑟(𝑥₀). Further 𝐹(𝑥^𝛿𝛼)+𝛼(𝑥^𝛿𝛼−𝑥₀)=𝑓^𝛿 and ‖‖𝑥_𝑛−𝑥^𝛿𝛼‖‖≤𝑞^2𝑛𝛾_𝜌(1−𝑞).(2.17)

Proof. Using the relation (b) and (c) of Theorem 2.4, we obtain ‖‖𝑥_𝑛+𝑚−𝑥_𝑛‖‖≤^𝑚−1_𝑖=0≤_𝑚−1‖‖𝑥_𝑛+𝑖+1−𝑥_𝑛+𝑖‖‖_𝑖=0(1+𝑞)𝑒_𝑛+𝑖≤^𝑚−1_𝑖=0(1+𝑞)𝑞^2(𝑛+𝑖)𝑒₀≤(1+𝑞)𝑞^2𝑛−𝑞^2𝑛+2𝑚1−𝑞²𝑒₀≤𝑞^2𝑛1−𝑞𝑒⁰≤𝑞^2𝑛1−𝑞𝛾^𝜌.(2.18) Thus 𝑥_𝑛 is a Cauchy sequence in 𝐵_𝑟(𝑥₀) and hence it converges, say, to 𝑥^𝛿𝛼∈𝐵_𝑟(𝑥₀). Observe that ‖‖𝐹𝑥_𝑛−𝑓^𝛿+𝛼𝑥_𝑛−𝑥₀‖‖≤=‖‖‖‖𝑅𝑅_𝛼𝛼𝑥₀𝑥_𝑛−𝑦_𝑛‖‖≤1𝑥₀‖‖‖‖𝑥_𝑛−𝑦_𝑛‖‖𝛼𝑞^2𝑛𝛾_𝜌.(2.19) Now by letting 𝑛→∞ in (2.19), we obtain 𝐹(𝑥^𝛿𝛼)−𝑓^𝛿+𝛼(𝑥^𝛿𝛼−𝑥₀)=0. This completes the proof.

3. Error Bounds under Source Conditions

The objective of this section is to obtain an error estimate for ‖𝑥_𝑛−𝑥̂‖ under the following assumption on 𝑥₀−𝑥̂.

Assumption 3.1 (see [4]). There exists a continuous, strictly monotonically increasing function 𝜑∶(0,𝑎]→(0,∞) with 𝑎≥‖𝐹′(𝑥̂)‖ satisfying lim_𝜆→0𝜑(𝜆)=0 and 𝑣∈𝑋 with ‖𝑣‖≤1 such that sup_𝜆≥0𝑥₀−𝑥̂=𝜑𝐹^(𝑥̂)𝑣,𝛼𝜑(𝜆)𝜆+𝛼≤𝑐^𝜑𝜑(𝛼),∀𝜆∈(0,𝑎].(3.1)

Remark 3.2. It can be seen that functions 𝜑(𝜆)=𝜆^𝜈,𝜆>0(3.2) for 0<𝜈≤1 and 𝜑(𝜆)=ln𝜆1^−𝑝0,,0ot<her𝜆w≤ise𝑒,^−(𝑝+1),(3.3) for 𝑝≥0 satisfy Assumption 3.1 (see [14]).

We will be using the error estimates in the following proposition, which can be found in [5], for our error analysis.

Proposition 3.3 (cf. [5], Proposition 3.1). Let 𝑥̂∈𝐷(𝐹) be a solution of (1.1) and let 𝐹∶𝐷(𝐹)⊆𝑋→𝑋 be a monotone operator in 𝑋. Let 𝑥_𝛼 be the unique solution of (1.4) with 𝑓 in place of 𝑓^𝛿 and let 𝑥^𝛿𝛼 be the unique solution of (1.4). Then ‖‖𝑥^𝛿𝛼−𝑥_𝛼‖‖≤𝛿‖‖𝑥_𝛼𝛼,−𝑥̂‖‖≤‖‖𝑥₀−𝑥̂‖‖.(3.4)

The following theorem can be found in [1].

Theorem 3.4 (see, [1], Theorem 3.1). Let 𝑥_𝛼 be the unique solution of (1.4) with 𝑓 in place of 𝑓^𝛿. Let the assumptions in the Proposition 3.3 and the Assumptions 2.1, 2.2, and 3.1 be satisfied. Then ‖‖𝑥_𝛼−𝑥̂‖‖≤𝑘₀𝑟+1𝑐_𝜑𝜑(𝛼).(3.5)

Combining the estimates in Proposition 3.3, Theorems 2.5, and 3.4 we obtain the following.

Theorem 3.5. Let 𝑥_𝑛 be as in (1.6) and let the assumptions in Proposition 3.3, Theorems 2.5, and 3.4 be satisfied. Then ‖‖𝑥_𝑛−𝑥̂‖‖≤𝑞^2𝑛𝛾_𝜌(1−𝑞)+max1,𝑘⁰𝑟+1𝑐_𝜑𝜑(𝛼)+𝛿𝛼.(3.6)

Let𝐶∶=max𝛾^𝜌1−𝑞+1,𝑘⁰𝑟+1𝑐_𝜑,(3.7) and let𝑛_𝛿∶=min𝑛∶𝑞^2𝑛≤𝛿𝛼.(3.8)

Theorem 3.6. Let 𝑥^𝛿𝛼 be the unique solution of (1.4) and let 𝑥_𝑛 be as in (1.6). Let the assumptions in Theorem 3.5 be satisfied. Let 𝐶 be as in (3.7) and let 𝑛_𝛿 be as in (3.8). Then ‖‖𝑥_𝑛𝛿−𝑥̂‖‖≤𝐶𝜑(𝛼)+𝛿𝛼.(3.9)

3.1. A Priori Choice of the Parameter

Note that the error estimate 𝜑(𝛼)+𝛿/𝛼 in (3.9) is of optimal order if 𝛼∶=𝛼_𝛿 satisfies, 𝜑(𝛼_𝛿)𝛼_𝛿=𝛿.

Now using the function 𝜓(𝜆)∶=𝜆𝜑⁻¹(𝜆), 0<𝜆≤𝑎, we have 𝛿=𝛼_𝛿𝜑(𝛼_𝛿)=𝜓(𝜑(𝛼_𝛿)), so that 𝛼_𝛿=𝜑⁻¹(𝜓⁻¹(𝛿)). In view of the above observations and (3.9), we have the following.

Theorem 3.7. Let 𝜓(𝜆)∶=𝜆𝜑⁻¹(𝜆) for 0<𝜆≤𝑎, and let the assumptions in Theorem 3.6 hold. For 𝛿>0, let 𝛼∶=𝛼_𝛿=𝜑⁻¹(𝜓⁻¹(𝛿)). Let 𝑛_𝛿 be as in (3.8). Then ‖‖𝑥_𝑛𝛿−𝑥̂‖‖=𝑂𝜓⁻¹(𝛿).(3.10)

3.2. An Adaptive Choice of the Parameter

In this subsection, we will present a parameter choice rule based on the adaptive method studied in [3, 9]. Let𝐷_𝑀(𝛼)∶=𝛼_𝑖=𝜇^𝑖𝛼₀,𝑖=0,1,…,𝑀,(3.11) where 𝜇>1, 𝛼₀>0, and let𝑛_𝑖∶=min𝑛∶𝑞^2𝑛≤𝛿𝛼_𝑖.(3.12) Then for 𝑖=0,1,…,𝑀, we have‖‖𝑥_𝑛𝑖−𝑥^𝛿𝛼_𝑖‖‖≤𝛿𝛼_𝑖,∀𝑖=0,1,…,𝑀.(3.13)

Let 𝑥_𝑖∶=𝑥^𝛿𝑛_{𝑖,𝛼𝑖}. In this paper, we select 𝛼=𝛼_𝑖 from 𝐷_𝑀(𝛼) for computing 𝑥_𝑖 for each 𝑖=0,1,…,𝑀.

Theorem 3.8 (cf. [9], Theorem 4.3). Assume that there exists 𝑖∈{0,1,2,…,𝑀} such that 𝜑(𝛼_𝑖)≤𝛿/𝛼_𝑖. Let assumptions of Theorems 3.6 and 3.7 hold and let 𝑙∶=max𝑖∶𝜑𝛼_𝑖≤𝛿𝑘∶=max𝑖∶‖‖𝑥_𝑖𝛼_𝑖<𝑀,−𝑥_𝑗‖‖≤4𝐶𝛿𝛼_𝑗,𝑗=0,1,2,…,𝑖−1.(3.14) Then 𝑙≤𝑘 and ‖‖𝑥̂−𝑥_𝑘‖‖≤𝑐𝜓⁻¹(𝛿),(3.15) where 𝑐=6𝐶𝜇.

4. Implementation of Adaptive Choice Rule

Following steps are involved in implementing the adaptive choice rule.(i)Choose 𝛼₀>0 such that 𝛿₀<3𝛼₀/4𝑘₀ and 𝜇>1.(ii)Choose 𝛼_𝑖∶=𝜇^𝑖𝛼₀,𝑖=0,1,2,….

Finally the adaptive algorithm associated with the choice of the parameter specified in Theorem 3.8 involves the following steps.

4.1. Algorithm

Step 1. Set 𝑖=0.

Step 2. Choose 𝑛_𝑖=min{𝑛∶𝑞^2𝑛≤𝛿/𝛼_𝑖}.

Step 3. Solve 𝑥_𝑖∶=𝑥^𝛿𝑛_{𝑖,𝛼𝑖} by using the iteration (1.5) and (1.6).

Step 4. If ‖𝑥_𝑖−𝑥_𝑗‖>4𝑐𝛿𝛼_𝑗,𝑗<𝑖, then take 𝑘=𝑖−1 and return 𝑥_𝑘.

Step 5. Else set 𝑖=𝑖+1 and go to Step 2.

5. Numerical Example

In this section, we consider the example considered in [4] for illustrating the algorithm considered in Section 4.1. We apply the algorithm by choosing a sequence of finite dimensional subspace (𝑉_𝑛) of 𝑋 with dim𝑉_𝑛=𝑛+1. Precisely we choose 𝑉_𝑛 as the linear span of {𝑣₁,𝑣₂,…,𝑣_𝑛+1}, where 𝑣_𝑖,𝑖=1,2,…,𝑛+1 are the linear splines in a uniform grid of 𝑛+1 points in [0,1].

Example 5.1 (see, [4], Section 4.3). Let 𝐹∶𝐷(𝐹)⊆𝐿²(0,1)→𝐿²(0,1) defined by 𝐹(𝑢)∶=¹⁰𝑘(𝑡,𝑠)𝑢³(𝑠)𝑑𝑠,(5.1) where 𝑘(𝑡,𝑠)=((11−−𝑠𝑡))𝑠𝑡,,00≤≤𝑠𝑡≤≤𝑠𝑡≤≤11,.(5.2) Then for all 𝑥(𝑡),𝑦(𝑡)∶𝑥(𝑡)>𝑦(𝑡), ⟨𝐹(𝑥)−𝐹(𝑦),𝑥−𝑦⟩=¹⁰¹⁰𝑘(𝑡,𝑠)𝑥³−𝑦³(𝑠)𝑑𝑠(𝑥−𝑦)(𝑡)𝑑𝑡≥0.(5.3)
Thus the operator 𝐹 is monotone. The Fréchet derivative of 𝐹 is given by 𝐹^′(𝑢)𝑤=3¹⁰𝑘(𝑡,𝑠)(𝑢(𝑠))²𝑤(𝑠)𝑑𝑠.(5.4)
Note that for 𝑢,𝑣>0, 𝐹^(𝑣)−𝐹^(𝑢)𝑤=×3∫¹⁰𝑘(𝑡,𝑠)(𝑢(𝑠))²𝑑𝑠₁₀𝑘(𝑡,𝑠)(𝑣(𝑠))²−(𝑢(𝑠))²𝑤(𝑠)𝑑𝑠∶=𝐹_∫¹⁰𝑘(𝑡,𝑠)(𝑢(𝑠))²𝑑𝑠(𝑢)Φ(𝑣,𝑢,𝑤),(5.5) where Φ(𝑣,𝑢,𝑤)=(∫¹⁰𝑘(𝑡,𝑠)((𝑣(𝑠))²−(𝑢(𝑠))²)𝑤(𝑠)𝑑𝑠)/(∫¹⁰𝑘(𝑡,𝑠)(𝑢(𝑠))²𝑑𝑠).
Observe that Φ(𝑣,𝑢,𝑤)=∫¹⁰𝑘(𝑡,𝑠)(𝑣(𝑠))²−(𝑢(𝑠))²𝑤(𝑠)𝑑𝑠=∫∫¹⁰𝑘(𝑡,𝑠)(𝑢(𝑠))²𝑑𝑠₁₀𝑘(𝑡,𝑠)(𝑢(𝑠)+𝑣(𝑠))(𝑣(𝑠)−𝑢(𝑠))𝑤(𝑠)𝑑𝑠∫¹⁰𝑘(𝑡,𝑠)(𝑢(𝑠))²𝑑𝑠.(5.6) So Assumption 2.2 satisfies with 𝑘₀≥‖(∫¹⁰𝑘(𝑡,𝑠)(𝑢(𝑠)+𝑣(𝑠))𝑑𝑠)/(∫¹⁰𝑘(𝑡,𝑠)(𝑢(𝑠))²𝑑𝑠)‖.
In our computation, we take 𝑦(𝑡)=(𝑡−𝑡¹¹)/110 and 𝑦^𝛿=𝑦+𝛿𝑣_𝑛+1. Then the exact solutions 𝑥̂(𝑡)=𝑡³.(5.7) We use 𝑥₀(𝑡)=𝑡³+356𝑡−𝑡⁸(5.8) as our initial guess, so that the function 𝑥₀−𝑥̂ satisfies the source condition: 𝑥₀−𝑥̂=𝜑𝐹^(𝑥̂)1,(5.9) where 𝜑(𝜆)=𝜆. Thus we expect to obtain the rate of convergence 𝑂(𝛿^1/2).
Observe that while performing numerical computation on finite dimensional subspace (𝑉_𝑛) of 𝑋, one has to consider the operator 𝑃_𝑛𝐹^(𝑥₀)𝑃_𝑛 instead of 𝐹^(𝑥₀), where 𝑃_𝑛 is the orthogonal projection on to 𝑉_𝑛. Thus incurs an additional error ‖𝑃_𝑛𝐹^(𝑥₀)𝑃_𝑛−𝐹^(𝑥₀)‖=𝑂(‖𝐹^(𝑥₀)(𝐼−𝑃_𝑛)‖).
Let ‖𝐹^(𝑥₀)(𝐼−𝑃_𝑛)‖≤𝜀_𝑛. For the operator 𝐹^(𝑥₀) defined in (5.4), 𝜀_𝑛=𝑂(𝑛⁻²) (cf. [15]). Thus we expect to obtain the rate of convergence 𝑂((𝛿+𝜀_𝑛)^1/2).
We choose 𝛼₀=(1.5)𝛿, 𝜇=1.5 and 𝑞=0.51. The results of the computation (four decimal places) are presented in Table 1. The plots of the exact solution and the approximate solution obtained are given in Figures 1 and 2.

Table 1 

(a)

(b)

(c)

(d)

Figure 1 

Curves of the exact and approximate solutions.

(a)

(b)

(c)

(d)

Figure 2 

Curves of the exact and approximate solutions.

The last column of Table 1 shows that the error ‖𝑥_𝑘−𝑥̂‖ is of 𝑂((𝛿+𝜀_𝑛)^1/2).

6. Conclusion

We considered a two step method for obtaining an approximate solution for a nonlinear ill-posed operator equation 𝐹(𝑥)=𝑓 where 𝐹∶𝐷(𝐹)⊆𝑋→𝑋 is a monotone operator defined on a real Hilbert space 𝑋 when the available data is 𝑓^𝛿 with ‖𝑓−𝑓^𝛿‖≤𝛿. The proposed method converges to a solution of the normal equation 𝐹(𝑥)+𝛼(𝑥−𝑥₀)=𝑓^𝛿 and hence is an approximation for the solution 𝑥̂, for properly chosen parameter 𝛼. We obtained optimal order error estimate by choosing the regularization parameter 𝛼 according to the adaptive method considered by Pereverzev and Schock [3]. The computational results provided endorse the reliability and effectiveness of our method.

Acknowledgment

S. Pareth thanks the National Institute of Technology Karnataka, Surathkal for the financial support.