Newton - Kantorovich iterative regularization and the proximal point methods for nonlinear ILL - posed equations involving monotone operators

MINISTRY OF EDUCATION VIETNAM ACADEMY AND TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY ............***............ NGUYEN DUONG NGUYEN NEWTON-KANTOROVICH ITERATIVE REGULARIZATION AND THE PROXIMAL POINT METHODS FOR NONLINEAR ILL-POSED EQUATIONS INVOLVING MONOTONE OPERATORS Major: Applied Mathematics Code: 9 46 01 12 SUMMARY OF MATHEMATICS DOCTORAL THESIS Hanoi - 2018 This thesis is completed at: Graduate University of Science and Technology - Vietnam Academy of Science and Technology Supervisor 1: Prof. Dr. Nguyen Buong Supervisor 2: Assoc. Prof. Dr. Do Van Luu First referee 1: . . . . . . Second referee 2: . . . . . . Third referee 3: . . . . . . The thesis is to be presented to the Defense Committee of the Gradu- ate University of Science and Technology - Vietnam Academy of Science and Technology on . . . . . . . . . . . . 2018, at . . . . . . . . . . . . o’clock . . . . . . . . . . . . The thesis can be found at: - Library of Graduate University of Science and Technology - Vietnam National Library Introduction Many issues in science, technology, economics and ecology such as image processing, computerized tomography, seismic tomography in engineering geophysics, acoustic sounding in wave approximation, problems of linear programming lead to solve problems having the following operator equation type (A. Bakushinsky and A. Goncharsky, 1994; F. Natterer, 2001; F. Natterer and F. Wu¨bbeling, 2001): A(x) = f, (0.1) where A is an operator (mapping) from metric space E into metric space E˜ and f ∈ E˜. However, there exists a class of problems among these problems that their solutions are unstable according to the initial data, i.e., a small change in the data can lead to a very large difference of the solution. It is said that these problems are ill-posed. Therefore, the requirement is that there must be methods to solve ill-posed problems such that the smaller the error of the data is, the closer the approximate solution is to the correct solution of the derived problem. If E˜ is Banach space with the norm ‖.‖ then in some cases of the mapping A, the problem (0.1) can be regularized by minimizing Tikhonov’s functional: F δα(x) = ‖A(x)− fδ‖2 + α‖x− x+‖2, (0.2) with selection suitable regularization parameter α = α(δ) > 0, where fδ is the approximation of f satisfying ‖fδ − f‖ ≤ δ ↘ 0 and x+ is the element selected in E to help us find a solution of (0.1) at will. If A is a nonlinear mapping then the functional F δα(x) is generally not convex. Therefore, it is impossible to apply results obtained in minimizing a convex functional to find the minimum component of F δα(x). Thus, to solve the problem (0.1) with A is a monotone nonlinear mapping, a new type of Tikhonov regular- ization method was proposed, called the Browder-Tikhonov regularization 2method. In 1975, Ya.I. Alber constructed Browder-Tikhonov regulariza- tion method to solve the problem (0.1) when A is a monotone nonlinear mapping as follows: A(x) + αJs(x− x+) = fδ. (0.3) We see that, in the case E is not Hilbert space, Js is the nonlinear map- ping, and therefore, (0.3) is the nonlinear problem, even if A is the linear mapping. This is a difficult problem class to solve in practice. In addition, some information of the exact solution, such as smoothness, may not be retained in the regularized solution because the domain of the mapping Js is the whole space, so we can’t know the regularized solution exists where in E. Thus, in 1991, Ng. Buong replaced the mapping Js by a linear and strongly monotone mapping B to give the following method: A(x) + αB(x− x+) = fδ. (0.4) The case E ≡ H is Hilbert space, the method (0.3) has the simplest form with s = 2. Then, the method (0.3) becomes: A(x) + α(x− x+) = fδ. (0.5) In 2006, Ya.I. Alber and I.P. Ryazantseva proposed the convergence of the method (0.5) in the case A is an accretive mapping in Banach space E under the condition that the normalized duality mapping J of E is sequen- tially weakly continuous. Unfortunately, the class of infinite-dimensional Banach space has the normalized duality mapping that satisfies sequen- tially weakly continuous is too small (only the space lp). In 2013, Ng. Buong and Ng.T.H. Phuong proved the convergence of the method (0.5) without requiring the sequentially weakly continuity of the normalized du- ality mapping J . However, we see that if A is a nonlinear mapping then (0.3), (0.4) and (0.5) are nonlinear problems. For that reason, another sta- ble method to solve the problem (0.1), called the Newton-Kantorovich it- erative regularization method, has been studied. This method is proposed by A.B. Bakushinskii in 1976 to solve the variational inequality problem involving monotone nonlinear mappings. This is the regularization method built on the well-known method of numerical analysis which is the Newton- Kantorovich method. In 1987, based on A.B. Bakushinskii’s the method, 3to find the solution of the problem (0.1) in the case A is a monotone mapping from Banach space E into the dual space E∗, I.P. Ryazantseva proposed Newton-Kantorovich iterative regularization method: A(zn) + A ′(zn)(zn+1 − zn) + αnJs(zn+1) = fδn. (0.6) However, since the method (0.6) uses the duality mapping Js as a regular- ization component, it has the same limitations as the Browder-Tikhonov regularization method (0.3). The case A is an accretive mapping on Ba- nach space E, to find the solution of the problem (0.1), also based on A.B. Bakushinskii’s the method, in 2005, Ng. Buong and V.Q. Hung studied the convergence of the Newton-Kantorovich iterative regularization method: A(zn) + A ′(zn)(zn+1 − zn) + αn(zn+1 − x+) = fδ, (0.7) under conditions ‖A(x)− A(x∗)− J∗A′(x∗)∗J(x− x∗)‖ ≤ τ‖A(x)− A(x∗)‖, ∀x ∈ E (0.8) and A′(x∗)v = x+ − x∗, (0.9) where τ > 0, x∗ is a solution of the problem (0.1), A′(x∗) is the Fréchet derivative of the mapping A at x∗, J∗ is the normalized duality mapping of E∗ and v is some element of E. We see that conditions (0.8) and (0.9) use the Fréchet derivative of the mapping A at the unknown solution x∗, so they are very strict. In 2007, A.B. Bakushinskii and A. Smirnova proved the convergence of the method (0.7) to the solution of the problem (0.1) when A is a monotone mapping from Hilbert space H into H (in Hilbert space, the accretive concept coincides with the monotone concept) under the condition ‖A′(x)‖ ≤ 1, ‖A′(x)− A′(y)‖ ≤ L‖x− y‖,∀x, y ∈ H,L > 0. (0.10) The first content of this thesis presents new results of the Newton- Kantorovich iterative regularization method for nonlinear equations in- volving monotone type operators (the monotone operator and the accretive operator) in Banach spaces that we achieve, which has overcome limita- tions of results as are mentioned above. 4Next, we consider the problem: Find an element p∗ ∈ H such that 0 ∈ A(p∗), (0.11) where H is Hilbert space, A : H → 2H is the set-valued and maximal monotone mapping. One of the first methods to find the solution of the problem (0.11) is the proximal point method introduced by B. Martinet in 1970 to find the minimum of a convex functional and generalized by R.T. Rockafellar in 1976 as follows: xk+1 = Jkx k + ek, k ≥ 1, (0.12) where Jk = (I + rkA) −1 is called the resolvent of A with the parameter rk > 0, e k is the error vector and I is the identity mapping in H. Since A is the maximal monotone mapping, Jk is the single-valued mapping (F. Wang and H. Cui, 2015). Thus, the prominent advantage of the proximal point method is that it varies from the set-valued problem to the single- valued problem to solve. R.T. Rockafellar proved that the method (0.12) converges weakly to a zero of the mapping A under hypotheses are the zero set of the mapping A is nonempty, ∑∞ k=1 ‖ek‖ 0, for all k ≥ 1. In 1991, O. Gu¨ler pointed out that the proximal point method only achieves weak convergence without strong convergence in infinite-dimensional space. In order to obtain strong convergence, some modifications of the proximal point method to find a zero of a maximal monotone mapping in Hilbert space (OA Boikanyo and G. Morosanu, 2010, 2012; S. Kamimura and W. Takahashi, 2000; N. Lehdili and A. Moudafi, 1996; G. Marino and H.K. Xu, 2004; Ch.A. Tian and Y. Song, 2013; F. Wang and H. Cui, 2015; H.K. Xu, 2006; Y. Yao and M.A. Noor, 2008) as well as of an accretive mapping in Banach space (L.C. Ceng et al., 2008; S. Kamimura and W. Takahashi, 2000; X. Qin and Y. Su, 2007; Y. Song, 2009) were investigated. The strong convergence of these modifications is given under conditions leading to the parameter sequence of the resol- vent of the mapping A is nonsummable, i.e. ∑∞ k=1 rk = +∞. Thus, one question arises: is there a modification of the proximal point method that its strong convergence is given under the condition is that the parameter sequence of the resolvent is summable, i.e. ∑∞ k=1 rk < +∞? In order to answer this question, the second content of the thesis introduces new 5modifications of the proximal point method to find a zero of a maximal monotone mapping in Hilbert space in which the strong convergence of methods is given under the assumption is that the parameter sequence of the resolvent is summable. The results of this thesis are: 1) Propose and prove the strong convergence of a new modification of the Newton-Kantorovich iterative regularization method (0.6) to solve the problem (0.1) with A is a monotone mapping from Banach space E into the dual space E∗, which overcomes the drawbacks of method (0.6). 2) Propose and prove the strong convergence of the Newton-Kantorovich iterative regularization method (0.7) to find the solution of the problem (0.1) for the case A is an accretive mapping on Banach space E, with the removal of conditions (0.8), (0.9), (0.10) and does not require the sequentially weakly continuity of the normalized duality mapping J . 3) Introduce two new modifications of the proximal point method to find a zero of a maximal monotone mapping in Hilbert space, in which the strong convergence of these methods are proved under the assumption that the parameter sequence of the resolvent is summable. Apart from the introduction, conclusion and reference, the thesis is com- posed of three chapters. Chapter 1 is complementary, presents a number of concepts and properties in Banach space, the concept of the ill-posed prob- lem and the regularization method. This chapter also presents the Newton- Kantorovich method and some modifications of the proximal point method to find a zero of a maximal monotone mapping in Hilbert space. Chap- ter 2 presents the Newton-Kantorovich iterative regularization method for solving nonlinear ill-posed equations involving monotone type operators in Banach spaces, includes: introducing methods and theorems about the convergence of these methods. At the end of the chapter give a numeri- cal example to illustrate the obtained research result. Chapter 3 presents modifications of the proximal point method that we achieve to find a zero of a maximal monotone mapping in Hilbert spaces, including the intro- duction of methods as well as results of the convergence of these methods. A numerical example is given at the end of this chapter to illustrate the obtained research results. Chapter 1 Some knowledge of preparing This chapter presents the needed knowledge to serve the presentation of the main research results of the thesis in the following chapters. 1.1. Banach space and related issues 1.1.1. Some properties in Banach space This section presents some concepts and properties in Banach space. 1.1.2. The ill-posed problem and the regularization method • This section mentions the concept of the ill-posed problem and the regularization method. • Consider the problem of finding a solution of the equation A(x) = f, (1.1) where A is a mapping from Banach space E into Banach space E˜. If (1.1) is an ill-posed problem then the requirement is that we must be used the solution method (1.1) such that when δ ↘ 0, the approximative solution is closer to the exact solution of (1.1). As presented in the Introduction, in the case where A is the monotone mapping from Banach space E into the dual space E∗, the problem (1.1) can be solved by Browder-Tikhonov type regularization method (0.3) (see page 2) or (0.4) (see page 2). The case A is an accretive mapping on Banach space E, one of widely used methods for solving the problem (1.1) is the Browder-Tikhonov type regularization method (0.5) (see page 2). Ng. Buong and Ng.T.H. Phuong (2013) proved the following result for the strong convergence of the method (0.5): Theorem 1.17. Let E be real, reflexive and strictly convex Banach space with the uniformly Gâteaux differentiable norm and let A be an m-accretive mapping in E. Then, for each α > 0 and fδ ∈ E, the equation (0.5) has a unique solution xδα. Moreover, if δ/α→ 0 as α→ 0 then the sequence {xδα} 7converges strongly to x∗ ∈ E that is the unique solution of the following variational inequality x∗ ∈ S∗ : 〈x∗ − x+, j(x∗ − y)〉 ≤ 0, ∀y ∈ S∗, (1.2) where S∗ is the solution set of (1.1) and S∗ is nonempty. We see, Theorem 1.17 gives the strong convergence of the regularization solution sequence {xδα} generated by the Browder-Tikhonov regularization method (0.5) to the solution x∗ of the problem (1.1) that does not require the sequentially weakly continuity of the normalized duality mapping J . This result is a significant improvement compare with the result of Ya.I. Alber and I.P. Ryazantseva (2006) (see the Introduction). Since A is the nonlinear mapping, (0.3), (0.4) and (0.5) are nonlinear problems, in Chapter 2, we will present an another regularization method, called the Newton-Kantorovich iteration regularization method. This is the regularization method built on the well-known method of the numerical analysis, that is the Newton-Kantorovich method, which is presented in Section 1.2. 1.2. The Kantorovich-Newton method This section presents the Kantorovich-Newton method and the conver- gence theorem of this method. 1.3. The proximal point method and some modifications In this section, we consider the problem: Find an element p∗ ∈ H such that 0 ∈ A(p∗), (1.3) whereH is Hilbert space andA : H → 2H is a maximal monotone mapping. Denote Jk = (I + rkA) −1 is the resolvent of A with the parameter rk > 0, where I is the identity mapping in H. 1.3.1. The proximal point method This section presents the proximal point method investigated by R.T. Rockafellar (1976) to find the solution of the problem (1.3) and the as- sertion proposed by O. Gu¨ler (1991) that this method only achieves weak convergence without strong convergence in the infinite-dimensional space. 81.3.2. Some modifications of the proximal point method This section presents some modifications of the proximal point method with the strong convergence of them to find the solution of the problem (1.3) including the results of N. Lehdili and A. Moudafi (1996), H.K. Xu (2006), O.A. Boikanyo and G. Morosanu (2010; 2012), Ch.A. Tian and Y. Song (2013), S. Kamimura and W. Takahashi (2000), G. Marino and H.K. Xu (2004), Y. Yao and M.A. Noor (2008), F. Wang and H. Cui (2015). Comment 1.6. The strong convergence of modifications of the proximal point method mentioned above uses one of the conditions (C0) exists constant ε > 0 such that rk ≥ ε for every k ≥ 1. (C0’) lim infk→∞ rk > 0. (C0”) rk ∈ (0;∞) for every k ≥ 1 and limk→∞ rk =∞. These conditions lead to the parameter {rk} of the resolvent is nonsummable, i.e. ∞∑ k=1 rk = +∞. In Chapter 3, we introduce two new modifications of the proximal point method that the strong convergence of these methods is given under the condition of the parameter sequence of the resolvent that is completely different from results we know. Specifically, we use the condition that the parameter sequence of the resolvent is summable, i.e. ∞∑ k=1 rk < +∞. Chapter 2 Newton-Kantorovich iterative regularization method for nonlinear equations involving monotone type operators This chapter presents the Newton-Kantorovich iteration regularization method for finding a solution of nonlinear equations involving monotone type mappings. Results of this chapter are presented based on works [2′], [3′] and [4′] in list of works has been published. 2.1. Newton-Kantorovich iterative regularization for nonlinear equations involving monotone operators in Banach spaces Consider the nonlinear operator equation A(x) = f, f ∈ E∗, (2.1) where A is a monotone mapping from Banach space E into its dual space E∗, with D(A) = E. Assume that the solution set of (2.1), denote by S, is nonempty and instead of f , we only know its approximation fδ satisfies ‖fδ − f‖ ≤ δ ↘ 0. (2.2) If A does not have strongly monotone or uniformly monotone properties then the equation (2.1) is generally an ill-posed problem. Since when A is the nonlinear mapping, (0.3) (see page 2) and (0.4) (see page 2) are nonlin- ear problems, to solve (2.1), in this section, we consider an another regu- larization method, called the Newton-Kantorovich iterative regularization method. This regularization method was proposed by A.B. Bakushinskii (1976) based on the Newton-Kantorovich method to find the solution of 10 the following variational inequality problem in Hilbert space H: Find an element x∗ ∈ Q ⊆ H such that 〈A(x∗), x∗ − w〉 ≤ 0, ∀w ∈ Q, (2.3) where A : H → H is a monotone mapping, Q is a closed and convex set in H. A.B. Bakushinskii introduced the iterative method to solve the problem (2.3) as follows:z0 ∈ H,〈A(zn) + A′(zn)(zn+1 − zn) + αnzn+1, zn+1 − w〉 ≤ 0, ∀w ∈ Q. (2.4) Based on the method (2.4), to find the solution of the equation (2.1) whenA is a monotone mapping from Hilbert spaceH intoH, A.B. Bakushin- skii and A. Smirnova (2007) proved the strong convergence of the Newton- Kantortovich type iterative regularization method: z0 = x + ∈ H,A(zn) + A′(zn)(zn+1 − zn) + αn(zn+1 − x+) = fδ, (2.5) with using the generalized discrepancy principle ‖A(zN)− fδ‖2 ≤ τδ < ‖A(zn)− fδ‖2, 0 ≤ n < N = N(δ), (2.6) and the condition ‖A′(x)‖ ≤ 1, ‖A′(x)− A′(y)‖ ≤ L‖x− y‖,∀x, y ∈ H. (2.7) Comment 2.1. The advantage of the method (2.5) is its linearity. This method is an important tool for solving the problem (2.1) in the case A is a monotone mapping in Hilbert space. However, we see that the condition (2.7) is fairly strict and should overcome such that the method (2.5) can be applied to the wider mapping class. When E is Banach space, to solve the equation (2.1) in the case instead of f , we only know its approximation fδn ∈ E∗ satisfying (2.2), in which δ is replaced by δn, I.P. Ryazantseva (1987, 2006) also developed the method (2.4) to propose the iteration: z0 ∈ E,A(zn) + A′(zn)(zn+1 − zn) + αnJs(zn+1) = fδn. (2.8) The convergence of the method (2.8) was provided by I.P. Ryazantseva under the assumption that E is Banach space having the ES property, the 11 dual space E∗ is strictly convex and the mapping A satisfies the condition ‖A′′(x)‖ ≤ ϕ(‖x‖),∀x ∈ E, (2.9) where ϕ(t) is a nonnegative and nondecreasing function. Comment 2.2. We see that lp and Lp(Ω) (1 < p < +∞) are Banach spaces having the ES property and the dual space is strictly convex. How- ever, since the method (2.8) uses the duality mapping Js as a regulariza- tion component, it has the same disadvantages as the Browder-Tikhonov regularization method (0.3) mentioned above. To overcome these drawbacks, in [3′], we propose the new Newton- Kantorovich iterative regularization method as follows: z0 ∈ E,A(zn) + A′(zn)(zn+1 − zn) + αnB(zn+1 − x+) = fδn, (2.10) where B is a linear and strongly monotone mapping. Firstly, to find the sol