Summary of mathematics doctoral thesis: Newton kantorovich iterative regularization and the proximal point methods for nonlinear ill posed equations involving monotone operators

The results of this thesis are: 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).

MINISTRY OF EDUCATION VIETNAM ACADEMY

GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY

***

NGUYEN DUONG NGUYEN

NEWTON-KANTOROVICH ITERATIVE REGULARIZATION AND THE PROXIMAL POINT METHODS FOR NONLINEAR ILL-POSED EQUATIONS INVOLVING

This thesis is completed at: Graduate University of Science andTechnology - Vietnam Academy of Science and Technology

Supervisor 1: Prof Dr Nguyen Buong

Supervisor 2: Assoc Prof Dr Do Van Luu

Gradu-The thesis can be found at:

- Library of Graduate University of Science and Technology

- Vietnam National Library

Many issues in science, technology, economics and ecology such as imageprocessing, computerized tomography, seismic tomography in engineeringgeophysics, acoustic sounding in wave approximation, problems of linearprogramming lead to solve problems having the following operator equationtype (A Bakushinsky and A Goncharsky, 1994; F Natterer, 2001; F.Natterer and F W¨ubbeling, 2001):

A(x) = f, (0.1)where A is an operator (mapping) from metric space E into metric space eEand f ∈ eE However, there exists a class of problems among these problemsthat their solutions are unstable according to the initial data, i.e., a smallchange in the data can lead to a very large difference of the solution It issaid that these problems are ill-posed Therefore, the requirement is thatthere must be methods to solve ill-posed problems such that the smallerthe error of the data is, the closer the approximate solution is to the correctsolution of the derived problem If eE is Banach space with the norm k.kthen in some cases of the mapping A, the problem (0.1) can be regularized

by minimizing Tikhonov’s functional:

Fαδ(x) = kA(x) − fδk2 + αkx − x+k2, (0.2)with selection suitable regularization parameter α = α(δ) > 0, where fδ isthe approximation of f satisfying kfδ− f k ≤ δ & 0 and x+ is the elementselected in E to help us find a solution of (0.1) at will If A is a nonlinearmapping then the functional Fαδ(x) is generally not convex Therefore, it isimpossible to apply results obtained in minimizing a convex functional tofind the minimum component of Fαδ(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

method In 1975, Ya.I Alber constructed Browder-Tikhonov tion method to solve the problem (0.1) when A is a monotone nonlinearmapping as follows:

regulariza-A(x) + αJs(x − x+) = fδ (0.3)

We see that, in the case E is not Hilbert space, Js is the nonlinear ping, and therefore, (0.3) is the nonlinear problem, even if A is the linearmapping This is a difficult problem class to solve in practice In addition,some information of the exact solution, such as smoothness, may not beretained in the regularized solution because the domain of the mapping Js

map-is the whole space, so we can’t know the regularized solution exmap-ists where

in E Thus, in 1991, Ng Buong replaced the mapping Js by a linear andstrongly 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 simplestform 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 ofthe method (0.5) in the case A is an accretive mapping in Banach space Eunder the condition that the normalized duality mapping J of E is sequen-tially weakly continuous Unfortunately, the class of infinite-dimensionalBanach 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 probleminvolving monotone nonlinear mappings This is the regularization methodbuilt on the well-known method of numerical analysis which is the Newton-Kantorovich method In 1987, based on A.B Bakushinskii’s the method,

to find the solution of the problem (0.1) in the case A is a monotonemapping from Banach space E into the dual space E∗, I.P Ryazantsevaproposed Newton-Kantorovich iterative regularization method:

A(zn) + A0(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-Tikhonovregularization 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 theconvergence of the Newton-Kantorovich iterative regularization method:

A(zn) + A0(zn)(zn+1− zn) + αn(zn+1− x+) = fδ, (0.7)under conditions

kA(x) − A(x∗) − J∗A0(x∗)∗J (x − x∗)k ≤ τ kA(x) − A(x∗)k, ∀x ∈ E

(0.8)and

A0(x∗)v = x+− x∗, (0.9)where τ > 0, x∗ is a solution of the problem (0.1), A0(x∗) is the Fréchetderivative 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) usethe Fréchet derivative of the mapping A at the unknown solution x∗, sothey are very strict In 2007, A.B Bakushinskii and A Smirnova provedthe 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 Hilbertspace, the accretive concept coincides with the monotone concept) underthe condition

kA0(x)k ≤ 1, kA0(x) − A0(y)k ≤ Lkx − yk, ∀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 accretiveoperator) in Banach spaces that we achieve, which has overcome limita-tions of results as are mentioned above

Next, 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 maximalmonotone mapping One of the first methods to find the solution of theproblem (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 = Jkxk+ ek, k ≥ 1, (0.12)where Jk = (I + rkA)−1 is called the resolvent of A with the parameter

rk > 0, ek 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 proximalpoint 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 thezero set of the mapping A is nonempty, P∞

k=1kekk < ∞ and rk ≥ ε > 0,for all k ≥ 1 In 1991, O G¨uler pointed out that the proximal pointmethod only achieves weak convergence without strong convergence ininfinite-dimensional space In order to obtain strong convergence, somemodifications of the proximal point method to find a zero of a maximalmonotone 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) aswell 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 vent of the mapping A is nonsummable, i.e P∞

resol-k=1rk = +∞ Thus, onequestion arises: is there a modification of the proximal point method thatits strong convergence is given under the condition is that the parametersequence of the resolvent is summable, i.e P∞

k=1rk < +∞? In order

to answer this question, the second content of the thesis introduces new

modifications of the proximal point method to find a zero of a maximalmonotone mapping in Hilbert space in which the strong convergence ofmethods is given under the assumption is that the parameter sequence ofthe resolvent is summable

The results of this thesis are:

1) Propose and prove the strong convergence of a new modification ofthe Newton-Kantorovich iterative regularization method (0.6) to solve theproblem (0.1) with A is a monotone mapping from Banach space E intothe dual space E∗, which overcomes the drawbacks of method (0.6).2) Propose and prove the strong convergence of the Newton-Kantorovichiterative 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, withthe removal of conditions (0.8), (0.9), (0.10) and does not require thesequentially weakly continuity of the normalized duality mapping J 3) Introduce two new modifications of the proximal point method to find azero of a maximal monotone mapping in Hilbert space, in which the strongconvergence of these methods are proved under the assumption that theparameter sequence of the resolvent is summable

Apart from the introduction, conclusion and reference, the thesis is posed of three chapters Chapter 1 is complementary, presents a number ofconcepts 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

com-to find a zero of a maximal monocom-tone mapping in Hilbert space ter 2 presents the Newton-Kantorovich iterative regularization method forsolving nonlinear ill-posed equations involving monotone type operators

Chap-in Banach spaces, Chap-includes: Chap-introducChap-ing methods and theorems about theconvergence of these methods At the end of the chapter give a numeri-cal example to illustrate the obtained research result Chapter 3 presentsmodifications of the proximal point method that we achieve to find a zero

of a maximal monotone mapping in Hilbert spaces, including the duction of methods as well as results of the convergence of these methods

intro-A numerical example is given at the end of this chapter to illustrate theobtained 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 theregularization 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 eE If (1.1)

is an ill-posed problem then the requirement is that we must be used thesolution 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 intothe dual space E∗, the problem (1.1) can be solved by Browder-Tikhonovtype 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 widelyused methods for solving the problem (1.1) is the Browder-Tikhonov typeregularization 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 spacewith the uniformly Gâteaux differentiable norm and let A be an m-accretivemapping in E Then, for each α > 0 and fδ ∈ E, the equation (0.5) has aunique solution xδα Moreover, if δ/α → 0 as α → 0 then the sequence {xδα}

converges strongly to x∗ ∈ E that is the unique solution of the followingvariational inequality

x∗ ∈ S∗ : hx∗ − x+, j(x∗ − y)i ≤ 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 regularizationsolution sequence {xδα} generated by the Browder-Tikhonov regularizationmethod (0.5) to the solution x∗ of the problem (1.1) that does not requirethe 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 nonlinearproblems, in Chapter 2, we will present an another regularization method,called the Newton-Kantorovich iteration regularization method This isthe regularization method built on the well-known method of the numericalanalysis, that is the Newton-Kantorovich method, which is presented inSection 1.2

1.2 The Kantorovich-Newton method

This section presents the Kantorovich-Newton method and the gence theorem of this method

conver-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)where H is Hilbert space and A : 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 G¨uler (1991) that this method only achieves weakconvergence without strong convergence in the infinite-dimensional space

1.3.2 Some modifications of the proximal point method

This section presents some modifications of the proximal point methodwith 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 proximalpoint 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

∞

P

k=1

rk < +∞

Chapter 2

Newton-Kantorovich iterative

regularization method for nonlinear equations involving monotone type operators

This chapter presents the Newton-Kantorovich iteration regularizationmethod for finding a solution of nonlinear equations involving monotonetype mappings Results of this chapter are presented based on works [20],[30] and [40] in list of works has been published

2.1 Newton-Kantorovich iterative regularization for nonlinear

equations involving monotone operators in Banach spacesConsider 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

kfδ − f k ≤ δ & 0 (2.2)

If A does not have strongly monotone or uniformly monotone propertiesthen the equation (2.1) is generally an ill-posed problem Since when A isthe 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 regularizationmethod This regularization method was proposed by A.B Bakushinskii(1976) based on the Newton-Kantorovich method to find the solution of

the following variational inequality problem in Hilbert space H: Find anelement x∗ ∈ Q ⊆ H such that

hA(x∗), x∗− wi ≤ 0, ∀w ∈ Q, (2.3)where A : H → H is a monotone mapping, Q is a closed and convexset in H A.B Bakushinskii introduced the iterative method to solve theproblem (2.3) as follows:

z0 = x+ ∈ H, A(zn) + A0(zn)(zn+1− zn) + αn(zn+1− x+) = fδ, (2.5)with using the generalized discrepancy principle

kA(zN) − fδk2 ≤ τ δ < kA(zn) − fδk2, 0 ≤ n < N = N (δ), (2.6)and the condition

kA0(x)k ≤ 1, kA0(x) − A0(y)k ≤ Lkx − yk, ∀x, y ∈ H (2.7)Comment 2.1 The advantage of the method (2.5) is its linearity Thismethod 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) + A0(zn)(zn+1− zn) + αnJs(zn+1) = fδ n (2.8)The convergence of the method (2.8) was provided by I.P Ryazantsevaunder the assumption that E is Banach space having the ES property, the

dual space E∗ is strictly convex and the mapping A satisfies the condition

kA00(x)k ≤ ϕ(kxk), ∀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 Banachspaces 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-Tikhonovregularization method (0.3) mentioned above

To overcome these drawbacks, in [30], we propose the new Kantorovich iterative regularization method as follows:

Newton-z0 ∈ E, A(zn) + A0(zn)(zn+1− zn) + αnB(zn+1− x+) = fδn, (2.10)where B is a linear and strongly monotone mapping

Firstly, to find the solution of the equation (2.1) in the case withoutperturbation for f , we have the following iterative method:

z0 ∈ E, A(zn) + A0(zn)(zn+1− zn) + αnB(zn+1− x+) = f (2.11)The convergence of the method (2.11) is given by the following theorem:Theorem 2.4 Let E be a real and reflexive Banach space, B be a linear,

mB-strongly monotone mapping with D(B) = E, R(B) = E∗ and A be amonotone, L-Lipschitz continuous and twice Fréchet differentiable mapping

on E with condition (2.9) Assume that the sequence {αn} and the initialpoint z0 in (2.11) satisfy the following conditions:

a) {αn} is a monotone decreasing sequence with 0 < αn < 1 and thereexists σ > 0 such that αn+1 ≥ σαn for all n = 0, 1, ;

b)

ϕ0kz0 − x0k2mBσα0

≤ q < 1, ϕ0 = ϕ(d + γ), (2.12)

d ≥ maxkB(x+ − x∗)k/mB + kx∗k, LkB(x+ − x∗)k/m2B ,

a positive number γ is found from the estimate

2mBσα0/ϕ0 ≤ γ, (2.13)where x∗ is the unique solution of the variational inequality

x∗ ∈ S, hB(x+ − x∗), x∗ − yi ≥ 0, ∀y ∈ S