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Parallel methods for regularizing systems of equations involving accretive operators

Pham Ky Anha, Nguyen Buongb & Dang Van Hieuaa

Department of Mathematics, Vietnam National University, Hanoi,

334 Nguyen Trai, Thanh Xuan, Hanoi 10000, Vietnam

b Vietnamese Academy of Science & Technology, Institute ofInformation Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi

10000, Vietnam

Published online: 20 Jan 2014

To cite this article: Pham Ky Anh, Nguyen Buong & Dang Van Hieu (2014) Parallel methods

for regularizing systems of equations involving accretive operators, Applicable Analysis: An

International Journal, 93:10, 2136-2157, DOI: 10.1080/00036811.2013.872777

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Vol 93, No 10, 2136–2157, http://dx.doi.org/10.1080/00036811.2013.872777

Parallel methods for regularizing systems of equations involving

accretive operators

Pham Ky Anha∗, Nguyen Buongband Dang Van Hieua

a Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi 10000, Vietnam; b Vietnamese Academy of Science & Technology, Institute of Information

Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi 10000, Vietnam

Communicated by Prof Boris Mordukhovich

(Received 21 September 2013; accepted 22 November 2013)

In this paper, two parallel methods for solving systems of accretive operatorequations in Banach spaces are studied The convergence analysis of the methods

in both free-noise and noisy data cases is provided

Keywords: uniformly smooth and uniformly convex Banach spaces; accretive

and inverse uniformly accretive operators; iterative regularization method;

Newton-type method; parallel computation

AMS Subject Classifications: 47J06; 47J25; 65J15; 65J20; 65Y05

1 Introduction, preliminaries, and notations

Various problems of science and engineering, including a multi-parameter identificationproblem, the convex feasibility problem, a common fixed point problem, etc., lead to asystem of ill-posed operator equations

of Hilbert spaces

In this paper, we study parallel methods extended to system (1) involving m-accretive

operators in the setting of Banach spaces In the sequel, we always assume that system (1)

is consistent, i.e the solution set S of (1) is not empty It is known that if A i (i = 1, , N)

are not strongly or uniformly accretive, then system (1) in general is ill-posed, i.e the

∗Corresponding author Email: anhpk@vnu.edu.vn

solution set S of (1) may not depend continuously on data In that case, a process known asregularization should be applied for stable solution of (1).

In what follows, for the reader’s convenience, we collect some definitions and resultsconcerning the geometry of Banach spaces and accretive operators, which are used in thispaper We refer the reader to [9 13] for more details

(1) strictly convex if the unit sphere S1(0) = {x ∈ X : ||x|| = 1} is strictly convex, i.e.

the inequality||x + y|| < 2 holds for all x, y ∈ S1(0), x = y;

(2) uniformly convex if for any given > 0 there exists δ = δ() > 0 such that for all

x , y ∈ X with x ≤ 1, y ≤ 1, x − y =  the inequality x + y ≤ 2(1 − δ)

Observe that if X is a real uniformly convex and uniformly smooth Banach space, then

the modulus of convexityδ Xis a continuous and strictly increasing function on the wholesegment[0, 2] (see, e.g [14])

Definition 1.3 A Banach space X possesses the approximation if there exists a directed family of finite dimensional subspaces X nordered by inclusion and a corresponding family

of projectors P n : X → X n , such that ||P n || = 1 for all n ≥ 0 and ∪ n X n is dense in X

Let X be a real Banach space with the dual space X∗ Throughout this paper, we assume

that the so-called normalized duality mapping J : X → X∗, satisfying the relation

2= J (x)2, ∀x ∈ X,

is single-valued This assumption will be fulfilled if X is smooth.

A Banach space X is said to have a uniformly Gateaux differetiable norm if for every

y ∈ S1(0) the limit lim t→0||x+ty||−||x|| t is attained uniformly for x ∈ S1(0) It is well known

(see [11]) that if the norm of X is uniformly Gateaux differentiable, then the normalized

duality mapping is single-valued and norm to weak star uniformly continuous on every

bounded subset of X

For the sake of simpicity, we will denote norms of both spaces X and X∗by the same

symbol. The dual product of f ∈ X∗and x

Besides, we putR+:= (0, ∞), R+

∗ := [0, ∞).

(1) accretive, if

(2) maximal accretive, if it is accretive and its graph is not the right part of the graph

of any other accretive operator;

(3) m-accretive, if it is accretive and R (A + αI ) = X for all α > 0, where I is the

identity operator in X;(4) uniformly accretive, if there exists a strictly increasing function ψ : R+

R+

∗, ψ(0) = 0, such that

(2)(5) strongly accretive, if there exists a positive constant c , such that in (2),ψ(t) = ct2;(6) inverse strongly accretive, if there exists a positive constant c , such that

2 ∀x, y ∈ X.

If X is a Hilbert space then J is an identity operator and accretive operators are also

called monotone

ϕ-inverse uniformly accretive (or simply, inverse uniformly accretive), if there exists a

functionϕ : R+× R+

∗ → R+

∗, which is continuous and strictly increasing in the second

variable andϕ(s, t) = 0 if and only if t = 0 for every fixed s > 0, such that

(3)

Example 1.6 Any inverse strongly accretive operator is inverse uniformly accretive, hence,

is accretive Indeed, let A be a c-inverse strongly accretive operator Then, A is Lipschitz continuous with the Lipschit constant c−1and the inequality (3) holds with the function

ϕ(s, t) = ct2

smooth Banach space X Then A := I − T is a Lipschitz continuous operator Moreover,

Observe that := A(x)−A(y) 4R ≤ 1 for any x, y ∈ X; x , y ≤ R and inequality (3)holds with the functionϕ(s, t) = L−1s2δ X

t 

, s ∈ R+; t ∈ [0; 2s].

Example 1.8 Now let X in Example1.7be one of the following Banach spaces L p , l p , W m

p,where 1< p < ∞ Then X is uniformly smooth and uniformly convex, and it is well known

(1) Hemicontinuous at a point x0 ∈ D(B), if B(x0+ t n h ) x0 as t n → 0 for any

vector h , such that x0+ t n h ∈ D(B) and 0 ≤ t n ≤ t(x0).

(2) Weakly continuous at x0∈ D(B), if D(B)  x x0implies that B (x) B(x0).

If B is hemicontinuous (weakly continuous) at every point of D (B), then B is said to

be hemicontinuous (weakly continuous), respectively

For regularizing accretive operator equations one needs the following fact.[9]

Le m m a 1.10 Suppose that the Banach space X possesses the approximation, A : X →

mapping J : X → X∗is sequentially weakly continuous and continuous Then the problem

where α is a fixed positive parameter and y ∈ X, is well-posed.

The unique solvability of (4) is established in [9] The continuous dependence of the

solution x α of (4) on the right-hand side y follows from the inequality ||x α,1 − x α,2|| ≤

where L is Figiel constant, (1 < L < 1.7).

Le m m a 1.12 [9] If X is a real uniformly smooth Banach space, then the inequality

x2≤ y2

≤ y2

holds for every x , y ∈ X.

Le m m a 1.13 [9] Let X be a uniformly smooth Banach space Then for x , y ∈ X

2+ C (x , y) ρ X (x − y),

Le m m a 1.14 [9] In a uniformly smooth Banach space X , for x , y ∈ X,

Le m m a 1.15 [9,17] Let {λ n } and {p n } be sequences of nonnegative numbers, {b n } be a

sequence of positive numbers, satisfying the inequalities

λ n+1≤ (1 − p n ) λ n + b n , ∀n ≥ 0, where p n ∈ (0; 1) , b n

p n → +∞ (n → +∞) and ∞n=1p n = +∞ Then λ n → 0 (n → +∞).

In Section3, when dealing with a parallel Newton-type regularization method, we needsome more results

Le m m a 1.16 [18] Suppose A : D(A) = X → X is a continuously Frechet differentiable

accretive operator and let L := A(h), h ∈ X, and α be a real positive number Then

An outline of the remainder of the paper is as follows: in Section2, we propose twoparallel iterative regularizations methods (PIRMs) for system (1), namely implicit PIRMand explicit PIRM The convergence of these PIRMs is established for both exact and noisydata cases Section3studies a parallel regularizing Newton-type method for system (1) Theconvergence analysis of the proposed method in exact and noisy data cases is also studied.

2 Equations with inverse uniformly accretive operators

In this section, we consider system (1) with inverse uniformly accretive operators Clearly, if

each operator A i isϕ i-inverse uniformly accretive, then it isϕ-inverse uniformly accretive

withϕ(s, t) := min i =1, ,N ϕ i (s, t) Thus, without loss of generality we can assume that all

the operators A i , i = 1, , N are ϕ-inverse uniformly accretive with the same funtion ϕ.

We begin with the following simple fact (cf [6])

Le m m a 2.1 Suppose A i , i = 1, 2, , N, are inverse uniformly accretive operators If system (1) is consistent, then it is equivalent to the operator equation

Proof Let the opeartors A i , i = 1, , N, be ϕ-inverse uniformly accretive with the same

funtionϕ Obviously, any solution of (1) is a solution of (5) Conversely, let y be a solution

where R = max {y , z} Thus, ϕ (R, A i (y) − A i (z)) = 0, and hence A i (y) =

In this section, we need the following lemma, where the sequential weak continuity ofthe normalized duality mapping is not required (cf [9, Theorem 2.7.1])

Le m m a 2.2 [20, Theorem 2.1] Let X be a real, reflexive and strictly convex Banach space with a uniformly Gateaux differentiable norm and let A be an m-accretive mapping

on X Then for each α > 0 and a fixed y ∈ X, Equation (4) possesses a unique solution

x α , and in addition, if the solution set S A of the equation A (x) = y is nonempty, then the net {x α } converges strongly to the unique element  x∗solving the following variational

x∗, J( x∗− x∗)≤ 0, ∀x∗∈ S A

α − x α || ≤ δ/α, where x δ

α is the unique solution of the equation

A (x) + αx = y δ , for any α > 0 and y δ ∈ X satisfying ||y δ − y|| ≤ δ.

In the remainder of Section2, we impose two sets of conditions on the space X , the duality mapping J and the operators A i , i = 1, 2, , N.

Conditions (AJX)

A1 A i , i = 1, 2, , N, are ϕ-inverse uniformly accretive operators with D(A i ) = X;

A2 the normalized duality mapping J is sequentially weakly continuous and

B2 X is a uniformly smooth and uniformly convex Banach space.

Together with Equation (5), we consider the following regularized one

N



i=1

where the regularization parameterα n → 0 as n → ∞.

Le m m a 2.3 Let conditions A1–A3 or B1–B2 be fulfilled Then the following statements hold:

(i) For every α n > 0, Equation (6) has a unique solution x∗

n ) −1R 6α n x∗2 i = 1, 2, , N, where R > 0 is a fixed number

satisfying an a-priori estimate R≥ 2 x∗ and ϕ−1

s denotes the inverse function of

ϕ (s, t) with respect to the second variable t for fixed s > 0.

(6) for Equation (5) with the accretive operator A= N

i=1A i For the proofs of statements

(i)–(iv) we refer the reader to [9] Concerning the last part (v) we observe that

Using the inverse uniform accretiveness of A i, from the last inequality we have

(2) Now suppose that conditions B1–B2 hold Since all A iare inverse uniformly accretive,

they are continuous, and hence locally bounded Besides, A i , i = 1, N, are m-accretive,

D (A i ) = X, and the spaces X and X∗are uniformly convex; then by [9, Theorem 1.15.22],

the operator A= N

i=1A i is also m-accretive Lemma2.2applied to Equation (5) ensures

the convergence of regularized solutions x∗

n–x∗ The remaining statements can be argued

whereα n > 0 and γ n > 0 are regularization and parallel splitting up parameters,

respec-tively, and defining the next approximation as an average of the regularized solutions x n i,

According to Lemmas2.2and2.3, all the problems (8) are well posed and independent

from each other; hence the regularized solutions x i ncan be found stably and simultaneously

ball with centerx∗and radius r Choose r > 0 sufficiently large such that r ≥  x∗ and

x0∈ B r ( x∗) Supposing for some n > 0, x n ∈ B r ( x∗), we will show that x n+1∈ B r ( x∗).

Indeed, from (8) and A i ( x∗) = 0, we get

By the accretiveness of A i, we get

Therefore, x n+1∈ B r ( x∗) Thus, {x n} is bounded 

Th e o r e m 2.5 Suppose conditions A1–A3 or B1–B2 are fulfilled Let {α n } and {γ n } be

real sequences, such that

If in addition, the function ϕ(s,t)

t is coercive in t for any fixed s > 0, i.e ϕ(s,t) t → +∞ as

t → +∞, then starting from arbitrary x0 ∈ X, the sequence {x n } defined by (8) and (9)

are bounded; hence, thesequences{e n} ande i n

are also bounded, i.e there exists a positive constant C > 0 such

where L ∈ (1, 1.7) is Figiel constant.

We show that ||A i (z)|| ≤ C i < +∞ for all ||z|| ≤ R0 := C + R; R ≥ 2||  x∗||

and i = 1, , N Indeed, suppose in contrary, that there exists a sequence {z n}, suchthat ||z n || ≤ R0, and ||A i (z n )|| → ∞ as n → ∞ Then t n := ||A i (z n ) − A i (0)|| ≥

||A i (z n )|| − ||A i (0)|| → ∞ as n → ∞ Since ϕ(R0, t n ) = ϕ(R0, ||A i (z n ) − A i (0)||) ≤

will tend to zero as n → +∞ We first prove that there exist positive integers m and n0,

such that for all n ≥ n0, h X (k0τ n ) ≤ 5m

k0h X (τ n ) Indeed, according to [10,p.65], we have

2≤ lim sup

τ→0+

ρ X (2τ)

ρ X (τ) ≤ 4.

Hence, there exists τ0 > 0, such that ρ X (2τ)

ρ X (τ) ≤ 5 for all τ ≤ τ0 Since τ n → 0

as n → +∞, we can find a number n0 such that k0τ n ≤ τ0 for all n ≥ n0 Let m be

a sufficiently large positive integer, such that 2m ≥ k0 Then for all n ≥ n0 we have

Example 2.6 Let A i , i = 1, , N be inverse strongly monotone operators on a real

Hilbert space X Then all the conditions A1–A3 and B1–B2 are satisfied Further, since

ϕ(s, t) = ct2, the function ϕ(s,t) t = ct is coercive Conditions (i) and (ii) on the parameters

α n , γ nhave been already stated in [5, Theorem 2.1] On the other hand, for a Hilbert space,

In the next two examples, we suppose that X = l p , 1 ≤ p < +∞ and A i = I − T i ,

where T i : X → X, i = 1, , N, are nonexpansive operators In this case, both sets of conditions A1–A3 and B1–B2 are fulfilled Observe that for proving the m- accretiveness

of A i one should use the identity A i + αI = (1 + α){I − (1 + α)−1T i} and the fact that

(1 + α)−1T i is a contraction for i = 1, , N.