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Journal of Science of Hanoi National University of Education Natural sciences, Volume 52, Number 4, 2007, pp. 47- 54 DISCREPANCY PRINCIPLE AND ILL-POSED EQUATIONS WITH M- ACCRETIVE PERTURBATIONS Dr. Nguyen Buong Vietnamse Academy of Science and Technology Institute of Information Technology 18, Hoang Quoc Viet, Cau Giay, Ha Noi E-mail: nbuong@ioit.ncst.ac.vn Ma.Vu Quang Hung Ministry of Industry 54 Hai Ba Trung Street Hoan Kiem, Ha Noi E.mail:hungvq@moi.gov.vn Abstract. In this paper, on the base of the discrepancy principle for regularization parameter choice, the convergence rates of the regularized solution as well as its Galerkin approximations for nonlinear ill-posed problems with m-accretive perturbations are established without demanding the weak continuity of the duality mapping of the Banach spaces. 1 2 1 Introduction Let X be a real uniformly convex Banach space having the property of approximations X ∗ , the dual space of X , be strickly convex. For the sake of simplicity, the ∗ ∗ ∗ norms of X and X will be denoted by the symbol k.k. We write hx, x i instead of x (x) ∗ ∗ for x ∈ X and x ∈ X . Let A be an m-accretive operator in X , i.e. (see [11]) i) hA(x + h) − A(x), J(h)i ≥ 0, ∀x, h ∈ X, ∗ where J is the normalized dual mapping of X , the mapping from X onto X satisfies the (see [11]) and condition hx, J(x)i = kJ(x)kkxk, kJ(x)k = kxk, ∀x ∈ X, and ii) R(A + λI) = X for X. each λ > 0, where R(A) denotes the range of A, and I is the identity operator in 1 Key words: Accretive operators, strictly convex Banach space, Fr²chet differentiable and Tikhonov regularization. 2 2000 Mathematics Subject Classification: 47H17; CR: G1.8. 47 NGUYEN BUONG AND VU QUANG HUNG We are interested in solving the operator equation A(x) = f, where A f ∈ X, (1.1) X. A is accretive (satisfies condition i) and demi-continuous or weak continuous, is an m-accretive operator in Note that if then it is m-accretive (see [3], [12]). The existence of solution of (1.1) is shown in [7], [11]. Without additional conditions on the structure of A, such as strongly or uniformly accretive property, equation (1.1) is, in general, ill-posed. Therefore, in order to to obtain approximative solutions for (1.1) we need to use stable methods. Among the class of stable methods there is well-known one named the Tikhonov regularization (see [10]). Its operator version is described by the following equation Ah (x) + α(x − x∗ ) = fδ , where α > 0, is the parameter of regularization, kfδ − f k ≤ δ → 0, Ah (1.2) are also the m-accretive operators satisfying the condition of approximation kA(x) − Ah (x)k ≤ hg(kxk), g(t) h → 0, (1.3) is a nonegative continuous and bounded (the image of bounded set is bounded) func- x∗ ∈ X \ S0 , the set of solutions of (1.1) which is assumed to be nonempty. The case x∗ = 0 was considered in [1]. Since the operators Ah are also m-accretive, equation (1.2) has a unique solution denoted τ by xα , τ = (δ, h), for each δ, h, α > 0. Moreover, by the similar argument as in [1] and [7], when J is sequential weak continuous and strong continuous, and S0 = {x0 } is the unique τ solution of (1.1), we can prove that xα converges to x0 , as (δ + h)/α, α → 0. Further, from (1.1)-(1.3) and the accretive property of Ah , A it is easy to obtain the estimates tion, and kxτα − x∗ k ≤ 2kx − x∗ k + (δ + hg(kxτα k))/α, (1.4) kxτα − x∗ k ≤ 2kx − x∗ k + (δ + hg(kxk))/α, (1.5) x ∈ S0 . In this paper, on the base of results in [2] and [6] for monotone operator A ∗ from Banach space X into X , we consider the problem of selecting the value α = α(δ, h) for the m-accretive and weakly continuous operator A in Banach space X . In particular, τ to obtain the similar result on convergence rates for xα we do not require the sequential weak continuity of J and the uniqueness of the solution of (1.1), as it has been demanded for any in [2]. Note that the problem of convergence and convergence rates when α is chosen a priori was investigated in [5]. Later, the symbols * → denote weak convergence and convergence a ∼ b is meant that a = O(b) and b = O(a). and spectively, and the notation In the following section we suppose that all above conditions are staisfied. 48 in norm, re- Discrepancy principle and ill-posed equation with m- accretive perturbation 2 Main Results For each fixed δ, h > 0, consider the function ρ(α) = αkxτα − x∗ k. As in [1], we can prove that the function for every fixed τ. ρ(α) is continuous and monotone nondecreasing Following [6] we show that the paprameter α can be chosen by the discrepancy principle as follows. Theorem 2.1. Let Ah be continuous at the point x∗ , and satisfy the condition kAh (x∗ ) − fδ k > [K + g(kx∗ k)](δ + h)p , K > 1, 0 < p ≤ 1. Then, there exists at least a value α (2.1) such that α ≥ (K − 1)(δ + h)p /(2kx − x∗ k), and ρ(α) = [K + g(kxατ k)](δ + h)p , xατ is the solution of (1.2) with α = α. Moreover, (δ + h)/α, α → 0, as τ → 0. Proof. With every fixed δ, h > 0 and x ∈ S0 we have where for the case 0 1, 0 < p ≤ 1 for sufficiently small α. Then, by virtue of (1.4) we obtain ρ(α) < (K − 1)(δ + h)p + δ + hg(kxτα k) < (K − 1)(δ + h)p + [1 + g(kxτα k)](δ + h)p < [K + g(kxτα k)](δ + h)p , δ + h < 1. Obviously, from (1.2), the m-accretive property of Ah and the property of (2.2) J we can see that xτα → x∗ , α → +∞. Therefore, as kxτα − x∗ k ≤ kAh (x∗ ) − fδ k/α. as α → +∞. Consequently, ρ(α) = kAh (xτα ) − fδ k → kAh (x∗ ) − fδ k, Now, consider the function d(α) = ρ(α) − [K + g(kxτα k)](δ + h)p for α ≥ α0 > 0. The function d(α) is also continuous, and lim d(α) = kAh (x∗ ) − fδ k − [K + g(kx∗ k)](δ + h)p . α→+∞ Thus, from (2.1 we have α such that d(α) < 0. limα→+∞ d(α) > 0. From (2.2) it implies that there is a value of Hence, the first conclusion of the theorem is proved. 49 NGUYEN BUONG AND VU QUANG HUNG α ≥ (K − 1)(δ + h)p /(2kx − x∗ k), then (δ + h)/α ≤ 2(δ + h)1−p kx0 − x∗ k/(K − 1) for 0 < p < 1. Consequently, (δ + h)/α → 0, as τ → 0. On the other hand, τ from (1.5) it follows the boundness of {x α(δ,h) }. Therefore, Further, since kx̃∗ − x∗ k ≤ lim inf kxτα(δ,h) − x∗ k, τ →0 x̃∗ is an weak cluster point of the set {xτα(δ,h) }. We shall prove that x̃∗ 6= x∗ . Indeed, x̃∗ = x∗ , then from where if kA(xτα(δ,h) ) − fδ k ≤ hg(kxτα(δ,h) k) + ρ(α(δ, h)) ≤ hg(kxτα(δ,h) k) + [K + g(kxτα(δ,h) k)](δ + h)p → 0, δ, h → 0, we can conclude x∗ k ≥ µ, µ > 0. Hence, as α= that A(x̃∗ ) = f . It contradicts [K + g(kxτα(δ,h) k)](δ + h)p kxτα(δ,h) k ≤ x∗ ∈ / S0 . Therefore, [K + g(kxτα(δ,h) k)](δ + h)p µ kxτα(δ,h) − . g(t) is a bounded continuous function, and {xτα(δ,h) } is a bounded set, there is a positive p constant C such that α ≤ C(δ + h) . Therefore, α = α(δ, h) → 0, as τ → 0. Theorem is As proved. As in [1], if J is continuous and sequential weak continuous, then the requirement of weak continuity of A is redundant. Now, consider the problem of convergence rates for that A satisfies the condition {xτα(δ,h) }. For this purpose assume kA(x) − A(x0 ) − J ∗ A0 (x0 )∗ J(x − x0 )k ≤ τ̃ kA(y) − A(x0 )k, where J∗ is normalized dual mapping of X ∗ , τ̃ ∀x ∈ X, is some positive constant, and (2.3) x0 is a solution of (1.1). Note that condition (2.3) is given in [9] for studying convergence rates of the regularized solutions for nonlinear ill-posed problems involving compact operator in Hilbert spaces. Theorem 2.2. Assume that the following conditions hold: (i) A is Frechet differentiable with (2.3); z ∈ X such that A0 (x0 )z = x∗ − x0 ; α = α(δ, h) is chosen by theorem 2.1. (ii) There exists an element (iii) The parameter Then, for 0 < p < 1, we have kxτα − x0 k = O((δ + h)θ ), θ = min{1 − p, p/2}. Proof. From (1.1)-(1.3) and the conditions of the theorem it follows kxτα − x0 k2 = hxτα − x0 , J(xτα − x0 )i 1 = hfδ − Ah (xτα ), J(xτα − x0 )i + hx∗ − x0 , J(xτα − x0 )i α 50 Discrepancy principle and ill-posed equation with m- accretive perturbation 1 (δ + hg(kx0 k))kxτα − x0 k + hz, A0 (x0 )∗ J(xτα − x0 )i. α ≤ (2.4) Since hz, A0 (x0 )∗ J(xτα − x0 )i ≤ kzkkA0 (x0 )∗ J(xτα − x0 )k where kA0 (x0 )∗ J(xτα − x0 )k = kJ ∗ A0 (x0 )∗ J(xτα − x0 )k ≤ (τ̃ + 1)kA(xτα ) − f k ≤ (τ̃ + 1)(kAh (xτα ) − fδ k + δ + hg(kxτα k)) ≤ (τ̃ + 1)(αkxτα − x∗ k + δ + hg(kxτα k)), from (2.4) it implies that 1 (δ + hg(kx0 k))kxτα − x0 k α + kzk(τ̃ + 1)(αkxτα − x∗ k + δ + hg(kxτα k)). kxτα − x0 k2 ≤ Because α = α(δ, h) is chosen by theorem 2.1 with 0 < p < 1, we can obtain kxτα − x0 k2 ≤ C1 (δ + h)1−p kxατ − x0 k + C2 (δ + h)p , where Ci 0 < δ + h < 1, are the positive constants. Now, by using the implication a, b, c ≥ 0, s > t, as ≤ bat + c =⇒ as = O(bs/(s−t) + c) we have got kxτα − x0 k = O((δ + h)θ ). Theorem is proved. Now, consider the problem of approximating (1.2) by the sequence of finite-dimensional problems Anh (x) + α(x − xn∗ ) = fδn , x ∈ Xn , (2.5) fδn = Pn fδ , xn∗ = Pn x∗ , Anh = Pn Ah Pn , Pn is the linear projection from X onto Xn , Pn x → x, ∀x ∈ X, kPn k ≤ c, where c is some positive constant, and {Xn } is the sequence of finite-dimensional subspaces of X such that where X1 ⊂ X2 ... ⊂ Xn ... ⊂ X. Without loss of generality, assume that It is clear that solution xτα,n Anh c = 1. is m-accretive. The aspects of existence and convergence of the of (2.5) to the solution xτα of (1.2), for each The question under which condition the sequence α, δ → 0 and n → +∞ {xτα,n } α > 0, has been studied in [11]. converges to the solution x0 , as will be showed in the rest of the paper. xτα,n is continuous with respect to α on [α0 , ∞), α0 > τ the function ρ̃(α) := kAh (xα,n ) − fδ k is also contin- As in [6] we can show that the solution 0 τ and xα,n → xn∗ , as uous with respect α → +∞. to α, and Thus, lim ρ̃(α) = kAh (xn∗ ) − fδ k α→+∞ 51 NGUYEN BUONG AND VU QUANG HUNG for each δ, h > 0 and n. Therefore, on the base of (2.1) and of verify that the relation xn∗ → x∗ , as n → ∞, we kAh (xτα,n ) − fδ k > [K + g(kxn∗ k)](δ + h)p holds for sufficiently large can (2.6) n. Set γn = γn (x0 ), In addition, suppose that J γn (x) = k(I − Pn )xk, satisfies the condition kJ(y) − J(x)k ≤ C(R̃)ky − xkν , where C(R̃), R̃ > 0 x ∈ X. 0<ν≤1 , is a positive increasing function on (2.7) R̃ = max{kxk, kyk} (see [8]). We can propose the following a posteriori parameter choice strategy based on the discrepancy principle. The rule: Let c1 , c2 > 1 and K1 > K . Then α = α(δ, h, n) ≥ α0 := (c1 (δ + h) + c2 γn )p (i) choose such that (2.6) and kAh (xτα,n ) − fδ k ≤ [K1 + g(kxn∗ k)](δ + h)p (2.8) hold; (ii) if there is no α ≥ α0 such that (2.8) holds, then choose Note that the similar rule for a compact operator A α = α0 . in Hilbert space X has been inves- tigated in [4]. Theorem 2.3. Suppose that the following conditions hold: (i) A is Frechet differentiable with condition (2.3); (ii) There exists an element z∈X such that A0 (x0 )z = x∗ − x0 ; (iii) The papameter α is chosen by the rule. Then, kxτα,n − x0 k = O((δ + h + γn )θ + γnν/2 ). n n Proof. Set x0 = Pn x0 . From (1.3) and the property J (x) = J(x), ∀x n ∗ J = Pn JPn is the normalized dual mapping of Xn (see [7]), it follows ∈ Xn , where kxτα,n − xn0 k2 = hxτα,n − xn0 , J n (xτα,n − xn0 )i 1 ≤ hfδn − Anh (xn0 ), J n (xτα,n − xn0 )i + hxn∗ − xn0 , J n (xτα,n − xn0 )i α 1 ≤ hPn (fδ − f + A(x0 ) − A(xn0 )), J n (xτα,n − xn0 )i α +hx∗ − x0 , J n (xτα,n − xn0 )i + hg(kxn0 k)kxτα,n − xn0 k. 52 (2.9) Discrepancy principle and ill-posed equation with m- accretive perturbation Since xn0 → x0 , as n → +∞, we have kA(xn0 ) − A(x0 )k ≤ kA0 (x0 )(Pn − I)x0 k + o(γn ) ≤ kA0 (x0 )(Pn − I)kγn + o(γn ). Therefore, from (2.9) we obtain kxτα,n − xn0 k2 ≤ δ + hg(kxn0 k) + kA0 (x0 )(Pn − I)kγn + o(γn ) τ kxα,n − xn0 k α +hx∗ − x0 , J n (xτα,n − xn0 )i. (2.10) Obviously, from(2.7) and condition (ii) of the theorem it implies that hx∗ − x0 ,J n (xτα,n − xn0 )i = hz, A0 (x0 )∗ (J n (xτα,n − xn0 ) − J n (xτα,n − x0 ))i + hz, A0 (x0 )∗ J n (xτα,n − x0 )i ≤ kA0 (x0 )kC(R1 )kzkγnν + kzkkA0 (x0 )∗ J(xτα,n − x0 )k where R1 R1 ≥ kx0 k, kxτα,n k. ∗ the property of J and is some positive constant: On the other hand, by virtue of (2.3) we can write kA0 (x0 )∗ J(xτα,n − x0 )k = kJ ∗ A0 (x0 )∗ J(xτα,n − x0 )k ≤ (1 + τ )kA(xτα,n ) − A(x0 )k ≤ (1 + τ )[kAh (xτα,n ) − fδ k + δ + hg(kxτα,n k)]. By virtue of the rule, for δ+h<1 we have kA0 (x0 )∗ J(xτα,n − x0 )k ≤ (1 + τ )[K1 + g(kxn∗ k) + 1 + g(kxτα,n k)](δ + h)p for the case that (2.8) holds. In the negative case, using the Hahn-Banach theorem there y ∗ ∈ X ∗ such that ky ∗ k = 1 and kAh (xτα,n ) − fδ k = hAh (xτα,n ) − fδ , y ∗ i. ∗ ∗ τ ∗ τ ∗ Denote by I the identity operator in X . Then, hAh (xα,n )− fδ , y i = hAh (xα,n )− fδ , (I − Pn∗ )y ∗ i+hAnh (xτα,n )−fδn , y ∗ i where Pn∗ denotes the adjoint of Pn . Since k(I ∗ −Pn∗ )y ∗ k ≤ 1/2 n τ n τ for sufficiently large n, we obtain the esimate kAh (xα,n ) − fδ k ≤ 2kAh (xα,n ) − fδ k = 2αkxτα,n − xn∗ k. Therefore, exists an element kA0 (x0 )∗ J(xτα,n − x0 )k ≤ (1 + τ )[2αkxτα,n − xn∗ k + δ + hg(kxτα,n k)] ≤ (1 + τ )[2(c1 (δ + h) + c2 γn )p kxτα,n − xn∗ k + δ + hg(kxτα,n k)] ≤ c̃(δ + h + γn )p , where c̃ is some positive constant. Thus, (2.10) has the form kxτα,n − xn0 k2 ≤ C̃1 (δ + h + γn )1−p kxτα,n − xn0 k + C̃2 ((δ + h + γn )p + γnν ), C̃i > 0. Hence, kxτα,n − xn0 k = O((δ + h + γn )θ + γnν/2 ). Consequently, kxτα,n − x0 k = O((δ + h + γn )θ + γnν/2 ). Theorem is proved. 53 NGUYEN BUONG AND VU QUANG HUNG Remark. The symbol can be replaced by Ah A in (2.3), theorems 2.2 and 2.3, and and zh z in the last two theorems respectively. 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