Discrepancy principle and ill-posed equations with M-accretive perturbations

Chia sẻ: Y Y | Ngày: | Loại File: PDF | Số trang:8

Thêm vào BST

Báo xấu

9
lượt xem 2
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

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.

Chủ đề:

Bình luận(0) Đăng nhập để gửi bình luận!

Lưu

Nội dung Text: Discrepancy principle and ill-posed equations with M-accretive perturbations

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 regulariza- tion parameter choice, the convergence rates of the regularized solution as well as its Galerkin approximations for nonlinear ill-posed problems with m-accretive per- turbations 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 (see [11]) andX ∗ , 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 condition hx, J(x)i = kJ(x)kkxk, kJ(x)k = kxk, ∀x ∈ X, and ii) R(A + λI) = X for each λ > 0, where R(A) denotes the range of A, and I is the identity operator inX. 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, f ∈ X, (1.1) where A is an m-accretive operator in X. Note that if A is accretive (satisfies condition i) and demi-continuous or weak continuous, 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 , kfδ − f k ≤ δ → 0, (1.2) where α > 0, is the parameter of regularization, Ah are also the m-accretive operators satisfying the condition of approximation kA(x) − Ah (x)k ≤ hg(kxk), h → 0, (1.3) g(t) is a nonegative continuous and bounded (the image of bounded set is bounded) func- tion, and 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 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) for any 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 in [2]. Note that the problem of convergence and convergence rates when α is chosen a priori was investigated in [5]. Later, the symbols * and → denote weak convergence and convergence in norm, re- spectively, and the notation a ∼ b is meant that a = O(b) and b = O(a). In the following section we suppose that all above conditions are staisfied. 48
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 ρ(α) is continuous and monotone nondecreasing for every fixed τ. 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. (2.1) Then, there exists at least a value α such that α ≥ (K − 1)(δ + h)p /(2kx − x∗ k), and ρ(α) = [K + g(kxατ k)](δ + h)p , where xατ is the solution of (1.2) with α = α. Moreover, 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. (2.2) Obviously, from (1.2), the m-accretive property of Ah and the property of J we can see that kxτα − x∗ k ≤ kAh (x∗ ) − fδ k/α. Therefore, xτα → x∗ , as α → +∞. Consequently, ρ(α) = kAh (xτα ) − fδ k → kAh (x∗ ) − fδ k, as α → +∞. 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 limα→+∞ d(α) > 0. From (2.2) it implies that there is a value of α such that d(α) < 0. Hence, the first conclusion of the theorem is proved. 49
NGUYEN BUONG AND VU QUANG HUNG Further, sinceα ≥ (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, from (1.5) it follows the boundness of {x τ α(δ,h) }. Therefore, x∗ − x∗ k ≤ lim inf kxτα(δ,h) − x∗ k, k˜ τ →0 where ˜∗ is an weak cluster point of the set {xτα(δ,h) }. We shall prove that x x ˜∗ 6= x∗ . Indeed, if x ˜∗ = x∗ , then from 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, as δ, h → 0, we can conclude that A(˜ x∗ ) = f . It contradicts x∗ ∈ / S0 . Therefore, kxτα(δ,h) − x∗ k ≥ µ, µ > 0. Hence, [K + g(kxτα(δ,h) k)](δ + h)p [K + g(kxτα(δ,h) k)](δ + h)p α= ≤ . kxτα(δ,h) k µ As 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 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 {xτα(δ,h) }. For this purpose assume that A satisfies the condition kA(x) − A(x0 ) − J ∗ A0 (x0 )∗ J(x − x0 )k ≤ τ˜kA(y) − A(x0 )k, ∀x ∈ X, (2.3) where J∗ is normalized dual mapping of X ∗ , τ˜ is some positive constant, and 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); (ii) There exists an element z ∈ X such that A0 (x0 )z = x∗ − x0 ; (iii) The parameter α = α(δ, h) is chosen by theorem 2.1. 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. (2.4) α 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 τ + 1)kA(xτα ) − f k ≤ (˜ τ + 1)(kAh (xτα ) − fδ k + δ + hg(kxτα k)) ≤ (˜ τ + 1)(αkxτα − x∗ k + δ + hg(kxτα k)), ≤ (˜ from (2.4) it implies that 1 kxτα − x0 k2 ≤ (δ + hg(kx0 k))kxτα − x0 k α τ + 1)(αkxτα − x∗ k + δ + hg(kxτα k)). + kzk(˜ 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 , 0 < δ + h < 1, where Ci 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) where 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 X1 ⊂ X2 ... ⊂ Xn ... ⊂ X. Without loss of generality, assume that c = 1. It is clear that Anh is m-accretive. The aspects of existence and convergence of the solution xτα,n of (2.5) to the solution xτα of (1.2), for each α > 0, has been studied in [11]. The question under which condition the sequence {xτα,n } converges to the solution x0 , as α, δ → 0 and n → +∞ will be showed in the rest of the paper. As in [6] we can show that the solution xτα,n is continuous with respect to α on [α0 , ∞), α0 > 0 τ and xα,n → xn∗ , as α → +∞. Thus, the function ρ ˜(α) := kAh (xτα,n ) − fδ k is also contin- uous with respect to α, and lim ρ˜(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 xn∗ → x∗ , as n → ∞, we can verify that the relation kAh (xτα,n ) − fδ k > [K + g(kxn∗ k)](δ + h)p (2.6) holds for sufficiently large n. Set γn = γn (x0 ), γn (x) = k(I − Pn )xk, x ∈ X. In addition, suppose that J satisfies the condition ˜ kJ(y) − J(x)k ≤ C(R)ky − xkν , 00 , is a positive increasing function on ˜ = max{kxk, kyk} R (see [8]). We can propose the following a posteriori parameter choice strategy based on the dis- crepancy principle. The rule: Let c1 , c2 > 1 and K1 > K . Then (i) choose α = α(δ, h, n) ≥ α0 := (c1 (δ + h) + c2 γn )p 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 α = α0 . Note that the similar rule for a compact operator A 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 ∈ Xn , where n ∗ J = Pn JPn is the normalized dual mapping of Xn (see [7]), it follows 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. (2.9) 52
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 δ + hg(kxn0 k) + kA0 (x0 )(Pn − I)kγn + o(γn ) τ kxτα,n − xn0 k2 ≤ 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 is some positive constant: R1 ≥ kx0 k, kxτα,n k. On the other hand, by virtue of ∗ the property of J and (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 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 A in (2.3), theorems 2.2 and 2.3, and z in the last two theorems can be replaced by Ah and zh respectively. This work was supported by the National Fundamental Research Program in Natural Sciences. T i li»u [1] Alber, Ya.I., On solution by the method of regularization for operator equation of the first kind involving accretive mappings in Banach spaces, Differential equations SSSR, XI (1975), 2242-2248. [2] Alber, Ya.I. and Ryazanseva I.P., On solutions of the nonlinear problems involving monotone discotinuous mappings, Differential equations SSSR XI (1975), 2242-2248. [3] Fitzgibbon, W.E., Weak continuous accretive operators, Bull.AMS. 79 (1973), 473- 474. [4] Jin Q.N., Application of the modified discrepancy principle to Tikhonov regularization of nonlinear ill-posed problems, SIAM J. Numer. Anal., v. 36 (1999), n.2, 475-490. [5] Nguyen Buong, Convergence rates in regularization for nonlinear ill-posed equations under accretive perturbations, Zh. Vychisl. Matem. i Matem. Fiziki 44 (2004), 397- 402. [6] Ryazanseva I.P., The regularization parameter choice for nonlinear equation involv- ing monotone perturbative operators, Izvestia Vyschix Uchebnix Zavedenii Ser. Math. SSSR 9, (1982), 49-53. [7] Ryazanseva I.P., On nonlinear operator equations involving accretive mappings, Izves- tia Vyschix Uchebnix Zavedenii Ser. Math. SSSR N. 1 (1985), 42-46. [8] Ryazantseva I.P., On an algorithm of solving nonlinear monotone equations with un- known error in the priori data, Zh. Vychisl. Matem. i Matem. Fiziki T. 29 (1989), n. 10, 1572-1576. [9] Tautenhann U., On general regularization scheme for nonlinear ill-posed problems, Technische Universitat Chemnitz-Zwickau, 10 (1994) (preprint). [10] Tikhonov A.N. and Arsenin V.Y., Solutions of ill-posed problems, New-York, Wiley 1977. [11] Vainberg M.M., Variational method and method of monotone operators, Moscow, Mir, 1972. [12] Webb J.R., On a property of dual mappings and a properness of accretive operators, Bull. London Math. Soc. 13 (1981), 235-238. 54