About convergence rates in regularization for ill posed operator equations of hammerstein type

Chia sẻ: Nguyễn Minh Vũ | Ngày: | Loại File: PDF | Số trang:9

Thêm vào BST

Báo xấu

30
lượt xem 2
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

The aim of this paper is to study convergence rates of the regularized solutions in connection with the finite-dimensional approximations for the operator equation of Hammerstein type x+ F2F1(x)=f in reflexive Banach spaces under the perturbations for not only the operators Fi,i=1,2, but also f.

Chủ đề:

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

Lưu

Nội dung Text: About convergence rates in regularization for ill posed operator equations of hammerstein type

’ Tap ch´ Tin hoc v` Diˆu khiˆ n hoc, T.23, S.1 (2007), 50—58 ı e e . . a ` . ABOUT CONVERGENCE RATES IN REGULARIZATION FOR ILL-POSED OPERATOR EQUATIONS OF HAMMERSTEIN TYPE NGUYEN BUONG1 , DANG THI HAI HA2 1 Vietnamse Academy of Science and Technology, Institute of Information Technology 2 Vietnamese Forestry University, Xuan Mai, Ha Tay Abstract. The aim of this paper is to study convergence rates of the regularized solutions in connection with the ﬁnite-dimensional approximations for the operator equation of Hammerstein type x + F2 F1 (x) = f in reﬂexive Banach spaces under the perturbations for not only the operators Fi , i = 1, 2, but also f . The conditions of convergence and convergence rates given in this paper for a class of inverse-strongly monotone operators Fi , i = 1, 2, are much simpler than those in the past papers. ´ ´ . . ’ T´m t˘t. Muc d´ cua b`i b´o n`y l` nghiˆn c´.u tˆc dˆ hˆi tu cua nghiˆm hiˆu chınh d˜ du.o.c o a ıch ’ a a a a e u o o o . ’ e e a . . . . ` ´p xı h˜.u han chiˆu cho phu.o.ng tr` to´n tu. loai Hammerstein x + F2 F1 (x) = f trong khˆng ’ . ’ u e ınh a o xˆ a . ˜ ` ’ o ’ a ’ a ’ ’ e e o . e o a ’ gian Banach phan xa v´.i nhiˆu khˆng chı c´ o. c´c to´n tu. Fi , i = 1, 2 m` ca o. f . Diˆu kiˆn hˆi tu . . . o ´ . . ` ´ v` tˆc dˆ hˆi tu trong b`i b´o n`y cho to´n tu. ngu.o.c do.n diˆu manh Fi , i = 1, 2 l` yˆu ho.n nhiˆu a o o o . e a a a a ’ e a e . . . .i c´c kˆt qua tru.´.c. ´ ’ so v´ a e o o 1. INTRODUCTION Let X be a reﬂexive real Banach space, and X ∗ be its dual which both are strictly convex. For the sake of simplicity the norms of X and X ∗ are denoted by the symbol . . We write x∗ , x or x, x∗ instead of x∗ (x) for x∗ ∈ X ∗ and x ∈ X . Concerning the space X , in addition assume that it possesses the property: the weak convergence and convergence of norms for any sequence follows its strong convergence. Let F1 : X → X ∗ and F2 : X ∗ → X be monotone, in general nonlinear, bounded (i.e. image of any bounded subset is bounded) and continuous operators. Our main aim of this paper is to study a stable method of ﬁnding an approximative solution for the equation of Hammerstein type x + F2 F1 (x) = f, f ∈ X. (1.1) Usually instead of Fi , i = 1, 2, and f we know their monotone continuous approximations Fih and fδ , such that h F1 (x) − F1 (x) hg( x ) ∀x ∈ X, h F2 (x∗ ) − F2 (x∗ ) g(t) M t + N, hg( x∗ ) ∀x∗ ∈ X ∗ , M, N 0, 51 ABOUT CONVERGENCE RATES IN REGULARIZATION FOR ILL-POSED where g(t) is a real nonegative, non-decreasing, bounded function (the image of a bounded set δ. Without additional conditions for the operators is bounded) with g(0) = 0, and fδ − f Fi such as the strongly monotone property, equation (1.1) is ill-posed (see the example at the end of the paper). To solve (1.1) we need to use stable methods. One of them is the operator equation h h x + F2,α F1,α (x) = fδ (1.2) h (see [1], [2]), where Fi,α = Fih + αUi , Ui , i = 1, 2, are the normalized dual mappings of X and ∗ , respectively (see [9]), and α > 0 is the small parameter of regularization. For every α > 0 X equation (1.2) has a unique solution xh,δ , and the sequence {xh,δ } converges to a solution x0 α α of (1.1) as (h + δ)/α, α → 0. Moreover, this solution xh,δ , for every ﬁxed α > 0, depends α continuously on Fih , i = 1, 2 and fδ , the ﬁnite-dimensional problems h h x + F2,α,n F1,α,n (x) = fδ,n , x ∈ Xn , (1.3) h h ∗ h ∗ h where F2,α,n = Pn F2,α Pn , F1,α,n = Pn F1,α Pn , fδ,n = Pn fδ , Pn is a linear projection from X onto its ﬁnite-dimensional subspace Xn such that Xn ⊂ Xn+1 , Pn x → x, as n → ∞ for every ∗ x ∈ X , and Pn is the dual of Pn with Pn c = constant, for all n, have a unique solution ˜ h,δ h,δ h,δ xα,n , and the sequence {xα,n } converges to xα , as n → ∞, without additional conditions on Fi , i = 1, 2. In the case of linearity for F2 and fδ = f for all δ > 0, the convergence rates for the sequences {xh,δ } and {xh,δ } are given in the paper [3] provided the existence of bounded α α,n −1 , where I denotes the identity operator in X . It is not diﬃcult to inversion (I + F2 F1 (x0 )) verify that this condition can be replaced by the bounded inversion of (I + F2 (x∗ )F1 (x0 ))−1 , 0 when F2 also is nonlinear, where x∗ = F1 (x0 ). The last requirement is equivalent to that 0 -1 is not an eigenvalue of the operator F2 (x∗ )F1 (x0 ) and is used in studying a method of 0 collocation-type for nonlinear integral equations of Hammerstein type (see [6]). In general case, i.e., when both the operators Fi , i = 1, 2, are nonlinear, it means that R, the range of the operator I + F2 (x∗ )F1 (x0 ), is the whole space X . It is natural to ask if we can estimate 0 the convergence rates for the sequences {xh,δ }, {xh,δ }, when R is not the whole space X . For α α,n this purpose, only demanding that R contains a necessary element of X , the convergence rates of {xh,δ } and {xh,δ } are estimated in [4], [5] on the base of the zero value of the derivatives α α,n of higher order for F1 and F2 at x0 and x∗ , respectively. This result is formulated in the 0 following theorem. Theorem 1.1. (see [4] or [5]). Let the following conditions hold: (i) F1 is Fr´chet diﬀerentiable at some neighbourhood U0 of x0 s1 −1-times if s1 = [s1 ], the e e integer part of s1 , [s1 ]-times if s1 = [s1 ], and F2 is Fr´chet diﬀerentiable at some neighbourhood V0 of x∗ s2 − 1-times, if s2 = [s2 ], [s2 ]-times if s2 = [s2 ], 0 (ii) there exists a constant L > 0 such that (k) (k) F1 (x0 ) − F1 (y) (k) (k) F2 (x∗ ) − F2 (y ∗ ) 0 (k) L x0 − y , ∀ y ∈ U 0 , L x∗ − y ∗ , ∀ y ∗ ∈ V 0 , 0 for Fi : k = si − 1 if si = [si ], k = [si ] if si = [si ], and if [si ] (k) (2) (k) F1 (x0 ) = 0, and F2 (x∗ ) = ... = F1 (x∗ ) = 0, 0 0 (2) 3, then F1 (x0 ) = ... = 52 NGUYEN BUONG, DANG THI HAI HA (iii) there exists an element x1 ∈ X such that I + F2 (x∗ )∗ F1 (x0 )∗ x1 = F2 (x∗ )∗ U1 (x0 ) − U2 (x∗ ), 0 0 0 if s1 = [s1 ] then L x1 < m1 s1 !, and if s2 = [s2 ] then L F1 (x0 )∗ x1 − U1 (x0 ) < m2 s2 ! Then, if α is chosen such that α ∼ (h + ε)ρ , 0 < ρ < 1, we have xω − x0 = O((h + ε)θ ), 1 − ρ + θ2 }, θ = min {θ1 , s1 − 1 1−ρ ρ , }, i = 1, 2. θi = min { si si In this paper, the convergence rates of {xh,δ } and {xh,δ } are established under much weaker α α,n conditions on Fi , i = 1, 2. These are the assumptions that R contains some element of X , and Fi , i = 1, 2, are inverse-strongly monotone, i.e. F1 (x) − F1 (y), x − y ∗ ∗ ∗ F2 (x ) − F2 (y ), x − y ∗ m1 F1 (x) − F1 (y) 2 , ˜ ∗ ∗ 2 m2 F2 (x ) − F2 (y ) , ˜ x, y ∈ X, x∗ , y ∗ ∈ X ∗ , (1.4) where mi , i = 1, 2, are some positive constants. Note that in [7] the inverse-strongly monotone ˜ property was used to estimate the convergence rates of the regularized solutions for ill-posed variational inequalities. Below, by “a ∼ b” we mean “a = O(b) and b = O(a)”. 2. MAIN RESULTS Assume that the normalized dual mappings Ui , i = 1, 2, of the spaces X and X ∗ satisfy the following conditions (see [8]) i i i i Ui (y1 ) − Ui (y2 ), y1 − y2 i i Ui (y1 ) − Ui (y2 ) i i mi y 1 − y 2 si , mi > 0, si i i ci (Ri ) y1 − y2 νi , 0 < νi 2, 1, (2.1) (2.2) i i where y1 , y2 ∈ X or X ∗ on dependence of i = 1 or 2, respectively, and ci (Ri ), Ri > 0, are i i the positive increasing functions on Ri = max { y1 , y2 }. The following theorem answers the question on convergence rates for {xh,δ }. α Theorem 2.1. Assume that the following conditions hold: (i) Fi , i = 1, 2, are inverse-strongly monotone and continuously Fr´chet diﬀerentiable at e ∗ , respectively, and some neighbourhoods U of x0 and V of x0 F1 (x) − F1 (x0 ) − F1 (x0 )(x − x0 ) ∗ F2 (x ) − F2 (x∗ ) − 0 F2 (x∗ )(x∗ 0 − x∗ ) 0 where τi , i = 1, 2, are some positive constants, τ1 F1 (x) − F1 (x0 ) , ∗ τ2 F2 (x ) − F2 (x∗ ) 0 , ∀x ∈ U , ∀x∗ ∈ V, ABOUT CONVERGENCE RATES IN REGULARIZATION FOR ILL-POSED 53 (ii) there exists an element x1 ∈ X such that I + F2 (x∗ )∗ F1 (x0 )∗ x1 = F2 (x∗ )∗ U1 (x0 ) − U2 (x∗ ). 0 0 0 Then, if α is chosen such that α ∼ (h + δ)ρ , 0 < ρ < 1, we have xh,δ − x0 α = O (h + δ)θ/s1 ), θ = min {ρ/2, 1 − ρ}. Proof. Set A = m1 xh,δ − x0 α s1 + m2 xh,δ,∗ − x∗ α 0 s2 h , xh,δ,∗ = F1,α (xh,δ ). α α It is easy to see that x0 is a solution of (1.1) iﬀ z0 = [x0 , x∗ ] is a solution of the system of 0 following operator equations F1 (x) − x∗ = 0, F2 (x∗ ) + x − f = 0. h,δ Similarily, xh,δ is a regularized solution of the operator equation (1.2) iﬀ zα = [xh,δ , xh,δ,∗ ] is α α α a solution of the system of following equations h F1 (x) + αU1 (x) − x∗ = 0, h F2 (x∗ ) + αU2 (x∗ ) + x − fδ = 0. Consider the space Z = X × X ∗ with the norm z 2 = x 2 + x∗ 2 , z = [x, x∗ ], x ∈ X, and x∗ ∈ X ∗ . Then, the two above systems of equations can be written, respectively, in form of equations A(z) = f , (2.3) Ah (z) ≡ Ah (z) + αJ(z) = f δ , α where A(z) = [F1 (x), F2 (x∗ )] + [−x∗ , x], h h Ah (z) = [F1 (x), F2 (x∗ )] + [−x∗ , x], J (z) = [U1 (x), U2 (x∗ )], f = [0, f ], (2.4) f δ = [0, fδ ]. It is easy to verify that A and Ah are the monotone operators from Z to Z ∗ = X ∗ × X , and the operator J is the normalized duality mapping of the space Z . Hence, from (2.1), (2.3), (2.4) and the monotone property of Ah it implies that A h,δ h,δ h,δ J(zα ) − J (z0 ), zα − z0 J(z0 ), z0 − zα 1 h,δ h,δ + [ f δ − f , zα − z0 + A(z0 ) − Ah (z0 ), zα − z0 ]. α (2.5) It is not diﬃcult to verify that Ah (z) − A(z) √ 2hg( z ). (2.6) 54 NGUYEN BUONG, DANG THI HAI HA Further, from (1.4) it follows h,δ h,δ A(zα ) − A(z0 ), zα − z0 = F1 (xh,δ ) − xh,δ,∗ − (F1 (x0 ) − x∗ ), xh,δ − x0 α α 0 α + F2 (xh,δ,∗ ) + xh,δ − (F2 (x∗ ) + x0 ), xh,δ,∗ − x∗ α α 0 α 0 = F1 (xh,δ ) − F1 (x0 ), xh,δ − x0 + F2 (xh,δ,∗ ) − F2 (x∗ ), xh,δ,∗ − x∗ α α α 0 α 0 m1 F1 (xh,δ ) − F1 (x0 ) ˜ α min{m1 , m2 }C 2 , ˜ ˜ 2 + m2 F2 (xh,δ,∗ ) − F2 (x∗ ) ˜ α 0 C 2 = F1 (xh,δ ) − F1 (x0 ) α 2 2 + F2 (xh,δ,∗ ) − F2 (x∗ ) 2 . α 0 On the other hand, from (2.3), (2.4)-(2.6) and the properties of A, Ah , J, g we have h,δ h,δ A(zα ) − A(z0 ), zα − z0 h,δ f δ − f , zα − z0 h,δ h,δ h,δ h,δ + α J(z0 ), z0 − zα + A(zα ) − Ah (zα ), zα − z0 , h,δ and {zα } is bounded, as (h + δ)/α → 0. Therefore, C2 Consequently, C √ 1 h,δ h,δ [δ + α J(z0 ) + 2hg( zα )] zα − z0 . min{m1 , m2 } ˜ ˜ √ O( h + δ + α). Hence, √ F1 (xh,δ ) − F1 (x0 ) O( h + δ + α), α √ F2 (xh,δ,∗ ) − F2 (x∗ ) O( h + δ + α). α 0 (2.7) h,δ Now, we shall estimate the value J(z0 ), z0 − zα . For this purpose, set x2 = U1 (x0 ) − F1 (x0 )∗ x1 . From condition (ii) of the theorem it follows that x1 and x2 (∈ X ∗ ) satisfy the system of following equalities F1 (x0 )∗ x1 + x2 = U1 (x0 ), F2 (x∗ )∗ x2 − x1 = U2 (x∗ ). 0 0 By virtue of h,δ J(z0 ), z0 − zα = U1 (x0 ), x0 − xh,δ + U2 (x∗ ), x∗ − xh,δ,∗ α 0 0 α = F1 (x0 )∗ x1 + x2 , x0 − xh,δ + F2 (x∗ )∗ x2 − x1 , x∗ − xh,δ,∗ α 0 0 α = xh,δ,∗ − x∗ − F1 (x0 )(xh,δ − x0 ), x1 α 0 α + x0 − xh,δ − F2 (x∗ )(xh,δ,∗ − x∗ ), x2 α 0 α 0 = F1 (xh,δ ) − F1 (x0 ) − F1 (x0 )(xh,δ − x0 ), x1 α α h + α U1 (xh,δ ), x1 + F1 (xh,δ ) − F1 (xh,δ ), x1 α α α + F2 (xh,δ,∗ ) − F2 (x∗ ) − F2 (x∗ )(xh,δ,∗ − x∗ ), x2 α 0 0 α 0 h + αU2 (xh,δ,∗ ) + f − fδ , x2 + F2 (xh,δ,∗ ) − F( xh,δ,∗ ), x2 , α α α we have h,δ J(z0 ), z0 − zα max{τ1 x1 , τ2 x2 } × ( F1 (xh,δ ) − F1 (x0 ) + F2 (xh,δ,∗ ) − F2 (x∗ ) ) + O(h + δ + α). α α 0