Continuous regularization method for ill posed operator equations of hammerstein type

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The aim of this paper is to study a method of approximating a solution of the operator equation of Hammerstein type x + F2F1(x) = f on the base of constructing a system of differential equations of the first order, where Fi , i = 1, 2, are the continuous monotone operators in real Hilbert space H. Then this method is considered in connection with finite-dimentional approximations for H.

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Nội dung Text: Continuous regularization method for ill posed operator equations of hammerstein type

’ Tap ch´ Tin hoc v` Diˆu khiˆn hoc, T.23, S.2 (2007), 99–108 ı a ` e e . . . CONTINUOUS REGULARIZATION METHOD FOR ILL-POSED OPERATOR EQUATIONS OF HAMMERSTEIN TYPE NGUYEN BUONG1 , DANG THI HAI HA2 1 Viˆn Cˆng nghˆ thˆng tin, Viˆn Khoa hoc v` Cˆng nghˆ Viˆt Nam e o e o e e e . . . . a o . . 2 Vietnamese Forestry University, Xuan Mai, Ha Tay Abstract. The aim of this paper is to study a method of approximating a solution of the operator equation of Hammerstein type x + F2 F1 (x) = f on the base of constructing a system of diﬀerential equations of the ﬁrst order, where Fi , i = 1, 2, are the continuous monotone operators in real Hilbert space H . Then this method is considered in connection with ﬁnite-dimentional approximations for H. ´ ´ ’ o a a ’ e ınh T´m t˘t. Muc d´ cua b`i b´o l` nghiˆn c´.u mˆt phu.o.ng ph´p xˆp xı nghiˆm cua phu.o.ng tr` o a a a a e u . . . ıch ’ ´ to´n tu. loai Hammerstein x + F2 F1 (x) = f du.a trˆn viˆc xˆy du.ng hˆ phu.o.ng tr` vi phˆn cˆp a ’ . e e a e ınh a a . . . . ’. dˆy c´c to´n tu. Fi , i = 1, 2, l` do.n diˆu v` liˆn tuc trong khˆng gian Hilbert H . Sau d´, ’ a a e a e . o o mˆt, o a a o . . ´ ´ ` ’ e e e o a a e a ’ u e phu.o.ng ph´p n`y du.o.c x´t liˆn kˆt v´.i viˆc xˆ p xı h˜.u han chiˆu cua H. . . . 1. INTRODUCTION Let H be a real Hilbert space with norm and scalar product denoted by . and x∗ , x , respectively. Let Fi , i = 1, 2, 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 ∈ R(I + F2 F1 ), (1.1) where I and R(A) denote the identity operator in H and the range of the operator A, respectively. Note that the solution set of (1.1), denoted by S0 , is closed convex (see [1]). Fih Usually instead of Fi , i = 1, 2, and f we know their monotone continuous approximations and fδ such that h F1 (x) − F1 (x) h F2 (x) − F2 (x) ∗ This hg( x ), hg( x ) ∀x ∈ H, work was supported by the Vietnamese Fundamental Research Program in Natural Sciences, MS 100506. 100 NGUYEN BUONG, DANG THI HAI HA where g(t) is a real nonnegative, non-decreasing, bounded function (the image of a bounded set is bounded), and fδ − f δ . Without additional conditions for the operators Fi such as the strongly monotone property, equation (1.1) is ill-posed. For example, consider the case H = E2 , the Euclidean space, and 1 −1 0 −1 F1 = , F2 = , x = (x1 , x2 ). 1 0 1 1 It is easy to verify that F1x, x = x2 0, and F2 x, x = x2 0∀x ∈ E2 . It means that 1 2 Fi , i = 1, 2, are monotone. Equation (1.1) has the form 0x1 = f1 , 2x1 = f2 with f = (f1 , f2). Obviously, this system of equations has a unique solution when f = (0, f2) for arbitrary f2 . δ δ When fδ = (f1 , f2) with f1 = 0 equation (1.1) in this case does not have solution. So, equation (1.1) with the monotone operators F1 , i = 1, 2, in general is ill-posed. To solve (1.1) we need use stable methods. One of the those is the operator equation h h x + F2,α F1,α (x) = fδ (1.2) h (see [1], [5]), where Fi,α = Fih + αI , α > 0 is the small parameter of regularization. For every α > 0 equation (1.2) has a unique solution xh,δ , and the sequence {xh,δ } converges to a α α solution x0 of satisfying x0 2 + x∗ 2 = min x 2 + F1(x) 2 , x∗ = F1 (x0), 0 0 x∈S0 as (h + δ)/α, α → 0. Moreover, this solution xh,δ , for every ﬁxed α > 0, depends continuously α on Fih , i = 1, 2 and fδ . Recently, the use of diﬀerential equations for regularizing ill-posed convex optimization and nonlinear monotone problems is intensively investigated (see [6]-[14] and references therein), because by discretiting them one can obtain much diﬀerent iterative processes. In this paper, this idea is developed for non-monotone, in general, Hammerstein equation, i.e., we ﬁnd a strong diﬀerentiable function u(t) : [t0, +∞) → H, t0 0, which is a solution of some diﬀeretial equation such that lim u(t) = x0 . (1.3) t→+∞ In Section 2, we give a system of diﬀerential equations with the solution u(t), u∗ (t) where u(t) satisﬁes (1.3). The Galerkin approximations un (t) for u(t) with the property lim un (t) = x0 , n,t→+∞ are considered in Section 3. Above and below, the symbols the norm, respectively. and → denote the weak convergence and convergence in 2. THE INFINITE-DIMENSIONAL CONTINUOUS REGULARIZATION Consider the system of diﬀerential equations du(t) h(t) + γ(t) F1 (u(t)) + α(t)u(t) − u∗ (t) = θ, dt du∗(t) h(t) + γ(t) F2 (u∗(t)) + α(t)u∗ (t) + u(t) − f (t) = θ, dt u(t0 ) = u0 , u∗(t0 ) = u∗ , t t0 0, 0 (2.1) CONTINUOUS REGULARIZATION METHOD FOR ILL-POSED OPERATOR EQUATIONS OF 101 where u0, u∗ are the ﬁxed elements in H , θ denotes the zero element, h = h(t), α = α(t) > 0 0, t 0, α(t) is a convex decreasing diﬀerentiable function, γ(t) is a nondecreasing positive and diﬀerentiable function such that lim α(t) = lim h(t) = 0, t→+∞ t→+∞ α (t) γ (t) h(t) = lim 2 = lim = 0. t→+∞ α (t)γ(t) t→+∞ α(t)γ 2 (t) t→+∞ α(t) (2.2) lim In order to prove that limt→+∞ u(t) = x0, we study the system of diﬀerential equations dy(t, τ ) + γ(t) F1 (y(t, τ )) + α(τ )y(t, τ ) − y ∗ (t, τ ) = θ, dt dy ∗ (t, τ ) (2.3) + γ(t) F2 (y ∗(t, τ )) + α(τ )y ∗ (t, τ ) + y(t, τ ) − f = θ, dt y(t0 , τ ) = u0 , y ∗(t0 , τ ) = u∗ , ∀t t0 0 depending on the parameter τ We have a result. t0 . Theorem 2.1. Assume that the following conditions hold: (i) problems (2.1) and (2.3) possess solutions in the class C 1 [t0, +∞) for any u0 , u∗ ∈ H 0 with u(t) , u∗(t) d1 , d1 > 0, t t0 . (ii) the functions α(t), h(t) and γ(t) satisfy the above conditions. Then, limτ →+∞ u(τ ) = x0 . Proof. Set r (t, τ ) = r1 (t, τ ) + r2(t, τ ), ˜ ˜ ˜ r1(t, τ ) = y(t, τ ) − xα (τ ) 2, ˜ r2(t, τ ) = y ∗(t, τ ) − x∗ (τ ) 2 , ˜ α where (xα(τ ), x∗ (τ )), x∗ (τ ) = F1 (xα (τ )), is the unique solution of the system of operator α α equations F1 (xα(τ )) + α(τ )xα(τ ) − x∗ (τ ) = θ, α (2.4) ∗ ∗ F2 (xα(τ )) + α(τ )xα(τ ) + xα (τ ) − f = θ, and limτ →+∞ xα (τ ) = x0 (see [1]). Since F1 is continuous, then x∗ = limτ →+∞ x∗ (τ ). Now, α 0 from (2.3) and (2.4) it follows d(y(t, τ ) − xα (τ )) , y(t, τ ) − xα (τ ) + γ(t) F1 (y(t, τ )) − F1 (xα (τ )), dt y(t, τ ) − xα(τ ) + α(τ )˜1 (t, τ ) + x∗ (τ ) − y ∗(t, τ ), y(t, τ ) − xα (τ ) = 0, r α d(y ∗ (t, τ ) − x∗ (τ )) ∗ α , y (t, τ ) − x∗ (τ ) + γ(t) F2 (y ∗(t, τ )) − F2 (x∗ (τ )), α α dt y ∗ (t, τ ) − x∗ (τ ) + α(τ )˜2(t, τ ) + y(t, τ) − xα (τ ), y ∗(t, τ ) − x∗ (τ ) = 0. r α α Substituting the two last equalities and using the relation dx(t) d x(t) 2 =2 , x(t) dt dt and the monotone property of Fi , i = 1, 2, we have got d˜(t, τ ) r + 2γ(t)α(τ )˜(t, τ ) 0. r dt 102 NGUYEN BUONG, DANG THI HAI HA Hence, t r (t, τ ) ˜ r (t0 , τ ) exp[−2α(τ ) ˜ γ(t)dt], (2.5) t0 where r(t0 , τ ) = y(t0, τ ) − xα(τ ) ˜ 2[ y(t0, τ ) 2[ u0 2 2 2 + y ∗ (t0 , τ ) − x∗ (τ ) α + xα (τ ) + u∗ 0 2 + x0 2 + y ∗ (t0, τ ) 2 2 2 + F1 (x0 ) 2 ]. + x∗ (τ ) 2] α Consequently, from (2.2), (2.5) and the properties of γ(t), α(t) we can obtain (see [13] for details) lim r (τ, τ ) = 0 ˜ τ →+∞ and the boundness of {y(t, τ )} and {y ∗ (t, τ )}. Therefore, lim y(τ, τ ) = x0 , τ →+∞ and there exists a positive constant d2 such that y(t, τ ) , y ∗(t, τ ) ˜ ˜ ˜ R(t, τ ) = R1 (t, τ ) + R2(t, τ ), d2 . Further, set ˜ R1 (t, τ ) = u(t) − y(t, τ ) 2, ˜ R2 (t, τ ) = u∗ (t) − y ∗ (t, τ ) 2. On the base of (2.1) and (2.3) we can write d(u(t) − y(t, τ )) h(t) , u(t) − y(t, τ ) + γ(t) F1 (u(t)) − F1 (y(t, τ )), dt u(t) − y(t, τ ) + α(t)u(t) − α(τ )y(t, τ ), u(t) − y(t, τ ) + y ∗(t, τ ) − u∗ (t), u(t) − y(t, τ ) = 0, d(u∗ (t) − y ∗ (t, τ )) ∗ h(t) , u (t) − y ∗ (t, τ ) + γ(t) F2 (u∗ (t)) − F2 (y ∗ (t, τ )), dt u∗ (t) − y ∗ (t, τ ) + α(t)u∗ (t) − α(τ )y ∗ (t, τ ), u∗(t) − y ∗ (t, τ ) + u(t)−y(t, τ ), u∗(t) − y ∗ (t, τ ) Thus, = 0. ˜ dR(t, τ ) ˜ +2γ(t)α(τ )R(t, τ ) dt γ(t)[h(t)g( y(t, τ ) ) + |α(t) − α(τ )| y(t, τ ) ] u(t) − y(t, τ ) + γ(t)[h(t)g( y ∗(t, τ ) ) + |α(t) − α(τ )| y ∗ (t, τ ) ] u∗(t) − y ∗ (t, τ ) . Hence, ˜ dR(t, τ ) Dγ(t)[h(t) + |α(t) − α(τ )|] − 2α(t)R(t, τ ), ˜ ˜ dt α(t) = γ(t)α(τ ), D = 2 max{g(d2 )(d1 + d2 ), d2}. ˜ It is not diﬃcult to verify that CONTINUOUS REGULARIZATION METHOD FOR ILL-POSED OPERATOR EQUATIONS OF ˜ R(τ, τ ) 103 R1 (τ ) + R2 (τ ) τ R1 (τ ) = D γ(t)h(t)ξ(t)dt/ξ(τ ), t0 τ R2 (τ ) = D γ(t)α (t)(t − τ )ξ(t)dt/ξ(τ ), t0 s ξ(s) = exp( α(t)dt. ˜ t0 Therefore, limt→+∞ R1 (τ ) = limt→+∞ R2 (τ ) = 0. Since x0 − u(τ ) x0 − xα(τ ) + xα (τ ) − y(τ, τ ) + y(τ, τ ) − u(τ ) , then limτ →+∞ u(τ ) = x0 . Theorem is proved. Remark. The solution existence of (2.1) or (2.3) is followed from [7], [15] and [16], when h(t) Fi are weakly continuous or Lipschitz continuous for each t t0 . 3. FINITE-DIMENSIONAL REGULARIZATION Consider the system of ﬁnite-dimensional problems dun (t) h(t) + γ(t) F1,n (un (t)) + α(t)un (t) − u∗ (t) = θ, n dt ∗ (t) dun h(t) + γ(t) F2,n (u∗ (t)) + α(t)u∗ (t) + un (t) − fn (t) = θ, n n dt ∗ ∗ un (t0 ) = Pn u0 , un (t0) = Pn u0 , h(t) h(t) h(t) (3.1) h(t) ∗ ∗ where F1,n = Pn F1 Pn , F2,n = Pn F2 Pn , fn (t) = Pn f (t), Pn is a linear projection from H onto its ﬁnite-dimensional subspace Hn such that Hn ⊂ Hn+1 , Pn x → x, as n → ∞ for ∗ every x ∈ H , and Pn is the dual of Pn with Pn c = constant, for all n, and un (t), u∗ (t) : ˜ n [t0 , +∞) → Hn . To prove lim un (t) = x0 , n,t→+∞ as in the Section 2, we use the system of ﬁnite-dimensional equations dyn (t, τ ) ∗ + γ(t) F1,n (yn (t, τ )) + α(τ )yn (t, τ ) − yn (t, τ ) = θ, dt ∗ dyn (t, τ ) ∗ ∗ + γ(t) F2,n (yn (t, τ )) + α(τ )yn (t, τ ) + yn (t, τ ) − fn = θ, dt ∗ yn (t0 , τ ) = Pn u0 , yn (t0 , τ ) = Pn u∗ , ∀t t0 , 0 depending on the parameter τ We have a result. (3.2) ∗ ∗ t0 , where F1,n = Pn F1 Pn , F2,n = Pn F2 Pn , and fn = Pn f. Theorem 3.1. Assume that the following conditions hold: (i) problems (3.1) and (3.2) possess solutions in the class C 1 [t0, +∞) for any u0 , u∗ ∈ H 0 with un (t) , u∗ (t) d3 , d3 > 0, t t0 . n (ii) the functions α(t), h(t) and γ(t) satisfy the above conditions. (iii) Fi , i = 1, 2, are Fr´chet diﬀerentiable with Lipschitz continuous derivatives ( common e Lipschitz constant L), there exist x1 and x2 such that