An Iteratively Regularized Projection Method with Quadratic Convergence for Nonlinear Ill-posed Problems

Transcription

1 Int. Journal of Math. Analysis, Vol. 4, 1, no. 45, 11-8 An Iteratively Regularized Projection Method with Quadratic Convergence for Nonlinear Ill-posed Problems Santhosh George Department of Mathematical and Computational Sciences National Institute of Technology Karnataka Surathkal, India Atef Ibrahim Elmahdy Department of Mathematical and Computational Sciences National Institute of Technology Karnataka Surathkal, India Abstract An iteratively regularized projection method, which converges quadratically, has been considered for obtaining stable approximate solution to nonlinear ill-posed operator equations F (x) = y where F : D(F ) X X is a nonlinear monotone operator defined on the real Hilbert space X. We assume that only a noisy data y δ with y y δ δ are available. Under the assumption that the Fréchet derivative F of F is Lipschitz continuous, a choice of the regularization parameter using an adaptive selection of the parameter and a stopping rule for the iteration index using a majorizing sequence are presented. We prove that under a general source condition on x ˆx, the error n, ˆx between the regularized approximation x h,δ n,, (x h,δ, := P hx where P h is an orthogonal projection on to a finite dimensional subspace X h of X) and the solution ˆx is of optimal order. Mathematics Subject Classification:65J,65J15, 47J6 Keywords: Nonlinear ill-posed operator, Monotone operator, Majorizing sequence, Regularized Projection method, Quadratic convergence.

2 1 S. George and A. I. Elmahdy 1 Introduction Let F : D(F ) X X be a nonlinear monotone operator (see [13]) defined on a real Hilbert space X. We consider the problem of approximately solving the nonlinear ill-posed operator equation; F (x) = y. (1) Throughout this paper we shall denote the inner product and the corresponding norm on X by.,. and. respectively. And we assume that (1) has a solution, namely ˆx and that y δ X are the available noisy data with y y δ δ. () Then the problem of recovery of ˆx from noisy equation F (x) = y δ is ill-posed, in the sense that the Fréchet derivative F (.) is not boundedly invertible (see [1], page 6). A well known method for regularizing (1), when F is monotone is the method of Lavrentiev regularization (see [13]). In this method approximation x δ is obtained by solving the singularly perturbed operator equation F (x) + (x x ) = y δ. (3) In practice, one has to deal with some sequence (x δ n,) n=1, converging to the solution x δ of (3). Many authors considered such sequences (see [, 3, 6, 7, 9, 8]). In [11], George and Elmahdy considered an iterative regularization method; x δ n+1, = x δ n, (F (x δ n,) + I) 1 (F (x δ n,) y δ + (x δ n, x )), (4) where x δ, = x and proved that (4) converges quadratically to the unique solution x δ of (3). Recall that, a sequence (x n ) is said to be converges quadratically to x, if there exist a positive number M, not necessarily less than 1, such that x n+1 x M x n x, for all n sufficiently large. And the convergence of (x n ) to x, is said to be linear if there exist a positive number M (, 1), such that x n+1 x M x n x. Note that regardless of the value of M quadratic convergent sequence will always eventually converges faster than a linearly convergent sequence. The following Assumptions are used for proving the results in [11] as well as the results in this paper.

3 An Iteratively Regularized Projection Method 13 Assumption 1.1 There exists r > such that B r (ˆx) D(F ) and F is Fréchet differentiable at all x B r (ˆx). Assumption 1. There exists a constant k > such that for every x, u B r (ˆx) and v X, there exists an element Φ(x, u, v) X satisfying [F (x) F (u)]v = F (u)φ(x, u, v), Φ(x, u, v) k v x u for all x, u B r (ˆx) and v X. Assumption 1.3 There exists a continuous, strictly monotonically increasing function ϕ : (, a] (, ) with a F (ˆx) satisfying lim ϕ(λ) = and λ v X with v 1 such that x ˆx = ϕ(f (ˆx))v and sup ϕ(λ) λ λ + c ϕϕ(), (, a]. The analysis in [11] as well as in this paper is based on majorizing sequences. Recall (see [5], Definition ) that a nonnegative sequence (t n ) is said to be a majorizing sequence of a sequence (x n ) in X if x n+1 x n t n+1 t n, n. The majorizing sequence gives an a priori error estimate which can be used to determine the number of iterations needed to achieve a prescribed solution accuracy before actual computation take place. The plan of this paper is as follows. In section we collect some results from [11] for the error estimates in this paper. In section 3 we considered an iteratively regularized projection method for obtaining a sequence (x h,δ n,) in a finite dimensional subspace X h of X and proved that x h,δ n, converges to x δ. Also in section 3 we obtained an estimate for n, x δ. Using an error estimate for x δ ˆx (see [11, 13]), we obtained an estimate for n, ˆx in section 4. The error analysis for the order optimal result using an adaptive selection of the parameter and a stopping rule using a majorizing sequence are also given in section 4. Implementation of the adaptive choice of the parameter and the choice of the stopping rule are given in section 5. Finally the paper ends with some concluding remarks in section 6.

4 14 S. George and A. I. Elmahdy Preliminaries In [11] the following majorizing sequence (t n ) defined iteratively by, t =, t 1 = η, and t n+1 = t n + 3k (t n t n 1 ) (5) where k, η,and q [, 1) are nonnegative numbers such that 3k η q (6) were used for proving the quadratic convergence of the sequence (x δ n,) to the unique solution x δ of equation (3). For proving the results in [11] as well as the results in this paper we use the following Lemma on majorization, which is a reformulation of Lemma in [5]. Lemma.1 Let (t n ) be a majorizing sequence for (x n ). If x = lim x n n exists and lim n t n = t, then x x n t t n, n. (7) The following Lemma based on the Assumption 1. will be used in due course. Lemma. For u, v B r (x ) 1 F (u) F (v) F (u)(u v) = F (u) Φ(v + t(u v), u, u v)dt. Proof.Using the Fundamental Theorem of Integration, for u, v B r (x ) we have and so, F (u) F (v) = F (u)(u v) = F (v + t(u v))(u v)dt F (u)(u v)dt F (u) F (v) F (u)(u v) = 1 [F (v + t(u v)) F (u)](u v)dt

5 An Iteratively Regularized Projection Method 15 so by Assumption 1. we have 1 F (u) F (v) F (u)(u v) = F (u) Φ(v + t(u v), u, u v)dt. This completes the proof of the Lemma. Here after we assume that x ˆx ρ and k ρ + ρ + δ η min{ q 3k, r (1 q)}. (8) Theorem.3 ([11], Theorem.1) Suppose Assumption 1. holds. Let < t η := t where η as in (8) and let (5) and (6) be satisfied. Then 1 q the sequence (x δ n,) defined in (4) is well defined and x δ n, B t (x ) for all n. Further (x δ n,) is a Cauchy sequence in B t (x ) and hence converges to x δ B t (x ) B t (x ) and F (x δ ) = y δ + (x x δ ). Moreover, the following estimate hold for all n, and x δ n+1, x δ n, t n+1 t n, (9) x δ n, x δ t t n qn η 1 q, (1) x δ n+1, x δ 3k xδ n, x δ. (11) Remark.4 Note that (11) implies (x δ n,) converges quadratically to x δ. 3 Iteratively Regularized Projection Method Let H be a bounded subset of positive reals such that zero is a limit point of H, and let {P h } h H be a family of orthogonal projections from X into itself. We assume that b h := (I P h )x (1) as h. The above assumption is satisfied if P h I pointwise. Let x h,δ n+1, = x h,δ n, (P h F (x h,δ n,) + I) 1 P h (F (x h,δ n,) y δ + (x h,δ n, x )), (13) where x h,δ, := P h x. Let Γ n,h := (I P h )F (x δ n,), (14) ϱ := F (P h x ),

6 16 S. George and A. I. Elmahdy γ n,h := F (x h,δ n,)(i P h ) (15) and let ( t n,h ), n be defined iteratively by t,h =, t 1,h = >, t n+1,h = t n,h + (1 + k ϱ t n,h + γ,h ) 3k ( t n,h t n 1,h ) (16) where k,, and r h [, 1) are nonnegative numbers. Lemma 3.1 Assume there exist nonnegative numbers k,, and r h [, 1 such that for all n (1 + k ϱ + γ,h ) 3k r h. (17) Then the sequence ( t n,h ) defined in (16) is increasing, bounded above by t h := 1 r h, and converges to some t h such that < t h 1 r h. Moreover, for n ; t n+1,h t n,h r h ( t n,h t n 1,h ) r n h, (18) 1+ 3k ϱ η h ) and t h t n,h rn h 1 r h. (19) Proof.Since the result is true for =, k = or r h =, we assume that, k and r h. Observe that t i,h t i 1,h for all i 1. If (1 + k ϱ t i,h + γ,h ) 3k ( t i,h t i 1,h ) r h () then the estimate (18) follows from (16). Thus we shall prove () by induction on i 1. For i = 1, () holds by (17). Suppose () holds for all i k for some k. Then by (16) we have (1 + k ϱ t k+1,h + γ,h ) 3k ( t k+1,h t k,h ) (1 + k ϱ t k+1,h + γ,h ) 3k (1 + k ϱ t k,h + γ,h ) 3k ( t k,h t k 1,h ) (1 + k ϱ( t k+1,h t k,h + t k,h ) + γ,h ) 3k (1 + k ϱ t k,h + γ,h ) 3k ( t k,h t k 1,h ) [(1 + k ϱ t k,h + γ,h ) 3k ( t k,h t k 1,h )]

7 An Iteratively Regularized Projection Method 17 + k ϱ( t k+1,h t k,h ) 3k Thus by induction () holds for all i 1 and for k, (1 + k ϱ t k,h + γ,h ) 3k ( t k,h t k 1,h ) [(1 + k ϱ t k,h + γ,h ) 3k ( t k,h t k 1,h )] + 3k ϱ [(1 + k ϱ t k,h + γ,h ) 3k ( t k,h t k 1,h ) ] rh + 3k ϱ r h( t k,h t k 1,h ) rh + 3k ϱ r h(rh k 1 ) r h. (1) t k+1,h t k,h + r h ( t k,h t k 1,h ) + r h + + r k h () Hence the sequence ( t n,h ), n is bounded above by so it converges to some t h 1 r h. Further t h t n,h = lim i t n+i,h t n,h This completes the proof of the Lemma. Hereafter we assume; = 1 rk+1 h < (3) 1 r h 1 r h lim i 1 i j= (1 + γ,h )(k (b h + ρ) + b h + ρ) + δ where C := 1 [ ( + γ k ϱ,h) + ( + γ,h ) + 8ϱ r h ]. 3 Remark 3. We need the above assumption because: 1 r h and nondecreasing, ( t n+1+j,h t n+j,h ) rn h. 1 r h min{c, r (1 r h )} (4) The assumption (4) implies t h = Assumption r h r and hence we can apply Equation (17) implies 1 k ϱ [ ( + γ,h) + ( + γ,h ) + 8ϱ r h ] = C. (5) 3

8 18 S. George and A. I. Elmahdy Theorem 3.3 Let the assumptions in Lemma 3.1 with as in (4) and Assumption 1. be satisfied. Then the sequence ( t n,h ) defined in (16) is a majorizing sequence of sequence (x h,δ n,) defined in (13) and x h,δ n, Bt h (P h x ) for all n. Proof. Let G(x) = x R (x) 1 [F (x) y δ + (x x )] where R (x) 1 = (P h F (x)p h + P h ) 1. Then for u, v B t h (P h x ), we have G(u) G(v) = u v R (u) 1 [F (u) y δ + (u x )] +R (v) 1 [F (v) y δ + (v x )] = R (u) 1 [R (u)(u v) (F (u) F (v)) (u v)] +(R (v) 1 R (u) 1 )(F (v) y δ + (v x )) = R (u) 1 [P h F (u)p h (u v) (F (u) F (v))] +R (u) 1 (R (u) R (v))r (v) 1 (F (v) y δ + (v x )) = R (u) 1 [P h F (u)p h (u v) (F (u) F (v))] R (u) 1 (R (u) R (v))(g(v) v). Now since G(x h,δ n,) = x h,δ n+1,, R (x) 1 = R (x) 1 P h = P h R (x) 1 and P h (x h,δ x h,δ n 1,) = (x h,δ n, x h,δ n 1,) we have (x h,δ n+1, x h,δ n,) = G(x h,δ n,) G(x h,δ n 1,) = R (x h,δ n,) 1 [F (x h,δ n,)(x h,δ n, x h,δ n 1,) (F (x h,δ n,) F (x h,δ n 1,))] R (x h,δ n,) 1 (R (x h,δ n,) R (x h,δ n 1,))(x h,δ n, x h,δ n 1,) = R (x h,δ n,) 1 F (x h,δ n,) Φ(x h,δ n, + t(x h,δ n 1, x h,δ n,), x h,δ n,, x h,δ n 1, x h,δ n,)dt R (x h,δ n,) 1 (F (x h,δ n,) F (x h,δ n 1,))(x h,δ n, x h,δ n 1,) = R (x h,δ n,) 1 F (x h,δ n,) Φ(x h,δ n, + t(x h,δ n 1, x h,δ n,), x h,δ n,, x h,δ n 1, x h,δ n,)dt +R (x h,δ n,) 1 F (x h,δ n,) n, Φ(x h,δ n 1,, x h,δ n,, x h,δ n, x h,δ n 1,)dt = R (x h,δ n,) 1 [F (x h,δ n,)p h + F (x h,δ n,)(i P h )] Φ(x h,δ n, + t(x h,δ n 1, x h,δ n,), x h,δ n,, x h,δ n 1, x h,δ n,)dt +R (x h,δ n,) 1 [F (x h,δ n,)p h + F (x h,δ n,)(i P h )]

9 An Iteratively Regularized Projection Method 19 Φ(x h,δ n 1,, x h,δ n,, x h,δ n, x h,δ n 1,)dt The last but one step follows from Lemma.. So by Assumption 1. and the relation we have Note that R (x h,δ n,) 1 [F (x h,δ n,)p h + F (x h,δ n,)(i P h )] 1 + γ n,h, (6) n+1, x h,δ n, (1 + γ n,h )3k xh,δ n, x h,δ n 1,. (7) γ n,h = F (x h,δ n,)(i P h ) = [F (x h,δ n,) F (P h x ) + F (P h x )](I P h ) F (P h x )(I P h ) + [F (x h,δ n,) F (P h x )](I P h ) γ,h + k F (P h x ) n, P h x γ,h + k ϱ n, P h x. (8) So by (7) we have n+1, x h,δ n, (1 + γ,h + k ϱ n, P h x ) 3k xh,δ n, x h,δ n 1,.(9) Now we shall prove that the sequence ( t n,h ) defined in (16) is a majorizing sequence of the sequence (x h,δ n,) and x h,δ n, Bt h (P h x ), for all n. Note that F (ˆx) = y, so 1, P h x = (P h F (P h x )P h + P h ) 1 P h (F (P h x ) y δ ) = (P h F (P h x )P h + P h ) 1 P h (F (P h x ) y + y y δ ) = (P h F (P h x )P h + P h ) 1 P h (F (P h x ) F (ˆx) + y y δ ) = (P h F (P h x )P h + P h ) 1 P h (F (P h x ) F (ˆx) F (P h x )(P h x ˆx) + F (P h x )(P h x ˆx) + y y δ ) (P h F (P h x )P h + P h ) 1 P h (F (P h x ) F (ˆx) F (P h x )(P h x ˆx)) + (P h F (P h x )P h + P h ) 1 P h F (P h x )(P h x ˆx) + (P h F (P h x )P h + P h ) 1 P h (y y δ ) (P h F (P h x )P h + P h ) 1 P h F (P h x ) Φ(ˆx + t(p h x ˆx), P h x, (P h x ˆx))dt

10 S. George and A. I. Elmahdy + (P h F (P h x )P h + P h ) 1 P h F (P h x )(P h x ˆx) + δ (P h F (P h x )P h + P h ) 1 P h [F (P h x )P h + F (P h x )(I P h )] Φ(ˆx + t(p h x ˆx), P h x, (P h x ˆx))dt + (P h F (P h x )P h + P h ) 1 P h [F (P h x )P h +F (P h x )(I P h )](P h x ˆx) + δ (1 + γ,h )(k P hx ˆx + P h x ˆx ) + δ (1 + γ,h )(k (b h + ρ) + b h + ρ) + δ. The last but one step follows from Assumption 1., (6) and the inequality P h x ˆx b h + ρ. So 1, P h x t 1,h t,h. Assume that for some k. Then i+1, x h,δ i, t i+1,h t i,h, i k (3) k+1, P hx k+1, xh,δ k, + xh,δ k, xh,δ k 1, + + xh,δ 1, P h x t k+1,h t k,h + t k,h t k 1,h + + t 1,h t,h = t k+1,h t h. So x h,δ i+1, B t h (P h x ) for all i k. Therefore by (9) and (3) we have k+, xh,δ k+1, 3k (1 + γ,h + k ϱ k+1, P hx 3k (1 + γ,h + k ϱ t k+1,h )( t k+1,h t k,h ) = t k+,h t k+1,h. ) k+1, xh,δ k, Thus by induction n+1, x h,δ n, t n+1,h t n,h for all n and hence ( t n,h ), n is a majorizing sequence of the sequence (x h,δ n,). In particular n, P h x t n,h t h, i.e., x h,δ n, Bt h (P h x ), for all n. Hence n, P h x t h 1 r h. (31) Lemma 3.4 Let x h,δ n, be as in (13) and x δ n, be as in (4). Let assumptions in Theorem.3 and Theorem(3.3) hold. Then n 1, x δ n 1, 1 r h + b h + η 1 q. (3)

11 An Iteratively Regularized Projection Method 1 Proof.Note that n 1, x δ n 1, = n 1, P h x + (P h I)x + x x δ n 1, [ This completes the proof. Let Q := k ( and n 1, P h x + (P h I)x + x x δ n 1, ] + b h + η 1 r h 1 q. + b h + η 1 r h 1 q ) (33) Q n,h := Γ n,h + Q F (x h,δ n,). (34) Lemma 3.5 Let Q n,h be as in (34) and assumptions in Theorem.3 be satisfied. Then for all n, Q n,h C h η where C h = (k + 1) F (x 1 q ) + Q(k 1 r h + 1)ϱ. Proof.Note that Γ n,h F (x δ n,) sup = v 1 F (x δ n,)v sup v 1 [F (x δ n,) F (x ) + F (x )]v sup v 1 [F (x δ n,) F (x )]v + sup v 1 F (x )v sup v 1 F (x )Φ(x δ n,, x, v) + F (x ) sup v 1 k F (x ) x δ n, x v + F (x ) k F (x ) x δ n, x + F (x ) k F (x ) η 1 q + F (x ) η (k 1 q + 1) F (x ). (35) Similarly one can proof that F (x h,δ n,) (k + 1)ϱ. (36) 1 r h Now the proof follows from (35), (36) and the relation Q n,h F (x δ n,) + Q F (x h,δ n,). Hereafter we assume that 1 < Q <, so that r h < 1 < Q and Q < 1.

12 S. George and A. I. Elmahdy Theorem 3.6 Let x h,δ n, be as in (13) and x δ n, be as in (4). Let assumptions in Theorem 3.3, Theorem.3 and Lemma 3.5 hold. Then we have the following estimate, ( ) ( ) Q n n, x δ n, b h + C Q n h ( Q r h). Proof.Note that x h,δ n, x δ n, = x h,δ n 1, x δ n 1, (P h F (x h,δ n 1,)P h + P h ) 1 P h where [F (x h,δ n 1,) y δ + (x h,δ n 1, x )] +(F (x δ n 1,) + I) 1 [F (x δ n 1,) y δ + (x δ n 1, x )] = x h,δ n 1, x δ n 1, [(P h F (x h,δ n 1,)P h + P h ) 1 P h (F (x δ n 1,) + I) 1 ][F (x h,δ n 1,) y δ + (x h,δ n 1, x )] (F (x δ n 1,) + I) 1 [F (x h,δ n 1,) F (x δ n 1,) + (x h,δ n 1, x δ n 1,)] = x h,δ n 1, x δ n 1, (F (x δ n 1,) + I) 1 [F (x h,δ n 1,) F (x δ n 1,) + (x h,δ n 1, x δ n 1,)] [(P h F (x h,δ n 1,)P h + P h ) 1 P h (F (x δ n 1,) + I) 1 ] [F (x h,δ n 1,) y δ + (x h,δ n 1, x )] = (F (x δ n 1,) + I) 1 [F (x δ n 1,)(x h,δ n 1, x δ n 1,) (F (x h,δ n 1,) F (x δ n 1,))] (F (x δ n 1,) + I) 1 [F (x δ n 1,)P h P h F (x h,δ n 1,)P h ](P h F (x h,δ n 1,)P h + P h ) 1 [F (x h,δ n 1,) y δ + (x h,δ n 1, x )] = (F (x δ n 1,) + I) 1 [F (x δ n 1,)(x h,δ n 1, x δ n 1,) (F (x h,δ n 1,) F (x δ n 1,))] (F (x δ n 1,) + I) 1 [F (x δ n 1,) P h F (x h,δ n 1,)](x h,δ n, x h,δ n 1,) = Γ 1 Γ (37) Γ 1 = (F (x δ n 1,)+I) 1 [F (x δ n 1,)(x h,δ n 1, x δ n 1,) (F (x h,δ n 1,) F (x δ n 1,))] and Γ = (F (x δ n 1,) + I) 1 [F (x δ n 1,) P h F (x h,δ n 1,)](x h,δ n, x h,δ n 1,) Thus by Lemma., Lemma 3.4 and Asssumption1. we have Γ 1 = (F (x δ n 1,) + I) 1 F (x δ n 1,)

13 An Iteratively Regularized Projection Method 3 and 1 k k Φ(x h,δ n 1, + t(x δ n 1, x h,δ n 1,), x δ n 1,, x δ n 1, x h,δ n 1,)dt x δ n 1, x h,δ n 1, n 1, + t(x δ n 1, x h,δ n 1,) x δ n 1, dt x δ n 1, x h,δ n 1, (t 1)(x δ n 1, x h,δ n 1,) dt k xh,δ n 1, x δ n 1, Q xh,δ n 1, x δ n 1,. (38) Γ = (F (x δ n 1,) + I) 1 ((I P h )F (x δ n 1,) +P h (F (x δ n 1,) F (x h,δ n 1,)))(x h,δ n, x h,δ n 1,) (F (x δ n 1,) + I) 1 (I P h )F (x δ n 1,)(x h,δ n, x h,δ n 1,) + (F (x δ n 1,) + I) 1 P h (F (x δ n 1,) F (x h,δ n 1,))(x h,δ n, x h,δ n 1,) (I P h)f (x δ n 1,) n, x h,δ n 1, + (F (x δ n 1,) + I) 1 P h F (x h,δ n 1,)Φ(x δ n 1,, x h,δ n 1,, x h,δ n, x h,δ n 1,) Γ n 1,h + k F (x h,δ n 1,) n 1, x δ n 1, n, x h,δ n 1, Γ n 1,h + Q F (x h,δ n 1,) n, x h,δ n 1, Q n 1,h xh,δ n, x h,δ n 1, (39) Therefore by (37), (38), (39) and Lemma 3.5 we have x h,δ n, x δ n, Q x h,δ n 1, x δ n 1, + C h x h,δ n, x h,δ n 1, ( ) Q n ( x h,δ, x δ Q, + ( ) Q Ch + + rn h + C h rn 1 h ( ) Q n ( ) Q n 1 ( C h Q b h + + ( ) Q Ch + + rn h + C h rn 1 h ) n 1 ( ) C h Q n η C h h + r h ) n C h r h

14 4 S. George and A. I. Elmahdy This completes the proof. ( ) Q n b h + C ( ) h Q n 1 ( ) Q n [ + r h ( ) Q + + rh n + rh n 1 ] ( ) ( ) Q n b h + C Q n h ( Q r h). 4 Error Bounds Under Source Conditions In view of Theorem.3 and Theorem 3.6; to obtain an error estimate for n, ˆx, it is enough to obtain an error estimate for x δ ˆx. It is known (cf. [13], Proposition 3.1) that x δ x δ (4) and (cf. [11], Theorem 3.1) that x ˆx (k r + 1)c ϕ ϕ(). (41) Combining the estimates in Theorem.3, and Theorem 3.6, (4) and (41) we obtaining the following, Theorem 4.1 Let x h,δ n, be as in (13) and let the assumptions in Theorem.3,Theorem 3.6 and Lemma 3.5 be satisfied. Then we have the following; n, ˆx n, x δ n, + x δ n, x δ + x δ x + x ˆx Let ( Q ) n b h + C h ( Q r h) C := max{1 + b h + ( Q ) n + qn η (1 q) + δ + (k r + 1)c ϕ ϕ(). C h ( Q r h) + η 1 q, (k r + 1)c ϕ } and let ( ) Q n n δ := min{n : δ}. (4) Note that for < < 1, δ δ/. Thus by Theorem 4.1 we have the following Theorem. Theorem 4. Let x h,δ n, be as in (13) an let the assumptions in Theorem.3, Theorem 3.6 and Lemma 3.5 be satisfied. Let n δ be as in (4). Then for < < 1 we have, n δ, ˆx C (ϕ() + δ ). (43)

15 An Iteratively Regularized Projection Method A Priori Choice of the Parameter Note that the error estimate ϕ() + δ in (43) is of optimal order if := δ = (δ) satisfies, δ ϕ( δ ) = δ. Now using the function ψ(λ) := λϕ 1 (λ), < λ a we have δ = δ ϕ( δ ) = ψ(ϕ( δ )), so that δ = ϕ 1 (ψ 1 (δ)). Hence by (43) we have the following. Theorem 4.3 Let ψ(λ) := λϕ 1 (λ) for < λ a, and assumptions in Theorem 4. holds. For δ >, let δ = ϕ 1 (ψ 1 (δ)). Let n δ be as in (4). Then n δ, ˆx = (ψ 1 (δ)). 4. An Adaptive Choice of the Parameter In this subsection, we will present a parameter choice rule based on the adaptive method studied in [1, 1]. In practice, the regularization parameter is often selected from some finite set D M () := { i = µ i, i =, 1,, M} (44) where µ > 1 and M is such that M < 1 M+1. We choose := δ, because we expect to have an accuracy of order ( δ) and from Theorem 4.3, it follows that such an accuracy cannot be guaranteed for < δ. Let ( ) Q n n M = min{n : δ}. (45) and let x i := x h,δ n M, i. The parameter choice strategy that we are going to consider in this paper, we selects = i from D M () and operates only with corresponding x i, i =, 1,, M. Theorem 4.4 Assume that there exists i {, 1,,, M} such that ϕ( i ) δ i. Let assumptions of Theorem 4. and Theorem 4.3 hold and let Then l k and l := max{i : ϕ( i ) δ i } < M, k := max{i : x i x j 4C δ j, j =, 1,,, i}. (46) ˆx x k cψ 1 (δ) where c = 6C µ. Proof. To see that l k, it is enough to show that, for each i {1,,, M}, ϕ( i ) δ i = x i x j 4C δ j, j =, 1,, i.

16 6 S. George and A. I. Elmahdy For j i, by (43) we have x i x j x i ˆx + ˆx x j δ δ C ϕ( i ) + C + C ϕ( j ) + C i j δ δ C + C i j δ 4C. j Thus the relation l k is proved. Next we observe that ˆx x k ˆx x l + x l x k δ δ C ϕ( l ) + C + 4C l l δ 6C. l Now since δ l+1 µ l, it follows that Thus δ l µ δ δ = µϕ( δ ) = µψ 1 (δ). ˆx x k 6C µψ 1 (δ) cψ 1 (δ) where c = 6C µ.this completes the proof of the theorem. 5 Implementation of Adaptive Choice Rule In this section we provide an algorithm for the determination of a parameter fulfilling the balancing principle (46) and also provide a starting point for the iteration (13) approximating the unique solution x δ of (3).The choice of the starting point involves the following steps: Choose = δ, µ > 1 and r h < 1. Choose x D(F ) such that x ˆx ρ and (1 + γ,h )( k (b h + ρ) + b h + ρ) + δ min{c, r (1 r h )}. Choose 1 < Q < where Q is as in (33). Choose n M such that n M = min{n : ( ) Q n δ}. Finally the adaptive algorithm associated with the choice of the parameter specified in Theorem 4.4 involves the following steps:

17 An Iteratively Regularized Projection Method Algorithm Set i Solve x i := x h,δ n M, i by using the iteration (13). If x i x j > 6C δ µ j, j i, then take k = i 1. Set i = i + 1 and return to step. 6 Concluding Remarks In this paper we considered a new iterative method in the finite dimensional setting for approximately solving the nonlinear ill-posed operator equation F (x) = y, when the available data y δ in place of the exact data y. It is assumed that F is Fréchet differentiable in a neighborhood of some initial guess x of the actual solution ˆx. The procedure involves finding the fixed point of the function G h (x) = x (P h F (x)p h + P h ) 1 (F (x) y δ + (x x )), in an iterative manner in a finite dimensional subspace X h of the Hilbert space X. Here x is an initial guess and P h is the orthogonal projection onto X h. For choosing the regularization parameter we made use of the adaptive method suggested by Pereversev and Schock in [1] and the stopping rule is based on a majorizing sequence. Acknowledgements The first author thanks National Institute of Technology Karnataka, India, for the financial support under seed money grant No. RGO/O. am/seed GRANT/16/9. The work of Atef I Elmahdy is supported by Indo-Egypt Cultural Exchange Programme 7-8, under the research fellowship of ICCR, India; BNG/171/7-8. References [1] A. G. Ramm, Inverse Problems: Mathematical and Analytical Techniques with Applications to Engineering, Springer, 5. [] A. Bakushinsky and A. Smirnova, On application of generalized discrepancy principle to iterative methods for nonlinear ill-posed problems, Numerical Func. Anal. and Optimization, 6(1), (5),

18 8 S. George and A. I. Elmahdy [3] B. Blaschke, A. Neubauer and O. Scherzer, On convergence rates for the iteratively regularized Gauss-Newton method, IMA Journal on Numerical Analysis, 17(3), (1997), [4] H. W. Engle and M. Hanke and A. Neubauer, Regularization of Inverse Problems, (Dordrecht: Kluwer), [5] I. K. Argyros, Convergence and Applications of Newton-type Iterations, Springer, 8. [6] Jin.Qi-Nian, Error estimates of some Newton-type methods for solving nonlinear inverse problems in Hilbert scales, Inverse problems, 16(1), (), [7] Jin.Qi-Nian, On the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed problems, Mathematics of Computation, 69(3), (), [8] M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 7(1), (1995), [9] P. Deuflhard, H. W. Engle and O. Scherzer, A convergenve analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions, Inverse problems, 14(5), (1998), [1] P. Mathe and S. V. Perverzev, Geometry of linear ill-posed problems in variable Hilbert scales, Inverse problems, 19(3) (3), [11] S. George and A. I. Elmahdy, A quadratic convergence yielding iterative method for nonlinear ill-posed operator equations,(1) (Communicated). [1] S. V. Perverzev and E. Schock, On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM J.Numer.Anal., 43(5), (5), [13] U. Tautanhahn, On the method of Lavrentiev regularization for nonlinear ill-posed problems, Inverse Problems, 18(1), (), Received: June, 1