Numerische Mathematik

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1 Numer. Math : Numerische Mathematik c Springer-Verlag 1999 On an a posteriori parameter choice strategy for Tikhonov regularization of nonlinear ill-posed problems Jin Qi-nian 1, Hou Zong-yi 2 1 Institute of Mathematics, Nanjing University, Nanjing 218, P.R. China; galgebra@netra.nju.edu.cn 2 Department of Mathematics, Fudan University, Shanghai 2433, P.R. China Received August 8, 1997 / Revised version received January 26, 1998 Summary. In the study of the choice of the regularization parameter for Tikhonov regularization of nonlinear ill-posed problems, Scherzer, Engl and Kunisch proposed an a posteriori strategy in To prove the optimality of the strategy, they imposed many very restrictive conditions on the problem under consideration. Their results are difficult to apply to concrete problems since one can not make sure whether their assumptions are valid. In this paper we give a further study on this strategy, and show that Tikhonov regularization is order optimal for each <ν 2 with the regularization parameter chosen according to this strategy under some simple and easy-checking assumptions. This paper weakens the conditions needed in the existing results, and provides a theoretical guidance to numerical experiments. Mathematics Subject Classification 1991: 65J2 1. Introduction A few regularization methods have been used to solve nonlinear ill-posed inverse problems which can be summed up into the form 1.1 F x =y, where F is a weakly closed, continuous and Fréchet differentiable nonlinear operator with domain DF in the Hilbert space X and with its range RF Correspondence to: Q.Jin

2 14 Q. Jin, Z. Hou in the Hilbert space Y and y RF. These methods include Tikhonov regularization [3, 13, 17, 2], Landweber iteration [5], asymptotical regularization [21], the iteratively regularized Gauss-Newton method [1] and Marti method [8] and so on. The reason why problem 1.1 receives so much attention is that many important inverse problems in natural science can lead to such ill-posed problems [2, 6]. Here we call problem 1.1 ill-posed in the sense that the solution of 1.1 does not depend continuously on the right hand side data which is often obtained by measurement and hence contains error. Let us assume that y δ is an observation data of y, and 1.2 y δ y δ with a given noise level δ>. Now the computation of the solution of 1.1 from the observation y δ becomes an important topic. Among those methods developed to solve nonlinear ill-posed problems, Tikhonov regularization is the most well-known one. In this method, the solution x δ α of the minimization problem 1.3 min x DF { F x y δ 2 + α x x 2} is used to approximate the solution of 1.1, where α>is the regularization parameter and x DF is an a-priori guess of the solution of 1.1. Under appropriate conditions on F, the stability of x δ α with respect to y δ can be guaranteed, and with a suitable choice of α, x δ α can be guaranteed to converge to an x -minimum -norm-solution x -MNS x of 1.1, i.e. converge to an element x X with the property 1.4 F x =y and x x = min { x x DF x : F x =y}. Now the regularization parameter α affects not only the convergence of x δ α but also the rates of convergence, and hence the choice of the regularization parameter is crucial. In [13] it has been proved that the best possible rate of convergence of x δ α to x is Oδ 2/3 and a priori parameter choice has been given to arrive at such rate. This a priori strategy has the disadvantage that the regularization parameter α depends not only on the noise level δ but also on the smoothness assumption of x x which is difficult to check in general in practice. And hence a wrong guess of the smoothness on x x will lead to a bad choice of α, and consequently to a bad approximation to the x -MNS of 1.1. Therefore, we hope to give a favorable a posteriori parameter choice strategy. Morozov s discrepancy principle has been used for nonlinear illposed problems in [3,11,15] to choose the regularization parameter and α is determined from the following nonlinear equation: 1.5 F x δ α y δ = cδ,

3 Tikhonov regularization of nonlinear ill-posed problems 141 where c 1 is a given constant. With such choice of α, however, the best possible convergence rate of x δ α to x is at most O δ see [15]. Therefore it is of interest to present an a posteriori strategy yielding convergence rate up to Oδ 2/3. Some researches have been done in [18]; with the inspection of the strategy developed by Gfrerer for linear ill-posed problems [4], the following principle has been proposed to choose the regularization parameter. Rule 1.1 Let c 1 be a given constant and x DF. i If F x y δ 2 cδ 2, then choose α =, i.e. take x as approximation; ii If F x y δ 2 >cδ 2, then choose α := αδ be the root of the equation 1.6 α F x δ α y δ, αi + F x δ αf x δ α 1 F x δ α y δ = cδ 2, where F x denotes the Fréchet derivative of F at point x DF and F x denotes the adjoint of F x. To prove the optimality of this rule, a series of restrictive conditions have been imposed on F see assumptions 1 14 and in [18], and most of them are difficult to verify for concrete problems. Therefore, for a given problem, we are not sure whether Rule 1.1 can be applied. Thus it is significant to derive some useful results under weaker and more easychecking conditions. This is the aim of the present paper and we will point out that the following assumption, which has been carefully interpreted in [18], is sufficient for our purpose. Assumption 1.2 Let x be an x -MNS of 1.1 such that there is a number p>3 x x such that B p x DF, and there exists a constant K such that for all x, z B p x and v X, there is kx, z, v X such that F x F zv = F zkx, z, v, where kx, z, v K x z v. In Sect. 2 we will prove the convergence of x δ α with α chosen by Rule 1.1 under Assumption 1.2, that is, we have the following result. Theorem 1.3 Let Assumption 1.2 be fulfilled and 2K x x < 1, let α := αδ be determined by Rule 1.1 and c>9.ifx is the unique x -MNS of 1.1, then lim δ xδ αδ x =.

4 142 Q. Jin, Z. Hou This result provides no convergence rates for x δ αδ. In general, the convergence of x δ αδ to x can be arbitrarily slow [19]. Therefore, to guarantee a suitable rate of convergence, some additional assumptions should be imposed on x, and these assumptions are called source conditions. The following one is frequently used for nonlinear ill-posed problems [5, 13]: There is a ν>and an element ω NF x X such that 1.7 x x =F x F x ν/2 ω. Theorem 1.4 Let Assumption 1.2 be fulfilled and 6K x x 1, let α := αδ be determined by Rule 1.1 and c>9. Ifthex -MNS x of 1.1 satisfies 1.7 with <ν 2, then for all δ>, there holds 1.8 where x δ αδ x C ν ω 1/1+ν δ ν/1+ν, C ν := 6 3c /1+ν c ν/1+ν 3 c 3. 7 This result is a satisfactory one since it weakens the assumptions required in [18] and expands the applied range of Rule 1.1. It suggests that Tikhonov regularization combining with Rule 1.1 defines a regularization method of optimal order for each <ν 2 see [12, 22]. Also it explains the reason why Rule 1.1 has elegant performance for the numerical examples in [18] which do not satisfy the assumptions therein. The proof of Theorem 1.4 is quite technical. By a careful investigation, we find the difficulty lies in the estimate of x αδ x, where here and below x α denotes a solution of the minimization problem 1.3 with y δ replaced by y. We overcome the difficulty by dominating x α x with x α x and α 1/2 αi+f x F x 1/2 F x α y α, where α α is chosen so that the two latter terms can be estimated easily. The upper bound provided by 1.8 is of uniform nature without special regard on y. In Sect. 3, however we will show that for typical instance the convergence of x δ αδ to x is faster than Theorem 1.4 claims. 2. Proofs of main results In this section we will contribute to the proofs of Theorem 1.3 and Theorem 1.4. Before doing this, we have to show the justification of Rule 1.1. A related

5 Tikhonov regularization of nonlinear ill-posed problems 143 result has been given in [18, Theorem 3.9], but there the noise level δ> is required to be sufficiently small. Therefore we discard the corresponding results in [18] and are ready to seek an alternative proof guaranteeing Rule 1.1 well defined for all δ>. To this end, we need only to consider the case F x y δ 2 >cδ 2. Lemma 2.1 Let Assumption 1.2 be fulfilled and 2K x x < 1, let c 2 and β := c 1δ2.If F x y δ 2 >cδ 2, then there exists an x x 2 α β satisfying 1.6. Proof. Under the given conditions, one can easily see that for all δ>and α β there holds x δ α x δ α + x x 1 c 1 +1 x x 2 x x. Therefore x δ α x 3 x x <pwhich implies that x δ α B p x is an interior point of DF. Hence we can use the similar method in the proof of [9, Lemma 1] to obtain 1 β δ α K + x x x δ α x δ β β δ β α 2 + x x β α for all α, β β. Since δ β + x x 2 x x and 2K x x < 1, we can immediately get x δ β xδ α as β α. Obviously, Assumption 1.2 implies the continuity of the mapping x F x on B p x. Therefore, the function α fα :=α F x δ α y δ, αi + F x δ αf x δ α 1 F x δ α y δ is continuous for α [β,. By noting that fβ F x δ β y δ 2 δ 2 + β x x 2 = cδ 2, we can use the intermediate value theorem to complete the proof Proof of Theorem 1.3 From the regualrization theory, we know, to guarantee the convergence of x δ α, the regularization parameter α should satisfy suitable conditions, especially it requires δ2 α as δ. Therefore to complete the proof of Theorem 1.3, let us prove that the α := αδ chosen by Rule 1.1 admits such property.

6 144 Q. Jin, Z. Hou Lemma 2.2 Let x be the unique x -MNS of 1.1, then F x α y lim =. α α Proof. Since the definition of x α implies F x α y 2 α x x 2 x α x 2 + x α x 2 =2αx α x,x x, assertion 2.1 follows immediately since the uniqueness of x -MNS x of 1.1 together with [3, Theorem 2.2] implies x α x as α. Lemma 2.3 Let Assumption 1.2 be fulfilled and K x x < 1. Then for all δ> and α β, we have F x δ α F x α 2δ. x δ 2 δ α x α. 1 K x x α Proof. Please refer to [7, Lemma 5] and its proof or [17]. Lemma 2.4 Let Assumption 1.2 be fulfilled and 2K x x < 1, let α := αδ be determined by Rule 1.1 and c>9.ifx is the unique x -MNS of 1.1, then δ lim δ αδ =. Proof. Suppose on the contrary, then there exists a positive constant τ and a sequence δ k > satisfying δ k as k such that 2.5 lim inf k δ 2 k αδ k τ. Obviously, this implies αδ k as k. According to the definition of αδ k we have F x δ k αδk yδ k 2 αδ k F x δ k αδk yδ k, αδ k I +F x δ k αδk F x δ k αδk 1 F x δ k αδk yδ k = cδ 2 k. Therefore by using 1.2 and Lemma 2.3 it follows c 3δ k F x αδk y.

7 Tikhonov regularization of nonlinear ill-posed problems 145 This together with Lemma 2.2 implies lim k δk 2 αδ k 1 c 3 2 lim F x αδk y 2 k αδ k =, which is a contradiction to 2.5, and hence the proof is complete. Now we are ready to give the proof of Theorem 1.3. Proof of Theorem 1.3. Let us carry out the proof by arguing for subsequences. First let δ k > satisfying δ k as k be such that αδ k. From F x δ k αδk yδ k 2 + αδ k x δ k αδk x 2 δ 2 k + αδ k x x 2 and Lemma 2.4 we know lim k F x δ k αδk = y and lim sup k x δ k αδk x x x. Therefore by standard techniques [3, 2] we can prove lim k x δ k αδk = x. Now let δ k > satisfying δ k as k be such that αδ k ɛ with some positive constant ɛ. In the following we proceed our argument by dividing into two cases. i Let αδ k be unbounded. In this case we can choose a subsequence δ kl of δ k such that α l := αδ kl. By using the definition of x l := x δ k l αδ kl we can easily obtain x l x as l. Let us define B l := F x l F x l, then we have 2.6 α l F x l y δ k l, α l I + B l 1 F x l y δ k l F x l y δ k l 2 =F x l y δ k l, α l I + B l 1 B l F x l y δ k l α l I + B l 1 B l F x l y δ k l 2 α l I + B l 1 B l F x y δ k l 2. Since Assumption 1.2 implies the continuity of the mapping x F x on B p x and since the proof of Lemma 2.1 implies x l B p x,wehave F x l F x, and hence B l is bounded and α l I + B l 1 B l as l. Therefore lim α lf x l y δ k l, α l I + B l 1 F x l y δ k l l = lim F x l y δ k l 2 = F x y 2. l From this, by utilizing the definition of α l := αδ kl it follows immediately that F x =y. This implies x is a solution of 1.1 and hence x = x. Therefore we obtain x δ k l αδ kl x.

8 146 Q. Jin, Z. Hou ii Let αδ k be bounded. Then there is a number α ɛ > and a subsequence δ kl of δ k such that α l := αδ kl α. Let x l := x δ k l αδ kl,we can use the manner similar to that in [3] to show x l x α and F x l F x α as l, where x α is a solution of the minimization problem 1.3 with y δ and α replaced by y and α respectively. Now using the definition of αδ kl and noting the continuity of the mappings x F x and x F x we can prove α F x α y, α I + F x α F x α 1 F x α y=, which implies F x α =y, that is, x α is a solution of 1.1. Since the definition of x α implies x α x x x, therefore x α also is an x -MNS of 1.1. By uniqueness we have x α = x and hence we obtained x δ k l αδ kl x again. Combining the above we have actually proved that for each sequence δ k > satisfying δ k, there always exists a subsequence δ kl of δ k such that x δ k l αδ kl x. Hence the proof is complete. Remark 2.5 From the proof of Theorem 1.3, one can easily see that if the x -MNS of 1.1 is not unique, then there holds the following claim: For each sequence δ k, there is a subsequence δ kl of δ k such that x δ k l αδ kl is convergent in X, and the limit of x δ k l αδ kl is an x -MNS of 1.1. Hence, under such circumstance, we will consider the convergence in the set value sense. Remark 2.6 It is worthy to point out that for the αδ determined by Rule 1.1, we have lim δ αδ =if F x y. In fact, from the proof of Theorem 1.3 we know if αδ has the cluster as δ, then x must be a solution of 1.1, which contradicts F x y. Ifαδ has a cluster <α < as δ, then the proof of Theorem 1.3 also tells us that x α satisfies F x α =y. On the other hand, Assumption 1.2 implies x α is an interior point of DF, and therefore we have the first order necessary optimality condition for x α as follows: F x α F x α y+α x α x =. This implies x α = x, and hence we obtain a contradiction again Proof of Theorem 1.4 In this subsection we consider the rates of convergence for x δ αδ and present the proof of Theorem 1.4. Since the process is quite technical, some necessary preparation should be given beforehand. We first present the lower

9 Tikhonov regularization of nonlinear ill-posed problems 147 bound for αδ in general conditions if the x -MNS x of 1.1 satisfies 1.7 with <ν 2. To this end, let us state some results first. Lemma 2.7 Let Assumption 1.2 be fulfilled and 2K x x < 1. Ifx satisfies 1.7 with <ν 2, then for all α> we have 2.7 x α x M 1 ω α ν/2, where M 1 := 1+2K x x 1 2K x x. Proof. See [9, Lemma 5] and its proof. Lemma 2.8 Let all the assumptions in Lemma 2.7 be fulfilled. Then for all α>there holds α F x α y, αi + F x F x 1 F x α y D ω 2 α 1+ν, 2.8 where D := 1 + K x x 2 M1 2 and M 1 is as in Lemma 2.7. Proof. According to Assumption 1.2 we have 2.9 F x α y = F x x α x + 1 k t dt with k t = kx +tx α x,x,x α x. Obviously, 1 k tdt K 2 x α x 2. Substituting 2.9 into the left hand side of 2.8, and letting A = F x F x, it follows α F x α y, αi +F x F x 1 F x α y 1 1 = α x α x + k t dt, αi + A 1 A x α x + k t dt α x α x + k t dt α 1+ K 2 2 x α x x α x 2 α1 + K x x 2 x α x 2. By using Lemma 2.7 it yields the assertion. Lemma 2.9 Let Assumption 1.2 be fulfilled and 6K x x 1, let α := αδ be determined by Rule 1.1 and c>9.

10 148 Q. Jin, Z. Hou 1 When <α<, there holds α F x α y, αi + F x F x 1 F x α y c δ 2 ; 3 2 When <α, there holds α F x α y, αi + F x F x 1 F x α y c +3 2 δ 2. When α =, the left hand side of 2.12 is to be understood as the limit F x y 2 as α. Proof. When α =, the assertion is trivial. In the following we consider the case α<. Since α is determined by Rule 1.1 we have cδ = α αi + F x δ αf x δ α 1/2 F x δ α y δ. From Lemma 2.3 and 1.2 it follows cδ α αi + F x δ αf x δ α 1/2 F x α y This implies α αi + F x δ αf x δ α 1/2 F x δ α F x α y δ + y F x δ α F x α + δ 3δ. c 3 2 δ 2 αf x α y, αi + F x δ αf x δ α 1 F x α y 2.13 c +3 2 δ 2. Now let a := αf x α y, αi + F x δ αf x δ α 1 F x α y, b := αf x α y, αi + F x F x 1 F x α y. We exploit the similar method in the proof of [18, Lemma 3.1] to proceed the argument. With the notations B := F x F x and Bα δ := F x δ αf x δ α, by using [18, Lemma 3.6] and noting K x δ α x 3K x x 1 2 we have a b = αf x α y, αi + B 1 αi + Bα δ 1 F x α y = αf x α y, αi + B 1 B BααI δ + Bα δ 1 F x α y = ααi + B 1/2 F x α y, αi + B 1/2 B Bα δ

11 Tikhonov regularization of nonlinear ill-posed problems 149 αi + Bα δ 1/2 αi + Bα δ 1/2 F x α y 2K α x δ α x αi + B 1/2 F x α y αi + Bα δ 1/2 F x α y 1 2 α{f x α y, αi + B 1 F x α y +F x α y, αi + B δ α 1 F x α y} = 1 a + b. 2 This implies b 3 a 3b, which together with 2.13 gives the assertion. Now we can give the lower bound for αδ. Lemma 2.1 Let Assumption 1.2 be fulfilled and 6K x x 1, let αδ be determined by Rule 1.1 and c>9. Ifthex -MNS x satisfies 1.7 with <ν 2, then for all δ> there holds 2.14 where D ν := αδ D ν δ/ ω 2/1+ν, 3 c 3 2/1+ν. 7 Proof. We first consider the case δ x < 1 6, from Assumption 1.2 and 1.7 we have ω 13 F x 1+ν 12 c 1. Noting that K x F x y δ Fx F x + δ 1 F x x x + kx + tx x,x,x x dt + δ F x 1+ K 2 x x x x + δ 13 F x 2.15 F x F x ν/2 ω + δ. 12 Since <ν 2, by interpolation inequality we can obtain 2.16 F x F x ν/2 ω ω 1 ν/2 F x F x ω ν/2 Inserting 2.16 into 2.15 gives F x ν ω. F x y δ 13 F x 1+ν ω 12 + δ cδ.

12 15 Q. Jin, Z. Hou This implies α = and hence assertion 2.14 is trivial. Therefore in δ the following we consider the case ω 13 F x 1+ν 12.Ifα = then c 1 assertion 2.14 is also trivial. Thus we can assume α <. By using Lemma 2.8 and 2.11 it gives c δ 2 D ω 2 α 1+ν. 3 Since D 49 9, the proof is complete. Lemma 2.1 provides a lower bound for αδ. Obviously, if we can give an upper bound for αδ analogous to the right hand side of 2.14, then Theorem 1.4 can be proved easily. Unfortunately, it is difficult to find such upper bound. Therefore we have to discard this line and search for other ways to arrive at our aim. The following inequality plays important role in the discussion below. Lemma 2.11 Let Assumption 1.2 be fulfilled and 6K x x 1, Then for all <α α< there holds x α x 8 x α x + 6 α1/2 αi + F x F x 1/2 F x α y α When α =, both sides of 2.18 are to be understood as the limits as α. Proof. The process is quite technical and is given in the appendix. Proof of Theorem 1.4. By applying the triangle inequality and Lemma 2.3 and noting that K x x 1 6 it gives 2.19 x δ α x x δ α x α + x α x 3δ + x α x. α 3 c 3 7 δ 2/1+ν. Let us choose α := ω then Lemma 2.1 implies α α. Therefore we can use Lemma 2.11 and α 1/2 αi +F x F x 1/2 F x α y 3 c +3δ, which is guaranteed by 2.12, to obtain x δ α x 6 3c +18 δ x α x. α Clearly δ α = 7 1/1+ν. 3 c 3 δ ω ν Using Lemma 2.7 and noting M 1 2 we also have x α x 2 ω α ν/2 =2 3 c 3 7 ω δ ν 1/1+ν. Combining these with 2.2 completes the proof. ν/1+ν

13 Tikhonov regularization of nonlinear ill-posed problems Some improvements on convergence rates It is well known that Rule 1.1 is just the parameter choice strategy proposed by Gfrerer when F is a linear operator [4]. Considering the convergence rates for linear ill-posed problems, one will ask for nonlinear ill-posed problems whether the estimates for convergence rates of x δ αδ in Theorem 1.4 can be improved. In this section we will give the positive answers. Lemma 3.1 Let Assumption 1.2 hold and x be an x -MNS of 1.1, F x y and K x x < 1. Then for all δ> there exists α δ > satisfying 3.1 x α δ x 2 = δ2 α δ. Moreover, α δ has the property lim δ α δ =. Proof. Since x α x is bounded, the function φα :=α x α x 2 satisfies lim α φα =. Since lim α x α = x,wehavelim α φα = due to F x y. From [15] we know φα is continuous for all α>. Therefore the intermediate value theorem guarantees the existence of α δ satisfying 3.1. The proof of the property lim δ α δ =can be carried out by following the lines in Remark 2.6. Now we can state an improved version of Theorem 1.4. Theorem 3.2 Let Assumption 1.2 be fulfilled, 6K x x < 1 and F x y, let α := αδ be determined by Rule 1.1 and c>9. Ifx satisfies 1.7 with <ν 2, then { x δ αδ oδ ν/1+ν if <ν<2 x Oδ 2/3 if ν =2 as δ. Proof. By checking the proof of [9, Lemma 5] carefully we have 1 ααi + A 1 A ν/2 ω x α x M 1 ααi + A 1 A ν/2 ω M 1 if x satisfies 1.7, where M 1 is defined as in Lemma 2.7 and A := F x F x.ifweuse{e λ } to denote the spectral family generated by A and let c ν α 2 := α 2 ν λ ν de α+λ 2 λ ω, ω, then ααi + A 1 A ν/2 ω = α ν/2 c ν α. Therefore for all α>there holds M 1 c ν αα ν/2 x α x M 1 c ν αα ν/2.

14 152 Q. Jin, Z. Hou Combining 3.2 with 2.1 it gives with D defined as in Lemma 2.8 that α F x α y, αi + F x F x 1 F x α y 3.3 Dα 1+ν c ν α 2. Since F x y, wehaveαδ < for suitable small δ>. Therefore by using 2.11 and 3.3, we have c δ 2 Dα 1+ν c ν α 2 3 Now from 2.19 and 3.4 it follows that 3.5 x δ αδ x Oδ ν/1+ν c ν αδ 1/1+ν + x αδ x. In the following we are in a position to estimate the term x αδ x. Let α δ be determined by Lemma 3.1, we carried out the argument by dividing into two cases. i If αδ α δ, then 3.2 gives x αδ x Oαδ ν/2 c ν αδ. Since α ν c ν α 2 = α 2 λ ν de α+λ 2 λ ω, ω and since for each fixed λ the function qα := α2 λ ν is increasing for α,, wehave α+λ 2 αδ ν c ν αδ 2 α δ ν c ν α δ 2, which implies 3.6 x αδ x Oα δ ν/2 c ν α δ. From the definition of α δ and 3.2 we have 1 δ 2 α δ, which implies M 2 1 α δ ν c ν α δ α δ Oδ 2/1+ν c ν α δ 2/1+ν. Substituting 3.7 into 3.6 gives 3.8 x αδ x Oδ ν/1+ν c ν α δ 1/1+ν. ii Now we consider the case αδ >α δ. We can use Lemma 2.11, 2.12 and 3.1 to obtain x αδ x K x α δ x δ = 2Kδ α δ α δ with a constant K independent of δ. Inserting 3.7 into 3.9 yields the same estimate as in 3.8.

15 Tikhonov regularization of nonlinear ill-posed problems 153 Combining 3.5 and 3.8, for the αδ determined by Rule 1.1 we always have x αδ x O δ ν/1+ν c ν α δ 1/1+ν + c ν αδ 1/1+ν. 3.1 Since F x y,wehaveαδ see Remark 2.6 and α δ see Lemma 3.1 as δ, therefore for <ν<2there hold c ν αδ = o1 and c ν α δ = o1 as δ and for ν =2there hold c ν αδ = O1 and c ν α δ = O1 as δ. Hence the proof follows from 3.1. Remark 3.3 Comparing with [4], one can find our results are the same as those therein if F is a linear operator, but our method is different from that. Therefore we also provide a new proof for those results in [4] for linear ill-posed problems. Under the assumption that x satisfies 1.7, Theorem 3.2 provides the estimates for the rates of convergence for x δ αδ. Now a question arises: Can we replace 1.7 with a weaker condition to obtain the same result? In the following we will use the recent results in [14] to point out the possibility. Let {E λ } be the spectral family of F x F x, then for <ν<2, instead of 1.7 we use the assumptions 3.11 µ d E λ x x 2 = Oµ ν and 3.12 µ d E λ x x 2 = oµ ν. Obviously, 3.11 and 3.12 hold if x x satisfies 1.7. Lemma 3.4 Let Assumption 1.2 be fulfilled and 4K x x 1. For each α>, let x α be a solution of the minimization problem 1.3 with y δ replaced by y, and let ˆx α be the solution of the linearized minimization problem 3.13 min { F x x q 2 + α x x 2 }, x X where q = F x x. Then x α x ˆx α x 2 x α x.

16 154 Q. Jin, Z. Hou Proof. Let A := F x F x. By the definition of ˆx α it is easy to see 3.15 ˆx α = x + ααi + A 1 x x. From 3.15 and A.3 in the appendix it follows ˆx α x α =αi + A 1 r α + s α. Therefore, the estimate A.8 in the appendix and the assumption 4K x x 1give ˆx α x α 1 2 x α x which implies Remark 3.5 Similar results have appeared in [16, 18], but there, addition to Assumption 1.2, some other restrictive conditions are required. Now our proof point out that this is not necessary. Remark 3.6 Under the assumptions in Lemma 3.4, the combination of Lemma 3.4 and [14, Theorem 2.1] gives the following converse results for <ν<2: x α x = Oα ν/2 x x satisfies 3.11, x α x = oα ν/2 x x satisfies Recently we have obtained further converse and saturation results for Tikhonov regularization of nonlinear ill-posed problems, and moreover, we have shown that Rule 1.1 is an optimal a posteriori parameter choice strategy. For detail information, please refer to [1]. Now we can give another improved result for Theorem 1.4. Theorem 3.7 Let Assumption 1.2 be fulfilled, 6K x x < 1 and F x y, let α := αδ be determined by Rule 1.1 and c>9. Then i if x x satisfies 3.11 with <ν<2, then x δ αδ x = Oδ ν/1+ν, ii if x x satisfies 3.12 with <ν<2, then x δ αδ x = oδ ν/1+ν. Proof. The proof can be carried out by using 3.16, 3.17 and the method to prove Theorem 3.2, and hence we omit the process.

17 Tikhonov regularization of nonlinear ill-posed problems Conclusions In this paper, the choice of the regularization parameter for Tikhonov regularization of nonlinear ill-posed problems has been considered, and the a posteriori parameter choice strategy, developed in [18], has been studied seriously again. From the discussion in the foregoing sections, it is found that most of the assumptions required in [18] are superfluous, and Assumption 1.2 is sufficient for our purpose. With Assumption 1.2, we have proved the convergence of Tikhonov regularized solution x δ α with α chosen by Rule 1.1, and shown that Rule 1.1 provides an order optimal a posteriori strategy for each <ν 2. Furthermore, we found the rates of convergence listed in Theorem 1.4 can be improved, and obtained a better result which is reduced to the one in [4] when F is a linear operator. This paper weakens the requirement in [18] and expands the applied range of Rule 1.1. It also explains the reason why Rule 1.1 has elegant performance for those numerical examples which can not be verified to satisfy the assumptions in [18]. Therefore, this paper provides a theoretical guidance to numerical experiments. Appendix. Proof of Lemma 2.11 We first consider the case α<. Since x α x 2 x x <p, x α is an interior point of DF. Therefore there holds the first order necessary optimality condition for x α : A.1 F x α F x α y+αx α x =. We can rewrite A.1 as follows F x α F x F x α y+f x F x x α x A.2 + F x F x α y F x x α x + αx α x =. Introducing the notation A := F x F x and letting s α := F x F x α y F x x α x, r α := F x α F x F x α y, then from A.2 it follows that A.3 x α = x + ααi + A 1 x x αi + A 1 r α + s α. By the same procedure we have for x α that A.4 x α = x + α α I + A 1 x x α I + A 1 r α + s α,

18 156 Q. Jin, Z. Hou where r α and s α are the expressions obtained from r α and s α by replacing x α with x α respectively. After subtracting A.4 from A.3, it reaches x α x α =ααi + A 1 α α I + A 1 x x A.5 αi + A 1 r α + s α +α I + A 1 r α + s α. Now we first give the estimates of αi + A 1 r α + s α and α I + A 1 r α + s α. From 2.9 we have 1 αi + A 1 s α =αi + A 1 A k t dt. By observing 1 k tdt K 2 x α x 2 and x α x x x we obtain αi + A 1 s α K A.6 2 x α x 2 K x x x α x. With the application of Assumption 1.2 and A.1 we can prove αi + A 1 r α = sup αi + A 1 r α,ω ω =1 = sup y F x α, F x F x α αi + A 1 ω ω =1 = sup F x α y F x α,kx,x α, αi + A 1 ω ω =1 = sup αx x α,kx,x α, αi + A 1 ω ω =1 sup α x x α kx,x α, αi + A 1 ω ω =1 K x α x x α x A.7 K x x x α x. Combining A.6 and A.7 it follows A.8 αi + A 1 r α + s α 2K x x x α x. Applying the same method we can show that A.9 α I + A 1 r α + s α 2K x x x α x. Therefore from A.5, A.8 and A.9 we obtain x α x α ααi + A 1 α α I + A 1 x x A.1 +2K x x x α x + x α x.

19 Tikhonov regularization of nonlinear ill-posed problems 157 In the following we concentrate on estimating the term ααi + A 1 α α I + A 1 x x. By letting I 1 := 1 α α α I + A 1 F x F x α y, I 2 := 1 α α α I + A 1 F x F x x α x F x α +y, I 3 := 1 α α α I + A 1 A ααi + A 1 x x x α x, one can see A.11 ααi + A 1 α α I + A 1 x x =I 1 + I 2 + I 3. Therefore in the following we need only to estimate I 1, I 2 and I 3.For I 1, if we use the abbreviations P := α I + A 1 2 F x and Q := α α α I + B 1 2 αi + B 1 2 with B := F x F x, then we can write I 1 = 1 α 1 α α PQα 1/2 αi + B 1/2 F x α y. Let {Êλ} be the spectral family generated by the self-adjoint operator B, then for all v Y we have Qv 2 α α + λ = αα + λ dêλv, v. Since α α, the function gλ := α α+λ αα +λ is decreasing with respect to λ on [, and attains its maximum 1 at point λ =. Therefore Qv 2 dêλv, v = v 2, v Y. This implies Q 1. By the same way we also have P 1. Noting that α α 1, we therefore obtain A.12 I 1 1 α 1 2 αi + B 1 2 F xα y. α To estimate I 2, we can use 2.9 to obtain I 2 1 α 1 α I + A 1 A k t dt α K A.13 2 x α x 2 K x x x α x. Finally we estimate I 3. Clearly we have A.14 I 3 ααi + A 1 x x x α x.

20 158 Q. Jin, Z. Hou Substituting A.3 into A.14 and using A.8 it follows that A.15 I 3 2K x x x α x. Now combining A.11 A.13 and A.15 we have ααi + A 1 α α I + A 1 x x 3K x x x α x + 1 α α 1 2 αi + B 1 2 F xα y. A.16 Inserting A.16 into A.1, and noting 6K x x 1, wehave x α x α 1 α 1 2 αi + B 1 2 F xα y + 5 α 6 x α x x α x. From this the proof can be carried out immediately. Now we consider the case α =. Ifα =, the assertion is trivial; otherwise, we can use the result established in the above and take the limit α to obtain the desired assertion. Acknowledgement. The authors are grateful to the referees for their careful reading of the manuscripts and some useful remarks. The work of Q. Jin and Z. Hou is supported by the National Natural Science Foundation of China under grant and References 1. Bakushinskii, A. B. 1992: The problem of the iteratively regularized Gauss-Newton method. Comput. Math. Math. Phys. 32, Engl, H. W. 1993: Regularization methods for the stable solutions of inverse problems. Surv. Math. Ind. 3, Engl, H. W., Kunisch, K., Neubauer, A. 1989: Convergence rates of Tikhonov regularization of nonliner ill-posed problems. Inverse Problems 5, Gfrerer, H. 1987: An a posteriori parameter choice for ordinary and iterated Tikhonov regularization leading to optimal convergence rates. Math. Comput. 49, Hanke, M., Neubauer, A., Scherzer, O. 1995: A convergence analysis of Landweber iteration of nonlinear ill-posed problems. Numer. Math. 72, Hofmann, B. 1986: Regularization of Applied Inverse and Ill-posed Problems. Leipzig, Teubner 7. Hou Zong-yi, Jin Qi-nian 1997: Tikhonov regularization for nonlinear ill-posed problems. Nonlinear Analysis: Theory, Methods and Applications 28, Jin Qi-nian, Hou Zong-yi 1996: Finite-dimensional approximations to the solutions of nonliner ill-posed problems. Appl. Anal. 62, Jin Qi-nian, Hou Zong-yi 1997: On the choice of the regularization parameter for ordinary and iterated Tikhonov regularization of nonlinear ill-posed problems. Inverse Problems 13,

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