A Family of Preconditioned Iteratively Regularized Methods For Nonlinear Minimization

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1 A Family of Preconditioned Iteratively Regularized Methods For Nonlinear Minimization Alexandra Smirnova Rosemary A Renaut March 27, 2008 Abstract The preconditioned iteratively regularized Gauss-Newton algorithm for the minimization of general nonlinear functionals was introduced by Smirnova, Renaut and Khan 2007). In this paper, we establish theoretical convergence results for an extended stabilized family of Generalized Preconditioned Iterative methods which includes M times iterated Tihonov regularization with line search. Numerical schemes illustrating the theoretical results are also presented. Keywords: Gauss-Newton method, stopping rule, ill-posed problem, regularization. AMS Subject Classification: 47A52, 65F22, 65J15, 65N21. 1 Introduction Consider a general ill-posed problem of minimizing a nonlinear functional Jq) := F δ q) 2 H 1 1.1) with a noisy operator F δ mapping between Hilbert spaces H and H 1, i.e., F : DF ) H H 1. Notice that this form of the nonlinear functional allows for error both in measurement data, and the forward nonlinear operator used for estimating the measured data. In particular, suppose that g δ are measured data and Cq) is a forward operator for obtaining estimates of g δ, then the nonlinear operator in which C is assumed error free is given by F δ = C g δ. Here we mae the more general assumption that C is also noise contaminated, probably due to a discretization process, and thus the subscript δ is omitted in the noise free case. When F δ is Fréchet differentiable in a neighborhood of a minimizer, one of the most used stabilizing numerical algorithms for solving 1.1) is the Iteratively Regularized Gauss-Newton IRGN) method BA93] q +1) = q F δ q ) )F δq ) ) + ) I) 1 F δ q ) ){F δ q ) ) F δq ) )q ) q), q H. 1.2) Georgia State University, Department of Mathematics and Statistics, Atlanta, GA Tel: , Fax: Supported by NSF grants DMS and DMS Arizona State University, Department of Mathematics and Statistics, Tempe, AZ Tel: , Fax:

2 2 IRGN, March 27, 2008 The element q +1) in 1.2) has the variational characterization DES98] F δ q ) ) F δq ) )q ) q) 2 H 1 + ) q q 2 H. For some inverse problems it proves to be extremely beneficial to impose the regularization in an alternative space. This suggests the use of a preconditioned IRGN method SRK07] q +1) = q F δ q ) )F δq ) ) + ) L L) 1 F δ q ) ){F δ q ) ) F δq ) )q ) q), q H, 1.3) with L LH, H 2 ) and F δ )F δ ) + L L being invertible for > 0. Here q +1) has the variational characterization F δ q ) ) F δq ) )q ) q) 2 H 1 + ) Lq q) 2 H 2, and the equivalence between the IRGN presented in SRK07] for the noise free case in C follows by taing F δ = C g δ. It was shown in SRK07] that method 1.3) is very effective for the diffusion optical tomography inverse problem, in which the operator L moves regularization from a B-spline coefficient space directly to the physical space. It may also be applied to allow an appropriate weighting on the elements of q ) to reflect the differing sensitivities of the operator with respect to different physical components. Assuming the necessary invertibility conditions on L L, and introducing the notation T := L L) 1/2 for self-adjoint operator T, F δ q) )T ) = T F δ q) ), 1.3) is rewritten as follows ] 1 q +1) = q T A δq ) )A δ q ) ) + ) I A δq ) ){F δ q ) ) F δq ) )q ) q), 1.4) where A δ q ) ) := F δq ) )T. 1.5) While scheme 1.4) is, in fact, preconditioned Tihonov s regularization combined with the Gauss-Newton algorithm, one can use other regularization methods in order to stabilize the Newton step EHN96]. In particular, replacing A δ q) )A δ q ) ) + ) I] 1 A δ q) ) in 1.4) by the more general operator Φ )A δ q ) ), yields q +1) = q T Φ )A δ q ) ){F δ q ) ) F δq ) )q ) q). 1.6) The class of methods described by 1.6) includes, amongst others, the M-times iterated preconditioned Tihonov s method, which is discussed in Section 3. For T = I, it has been shown in BA95] that through their incorporation in the Gauss-Newton process, some of these regularization techniques result in better convergence rates for sufficiently smooth solutions. Algorithm 1.6) is even more robust and efficient when we introduce a line search procedure with variable step size α ), 0 < α α ) 1 ] q +1) = q ) + α q ) q ) T Φ )A δ q ) ){F δ q ) ) F δq ) )q ) q). 1.7)

3 RSV, March 27, The family of methods 1.7) with T = I no preconditioning) and α ) = 1 no line search) was suggested by A.Baushinsy in BA95], and investigated further in K97], H97], BS05], BKA06], KN06]. Additional savings may also be realized by reuse of the Jacobian for M M,, inner iteration steps at the outer iteration step as follows: For m = 0 to M 1 Do ] q,m+1) = q,m) + α q,m) q,m) T Φ,m)A δ q,0) ){F δ q,m) ) F δq,0) )q,m) q) 1.8) End For Here the inner steps are initialized with q,0) = q 1,) and the line search parameter is still constrained by 0 < α α,m) 1. An alternative algorithm which uses inner iterations to limit the computational cost uses regularized Landweber iterations for the inner steps, hence avoiding matrix inversion in the inner steps, K97]. A ey to the convergence analysis of the methods presented in this paper, along with the local Lipschitz continuity of F, is the modified source condition L L)ˆq q) = F ˆq)v for some v H ) This source condition was introduced in KR93] and used in SRK07] for the analysis of schemes with preconditioning operator L. Here ˆq is a, possibly nonunique, solution to the noise free equation F q) = 0. Clearly, for T = I, 1.9) taes the form ˆq q = F ˆq)v, v H 1, which is equivalent to the Hölder source condition ˆq q = F ˆq)F ˆq)) 1/2 w, w H. A discussion of the advantages of 1.9) for convergence, and the associated stopping rule, as compared to other adopted convergence conditions and the Lepsij-type a posteriori stopping rule L90], BH05], was presented in SRK07]. Here, we emphasize that our results extend the methods in SRK07] to both the use of the more general operator Φ, as well as introducing the use of the inner iterations 1.8) for both T = I and the more general T = L L) 1/2. The paper is organized as follows: Theorems on the convergence of the iterations 1.7) and 1.8) are presented in Section 2. The resulting numerical schemes are discussed in Section 3. 2 Convergence Analysis We present the details of the basic convergence result for 1.7), followed by the result for 1.8) highlighting only the crucial aspects in which the proofs differ.

4 4 IRGN, March 27, 2008 Theorem 1. Assume that 1. F : DF ) H H 1 with H and H 1 being Hilbert spaces. The equation F q) = 0 is solvable maybe nonuniquely) and ˆq DF ) is a solution. 2. The operators F and F δ are Fréchet differentiable in U η ˆq), where U η ˆq) = {q H : ˆq q η DF ), and η = l 0) with l defined in 2.10) below. There is a positive constant M such that F δq 1 ) F δq 2 ) M q 1 q 2 for any q 1, q 2 U η ˆq). 2.1) 3. T LH, H) is a linear self-adjoint operator, T := t, and for some element v H 1, v := ε, the source condition holds ˆq q = T A ˆq)v, where Aˆq) := F ˆq)T, q H. 2.2) 4. The operator F δ approximates F to the following level of accuracy F δ ˆq) δ 1, F ˆq) F δˆq)) v δ ) 5. The regularization sequence { ) and the step size sequence {α ) satisfy the conditions ) > 0, ) ) 0, sup N {0 := d <, 0 < α +1) α) ) 6. For all G LH, H 1 ) and 0 < the regularizer Φ in 1.7) is constrained by Φ G)G I C 1, 2.5) Φ G)G I)G C 2, 2.6) To simplify the presentation we tae C = max{c 1, C 2, C 3 in the analysis. 7. Iterative process 1.7) is terminated according to the a priori stopping rule Φ G) C 3, > ) tδ 2 ) + δ 1 ) ρ < tδ 2 K) + δ 1 K), 0 < K = Kδ 1, δ 2 ), ρ > ) 8. For the constants associated with the operator F and iterations 1.7) the following conditions are fulfilled d 1 dα + t2 CMε + tc 2Mρ + ε) 1, 2.9) q 0) ˆq 0) 2tCρ + ε) 1 d 1 dα := l. 2.10) t2 CMε

5 RSV, March 27, Then the preconditioned iteratively regularized Gauss-Newton iterations with line search 1.7) satisfy q ) ˆq ) l, = 0, 1,..., Kδ 1, δ 2 ), 2.11) and q K) ˆq = O δ), 2.12) where δ = maxδ 1, δ 2 ). Proof. Tae arbitrary < Kδ 1, δ 2 ) and suppose that for any j such that 0 j < Kδ 1, δ 2 ) the induction assumption holds. Then, one has σ j) := qj) ˆq j) l. 2.13) q +1) ˆq = α ) q ˆq + 1 α ) )q ) { α ) T Φ )A δ q ) )) F δ q ) ) F δ ˆq) F δq ) )q ) ˆq) F δq ) )ˆq q) + F δ ˆq) = J 1 + J 2 + J 3, 2.14) where { J 1 = α ) T Φ )A δ q ) )) F δ q ) ) F δ ˆq) F δq ) )q ) ˆq), J 2 = α {ˆq ) q T Φ )A δ q ) ))F δq ) )ˆq q) and J 3 = 1 α ) )q ) ˆq) α ) T Φ )A δ q ) ))F δ ˆq). 2.15) We now estimate J i, i = 1, 2, By assumption 2.1), see for example ) EHN96]), F δ q ) ) F δ ˆq) F δq ) )q ) ˆq) M 2 q) ˆq 2. Thus, from 2.7) one derives J 1 α) tc ) M 2 q) ˆq ) 2. Source condition 2.2) yields ˆq q T Φ )A δ q ) ))F δq ) )ˆq q) = ] I T Φ )A δ q ) ))F δq ) ) T A ˆq)v. Adding and subtracting relevant terms, as well as applying 2.2) for F δ, gives ] I T Φ )A δ q ) ))F δq ) ) T A ˆq)v ]{ = T I Φ )A δ q ) ))A δ q ) ) Aˆq) A δ ˆq)) + A δ ˆq) A δ q ) )) + A δq ) ) v,

6 6 IRGN, March 27, 2008 which has the two terms T I Φ )A δ q ) ))A δ q ) ) ]{T F ˆq) F δˆq)) + T F δˆq) F δq ) )) v and ] T I Φ )A δ q ) ))A δ q ) ) A δq ) )v. Hence, assumptions 2.5) and 2.6), with 2.1) and 2.3), imply J 2 α ) t 2 Cδ 2 + α ) t 2 CM q ) ˆq ε + α ) tc ) ε. 2.17) 3. Finally, by 2.7) J 3 1 α ) ) q ) ˆq + α ) t Cδ 1 ). 2.18) Summarizing 2.16)-2.18) one concludes q +1) ˆq α) tcm 2 q ) ˆq α ) + α ) t 2 CMε] q ) ˆq ) { + α ) δ1 tc ) + tδ 2 + ε. 2.19) ) ) Because < Kδ 1, δ 2 ), combining 2.8), 2.19), 2.4) and 2.13), σ +1) α) tcmdl α ) + α ) t 2 CMε]dl + α ) tcdρ + ε). 2.20) Assumptions 2.9) and 2.10), together with 2.20), yield σ +1) l, hence proving inequality 2.11). 2.12) follows from stopping rule 2.8). Corollary 1. Assume F δq) N for any q U η ˆq), 2.21) and all conditions of Theorem 1, except 1. stopping rule 2.8) is replaced by the a posteriori stopping rule F δ q K) ) µ δ < F δ q ) ), 0 < Kδ), µ > 1, δ = max{δ 1, δ 2, and 2.22) ) and 2.10) are respectively replaced by { d 1 dα + t2 CMε + 2tC t 0) + 1)N 2 µ 1) 2 + M ε 1, 2.23) 2 and q 0) ˆq 0) 2tCε 1 d 1 dα t2 CMε := l. 2.24)

7 RSV, March 27, Then the iterations 1.7) satisfy q ) ˆq ) l, = 0, 1,..., Kδ), 2.25) and the sequence {Kδ) is admissible. Proof. From 2.22) it is immediate that for any < Kδ) one has µ δ F δ q ) ) F δ ˆq) + F δ ˆq) N q ) ˆq + δ, and therefore µ δ δ N q ) ˆq. Without loss of generality, one can set δ < 1. Thus and from 2.19) one obtains δ N 2 q ) ˆq 2 µ 1) ) q +1) ˆq α) tct 0) + 1)N 2 q ) ˆq 2 + α) tcm ) µ 1) 2 2 q ) ˆq 2 ) + 1 α ) + α ) t 2 CMε] q ) ˆq + α ) tcε ). 2.27) Using 2.23) and 2.24), result 2.25) now follows. While estimate 2.12) does not follow from stopping rule 2.22), the sequence K = Kδ) is nondecreasing as δ 0. Two cases are possible: 1. Kδ) = K 0 for any δ δ 0. Then by 2.3) and 2.22) q K0) F δ ) converges to a solution of the equation F q) = 0 in the norm of H as δ Kδ) as δ 0. Then q Kδ)) ˆq l Kδ)) 0 as δ 0. Sequence {Kδ) is, therefore, admissible. Remar 1. As opposed to the convergence results for iteratively regularized methods 1.6) with T = I, see BA95], K97], BS05], BKA06], KN06], in our convergence theorem ε, the norm of v in the source condition, does not have to be small for inequality 2.9) to be satisfied. Instead, t 2 ε with t := T must be small. While element q in 1.7) does not need to be close to the solution ˆq, condition 2.10) must hold for an appropriate choice of 0). Remar 2. When the forward operator F δ is not contaminated by noise, the constant δ 2 in 2.3) is zero, and the a priori stopping condition 2.8) simplifies accordingly. Scheme 1.8), which is just 1.7) when M = 1, permits reuse of the operator and has the potential to reduce the overall iteration cost, while possibly increasing the total number of iterations,. In the following note that the, m) element of the sequence corresponds to the i th element where i = 1 p=0 M p + m.

8 8 IRGN, March 27, 2008 Theorem 2. Assume all conditions of Theorem 1 but with the definitions 2.9) and 2.10) replaced by Then the iterations 1.8) satisfy d 1 dα + t2 CMεd + tc q 0,0) ˆq 0,0) q,m) ˆq,m) ˆl, 10Md ρ + ε) 1, 2.28) 2tCρ + ε) 1 d 1 dα t2 CMεd := ˆl. 2.29) 1 M p + m = 0, 1,..., Kδ 1, δ 2 ). 2.30) p=0 Proof. The result follows very similarly to the proof of Theorem 1. Assume that for the first i elements of the sequence, σ,m) := q,m) ˆq,m) ˆl, 1 0 M p + m i < Kδ 1, δ 2 ). 2.31) The i + 1) st element of the sequence is q,m+1) if m < M ), or q +1,0) if m = M ). Let m < M. Expression 2.14) is replaced by { q,m+1) ˆq = α,m) q ˆq + 1 α,m) )q,m) α,m) T Φ,m)A δ q,0) )) F δ q,m) ) F δ ˆq) F δq,0) )q,m) ˆq) F δq,0) )ˆq q) + F δ ˆq). 2.32) p=0 Notice now that { J 1 = α,m) T Φ,m)A δ q,0) )) F δ q,m) ) F δ ˆq) F δq,m) )q,m) ˆq) + F δq,m) ) F δq,0) ))q,m) ˆq). 2.33) Thus J 1 α,m) tcm 2 q,m) ˆq 2 + α,m) tcm q,m) ˆq q,m) q,0),m),m) 3α,m) tcm 2 q,m) ˆq 2 + α,m) tcm q,0) ˆq q,m) ˆq. 2.34),m),m) The bound of J 2 is obtained as for 2.17) yielding J 2 α,m) t 2 Cδ 2 + α,m) t 2 CM q,0) ˆq ε + α,m) tc,m) ε, 2.35) and we also immediately obtain Therefore J 3 1 α,m) ) q,m) ˆq + α,m) t Cδ 1,m). 2.36) q,m+1) ˆq 3α,m) tcm 2 q,m) ˆq α,m) ] q,m) ˆq + α,m) t 2 CMε q,0) ˆq,m) { + α,m) δ1 tc,m) + tδ 2 + ε + α,m) tcm q,m) ˆq q,0) ˆq.2.37),m),m),m)

9 RSV, March 27, By the induction, and using result d d M in order to combine first and fith terms of 2.37), one arrives at σ,m+1) 5α,m) tcmd Mˆl 2 2 and the result follows. Finally, if m = M, the result holds by Theorem 1. 3 Discussion of the Numerical Schemes + d dα ) + α,m) t 2 CMεd M ]ˆl + α,m) tcdρ + ε), 2.38) In this section we consider examples of generating operators Φ G), G LH, H 1 ), motivated by different regularization techniques merged with Gauss-Newton iterations. Example 1. From 1.3) and 1.4) it is clear that for Φ G) := G G + I] 1 G 3.1) algorithm 1.7) taes the form: q +1) = q ) + α ) p ), 3.2) where the search direction p ) is the solution of F δ q ) )F δq ) ) + ) L L]p ) = F δ q ) )F δ q ) ) + ) L Lq ) q)]. 3.3) This line search algorithm with search direction obtained from 3.3) was introduced and analyzed in SRK07]. Here, we extend its use by adopting the appropriately initialized inner iterations 1.8) For m = 0 to M 1 Do q,m+1) = q,m) α,m) F δ q ) )F δq ) )+,m) L L] 1 F δ q ) )F δ q,m) )+,m) L Lq,m) q)] End For Each inner update requires a system solve with system matrix G G +,m) I] which can be obtained cheaply provided that a method such as preconditioned conjugate gradients is adopted at every step and for M > 1 the solution is initialized with a good initial starting value, presumably the last value from the previous iteration. Example 2. In order to further stabilize the Gauss-Newton step, we may mae use of M-times repeated Tihonov regularization, BA95], Section 5.1 EHN96]). The search direction p ) in 3.2) is computed as the result of inner iterations in which the solution is prevented from stepping too far from the previous mapped value of q ). Calculate p,1) by 3.3) For End For m = 1 to M 1 Do p,m+1) = F δ q ) )F δq ) ) + ) L L] 1 F δ q ) )F δ q ) ) ) L Lp,m) ] 3.5) q +1) = q ) + α ) p,m) 3.4)

10 10 IRGN, March 27, 2008 Practically the algorithm is implemented as given, but to see how it differs from 3.4) notice that the inner updates, for m < M 1, can be written in the equivalent form q +1,m+1) = q ) F δ q ) )F δq ) ) + ) L L] 1 F δ q ) )F δ q ) ) + ) L Lq ) q +1,m) )] 3.6) with the line search parameter α ) introduced for m = M 1. We now show that this iteration is equivalent to 1.7) for the operator Introduce the notation Φ G) := i G G + I] i+1) G. 3.7) D := F δ q ) )F δq ) ) + ) B], g := F δ q ) )F δ q ) ), B = L L, 3.8) then and p,m+1) = ) D 1 Bp,m) D 1 g, m 1 3.9) D 1 B = F δ q ) )F δq ) ) + ) B] 1 B = T A δq ) )A δ q ) ) + ) I] 1 T 1 := Y. Noticing from 3.3) that we can define p,0) = q q ), one obtains Identity 3.11) yields p,m) = )) M Y M Y M q q ) ) )) i Y i D 1 g. 3.10) = T Λ M T 1, where Λ := A δq ) )A δ q ) ) + ) I. 3.11) p,m) = = )) M T Λ M )) M T Λ M )) i T Λ i+1) T 1 q q ) ) T 1 q q ) ) )) i T Λ i+1) T F δ q ) )F δ q ) ) )) i T Λ i+1) A δq ) ){F δ q ) ) F δq ) )q ) q). A δq ) )F δq ) )q ) q) But now, by induction, the first two terms in 3.10) simplify to q q ), and 3.10) becomes p,m) = q q ) T )) i Λ i+1) A δq ) ){F δ q ) ) F δq ) )q ) q), 3.12) from which 3.7) follows. It remains to verify that 3.7) meets conditions 2.5)-2.7) of Theorem Φ G)G I = i G G + I] i+1) G G I sup λ 0, ) i λ + ) i+1) λ 1

11 RSV, March 27, ] M = sup 1. λ 0, ) λ + 2. Φ G)G I)G = ) ] M i G G + I] i+1) G G I G sup λ λ 0, ) λ + = 1 2 2M 1) M 1 2 2M) M 2M. 3. Finally, Φ G) = i G G + I] i+1) G = sup λ 0, ) λ λ + ] i = sup λ + λ 0, ) To complete the estimate, we apply Lemma 3.1 BS05] With a = sup a 0,1] λ λ+ and P = M 1 one obtains Φ G) { sup λ 0, ) i λ + ) i+1) λ P 1 a) i a 2P + 1). 2M. 1 λ ] i λ λ + λ + 1 λ +. Example 3. A modification of 3.5) replaces the dependence of the iterative step in 3.5) on ) by the constant parameter σ > 0, F δ q ) )F δq ) ) + σl L]p,m+1) = F δ q ) )F δ q ) ) + σl Lp,m). It is immediate by the derivation of 3.12) that in this case p,m) = q q ) T M 1 σ i A δq ) )A δ q ) )+σi] i+1) A δq ) ){F δ q ) ) F δq ) )q ) q). 3.13) Now taing M as to satisfy the condition ) 0 for ) := 1/M, gives 1.7) with operator Again, it remains to verify conditions 2.5)-2.7). 1. For 2.5) Φ G)G I sup λ 0, ) 1 1 Φ G) := σ i G G + σi] i+1) G. 3.14) ] 1 σ λ+σ 2. For 2.6) Φ G)G I)G sup λ 0, ) σ λ+σ 1. ] 1 λ = σ ) σ 2, provided < 2, which is immediate for M To obtain 2.7) apply again Lemma 3.1 BS05] with a = λ λ+σ and P = 1 1, yielding Φ 2 G) σ.

12 12 IRGN, March 27, 2008 Remar 3. From Example 3, it is now evident that introducing a line search parameter in 3.6) q +1,m) = q ) + α,m) p,m), replaces σ on the right hand side in 3.5) by σ,m) = σ/α,m). An equivalent modification occurs in 3.5), and the distinction between 3.12) and 3.13) is lost. Moreover, it is not possible to reformulate the new parameter dependent scheme in terms of 1.7), and we conclude that line searching in the inner iterations is not feasible unless we utilize 1.8). 4 Conclusions Theoretical convergence results and numerical implementation of a family of preconditioned iteratively regularized Gauss-Newton schemes for solution of general ill-posed nonlinear minimization have been presented. These extend the initial theory of SRK07] for application of preconditioning to a larger class of iterative schemes with line search, and have potential application in a wide variety of ill-posed problems. References BA93] BA95] BS05] BH05] BKA06] DES98] EHN96] H97] Baushinsy, A. B., Iterative methods for nonlinear operator equations without regularity, new approach, Dol. Russian Acad. Sci , 1993). Baushinsy, A. B., Iterative methods without saturation for solving degenerate nonlinear operator equations, Dol. Russian Acad. Sci , 1995). Baushinsy, A. B. and Smirnova, A. On application of generalized discrepancy principle to iterative methods for nonlinear ill-posed problems, Numerical Functional Analysis and Optimization, 26, N1, 35-48, 2005). Bauer, F. and Hohage, T. A Lepsij-type stopping rule for regularized Newton methods, Inverse Problems, 21, , 2005). Burger M., and Kaltenbacher B., Regularizing NewtonKaczmarz Methods for Nonlinear Ill-Posed Problems, SIAM J. Num Anal., 44, N1, , 2006). Deuflhard, P., Engl, H.W. and Scherzer, O. A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions, Inv. Probl., 14, , 1998). Engl, H., Hane, M. and Neubauer, A. Regularization of Inverse Problems, Kluwer Academic Publisher, Dordecht, Boston, London, 1996). Hohage, T. Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and inverse scattering problem, Inverse Problems, 13, , 1997).

13 RSV, March 27, K97] KN06] KR93] Kaltenbacher, B., Some Newton-type methods for the regularization of nonlinear ill-posed problems, Inverse Problems, 13, , 1997). Kaltenbacher, B. and Neubauer, A. Convergence of projected iterative regularization methods for nonlinear problems with smooth solutions. Inverse Problems, 22, N3, , 2006). Kunisch K. and Ring N. Regularization of nonlinear illposed problems with closed operators. Numerical Functional Analysis and Optimization, 14, , 1993). L90] Lepsij, O.V. On a problem of adaptive estimation in Gaussian white noise, Theory Probab. Appl., 35, , 1990). SRK07] Smirnova, A.B., Renaut R.A., and Khan, T. Convergence and application of a modified iteratively regularized GaussNewton algorithm. Inverse Problems, 23, N4, ).

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

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