On The Iterated Tikhonov Regularizationg For Ill-Posed Problem

After a general discussion of the term ill-posed problems, examples of various classes of of ill-posed problems were arised in various application fields including Geological prospecting, Computer tomography, Backwards heat equation etc. Especially, a lot of in-verse problems are ill-posed.In this paper, we introduce the general knowledge of regularization methods for ill-posed problems and discuss the linear operator equation Kx=y(0.0.7) where, K is a linear compact operator between Hilbert spaces X and Y.We consider the iterated Tikhonov regularization method x0α=0,(αI+K*K)xmα= K*y+αxαm-1,m=1,2,…(0.0.8) and its perturbed equation x0,δα= 0, (αI+K*K)xαm,δ= K*yδ+αxαm-1,δ,m=1,2,….(0.0.9)The parameter m plays the role of the regularization parameter when the parameter a> 0 is fixed in this method. we deduce the property of regularizing filter function, give a priori optimal choice of m(α,δ)= O(αδ-2/2r+1), r≥0 and obtain optimal order of conver-gence. In practice, it is more convenient than viewing a as the regularization parameter for computation. Then, we induce a posteriori optimal choice of m(δ) and obtain optimal order of convergence. Finally, numerical examples are included to verify the theoretical results. We can conclude that the error percision of iterated Tikhonov regularization method with parameter m is the highest in the three methods. The main results in this paper are listed here: Theorem 1 The filter function q(m,μ)=1-(α/α+μ2)m,m=1,2,…obtained by(0.0.8) is a regularizing filter. Define the linear and bounded operators Rαm:Yâ†'X a regulariza-tion strategyTheorem 2 (Priori estimation)Let K:Xâ†'Y be linear compact operator.(â…°)Define the linear and bounded operators Rαm:Yâ†'X by(0.0.10)is a regularization strategy,and‖Rαm‖≤(?).The sequence xα,δm=Rαmyδis computed by the iteration (0.0.9).Every strategy m(α,δ)â†'∞(δâ†'0)withδ2m(α,δ)â†'0(δâ†'0)is admissi-ble.(â…±)Let x=(K*K)rz∈(K*K)r(X)with‖z‖≤E and 0<c1<c2,letα> (r/c1)(δ/E)2/2r+1.For every choice m(α,δ)with c1α(E/δ)2/2r+1≤m(α,δ)≤c2α(E/δ)2/2r+1, the following estimate holds: for some constant c>0 depending on c1,c2 and r.Theorem 3 Let K:Xâ†'Y be linear,compact and one-to-one with dense range.LetÏ„>1 and yδ∈Y,be perturbed with‖y-yδ‖≤δ,‖yδ‖≥τδ.Let xαm,δ=Rαmyδ, m=0,1,2,….be determined by(0.0.9),Then the following assertions hold:(â…°)(?)‖Kxαm,δ-yδ‖=0,for everyδ>0.i.e.,the following stopping rule is well-defined:Let m=m(δ)∈N0 be the smallest integer with‖Kxαm,δ-yδ‖≥τδ. (â…±)δ2m(δ)â†'0 forδâ†'0.i.e.,the choice of m(δ)is admissible.Therefore,the sequence xαm(δ),δconverges to x.Theorem 4(Posteriori estimation)Let all conditions in Theorem 3 be satisfied,andα>α0.If x:(K*K)rz∈(K*K)r(X),‖z‖≤E,then we have the following order of convergence: for some constant C>0,i.e.,this choice of m(δ)by the stopping rule is optimal.