The Methods For Solving Ill-posed Problems

There are many kinds of methods to solves the ill-posed equation , usually use the Tikhonov regularization method, the Tikhonov regularization iteration method, the Landweber iteration method as well as the dynamic system method, and so on. In the real life, the ill-posed equation's origin is quite widespread, including sick-state linear equation, natural survey, scanning image formation, counter time inversion and so on. This stud}' of kind of question has three basic questions: (1) solubility, namely solution existence. (2) solution uniqueness. (3) solution stability.First, in this article,we use two-step linear stationary iteration equationwith x^0,α,β= 0, x^-1,α,β= 0. (m= 0,1,2,...).and perturbation equationwith x^{0,α,β,δ}=0,x^{-1,α,β,δ}=0,( m= 0,1,2,...).discusses ill-posed equationthe regularity solution, the error estimate and the stopping criteria, the main conclusion is a.s follows:theorem 1 Filler function q(m.μ) get by the equation (0.0.1)is the regular filter function, namely definition operator R_m:is the regularization strategy. and||R_m||≤(?),wiith 0 <β< 1,0 <α<(?),a, b are the two different root for equation x² - (1+(?)-αμ²)x + (?)- 0.We get a =(?)b=(?) theorem 2(Prior estimate) K:X→Y is linear compact operator. (1):0<β<1,0<α<(?).define the linear and operator R_m:Y→X by (0.0.4).These operator R_m define a regularization strategy,with ||R_m||≤(?).The sequencex^m.δ=R_my^δ is computed by the iteration (0.0.2).Every strategy m(δ)→∞(δ→0) withδ²m(δ)→0(δ→0) is admissible. (2):Now let x = (K^*K')^σz∈(K^*K)^σ(X),||z||≤E,existence c >0, 01(σ)2(σ).For every choice m(δ) with c₁(σ)(E/δ)^?(δ)≤c₂(σ)(E/δ)^?.we havefor some c depending on c₁(σ),c₂(σ).theorem 3 Let K:X→Y be linear, compact, and one-to-one operator. LetÏ„>c and y^δ∈Y be perturbations with ||y-y^δ||≤δand ||y^δ||≥τδfor allδδ∈(0,δ₀), c = 1+2e^-1Let the sequence x^m,δ = R_my^δm = 0,1,2,...,0<β<1,0 <α<(?), be determined by (0.0.2).Then the following assertions hold:(1) lim_{m→∞}||Kx^m,δ-y^δ||=0. i.(?). the following stopping rule is well-defined: Let m = m(δ)∈N₀,be the smallest integer with || Kx^m,δ-y^δ||δ≤τδ.(2)δ²m(δ)δ→0,i.e.,this choice of m(β) is admissible, the sequences x^m(δ),δ converges to x. (3)If x = K^*z∈K^*(Y) and x = K^*Kz∈K^*K(X).for some z with ||z||≤E .Then we have the following orders of convergence:respectively, for some C>0. This means that this choice of m.(δ) is optimal.theorem 4(Posterior estimate) All conditions of theorem 3 arc Satisfied, if x =(K K)^?(?). for some z with ||z||≤E. Then we have. the following orders of convergence.with C>0. This means that this choice of m(δ) is optimal.Second.dynamical system methodwith its perturbation equation with u :=((du(t))/(dt)),u_δ:= ((du_δ(t))/(dt)),||f-f_δ||≤δ.α>0, u₀ be arbitrary number, discusses ill-posed equationthe convergence of solution, the error estimate and the stopping criteria, the main conclusionis as follows:theorem 5 Equation (0.0.5) has a unique global solution u(t) and u(∞)=lim_{t→∞}u(t) = y. Equation (0.0.6) has a unique global solution u_δ(t) with t_δsuch thatThis t_δcan be chosen, for example,theorem 6 (Prior estimate) Assume the smooth conditionIf t is chosen by the a priori parameter choice t =α^-1(8E²Î´^-2k²(v)v²)^1/(2v+1)withk(v) = max(v^v', 1}, then we haw the error estimatewhere c(v) = ((8v²)^v/(2v-1)+ (8v²)^{1/(2(2v+1))2-(1/2)}k(v)^1/(2v+1). the order optimality under the prior condition .theorem 7 u_δ(t) is solution of equation (0.0.6). Equation (0.0.7) has a unique smallestnorm solution. ||Au₀-f_δ||>τδ>0,τ≥2. Then exist a unique solution T <∞bysuch that h(T) = 0.theorem 8 u_δ(t) is solution of equation (0.0.6). Equation (0.0.7) has a unique smallcstnorm solution.||Au₀ f_δ||>τδ> 0,τ≥2. Satisfy y u₀=(A^*A)^?zwith||z||≤Eandv > 0. Where t = T is solution by h(T) = 0.Then(1) u_δ(T)→y,δ→0.(2) ||u_δ(T)-y||≤D(v)E^?δ^?, D(v) = (?)(?)^?+ (Ï„+2)^?. the order optimality under the posterior condition . Finally, we give the number example.