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25 Affine Regularization Algorithms in Hilbert Space The foregoing considerations treat the case where the input data (A, f ) of equation (1) are known without errors. Now suppose these data are given approximately. Let A ∈ L(X1 , X2 ). Assume that instead of the exact data (A, f ) in (1) their approximations (Ah , fδ ) ∈ F = L(X1 , X2 ) × X2 are available such that fδ − f X2 ≤ δ. (34) Ah − A L(X1 ,X2 ) ≤ h, In accordance with (11), an approximation to the quasisolution x∗ = PX1∗ (ξ) can be defined as follows: (h,δ) xα(h,δ) = Rα(h,δ) (Ah , fδ ), (35) where Rα(h,δ) (Ah , fδ ) = = (E − Θ(A∗h Ah , α(h, δ))A∗h Ah )ξ + Θ(A∗h Ah , α(h, δ))A∗h fδ .

For the first term in the right part of (23) we have (Θ(Ah , α)Ah −Θ(A, α)A)Am (A + εE)µ v X ≤ 1 ≤ v X |1 − Θ(λ, α)λ|· 2π Γα · (R(λ, A) − R(λ, Ah ))Am (A + εE)µ L(X) |dλ|. 12) and (16) it follows that (R(λ,A) − R(λ, Ah ))Am (A + εE)µ L(X) ≤ ∞ ≤ ( R(λ, A) k L(X) h) R(λ, A)Am L(X) (A + εE)µ L(X) ≤ k=1 ≤ c13 h R(λ, A)Am (1 − ω0 )|λ| L(X) ∀λ ∈ Γα Using the identity R(λ, A)A = −E + λR(λ, A), we obtain R(λ, A)Am L(X) ≤ c14 νp (|λ|), ∀α ∈ (0, α0 ]. 43 Regularization Algorithms in Banach Space where, by definition, νp (t) = t−1 , [p] = 0, 1, [p] > 0, t > 0.

25) The scheme (11), (23) can be implemented in the similar way: xα = u 1 , α α ∈ (0, α0 ]; (26) 24 Regularization Methods for Linear Equations du (27) + A∗ Au = A∗ f, u(0) = ξ. dt The methods (24)–(25) and (26)–(27) are often referred to as stabilization methods. 4. Let g : [0, M ] → R be a bounded and Borel measurable function, continuous at λ = 0. Suppose that sup |1 − λg(λ)| < 1 ∀ε ∈ (0, M ]. (28) λ∈[ε,M ] Then the function  1   [1 − (1 − λg(λ))1/α ], Θ(λ, α) = λ g(0)   , α λ = 0, (29) λ=0 defined for the discrete set of values α = 1, 1/2, .