International Conference «Inverse and Ill-Posed Problems of Mathematical Physics»,

We consider an operator equation) (, A R f f Au ∈ = , (1) where is the linear continuous operator between real Hilbert spaces H and F. In general our problem is ill-posed: the range R(A) may be non-closed, the kernel N(A) may be non-trivial. We suppose that instead of exact right-hand side f we have only an approximation) , (F H L A ∈ F f ∈ δ , δ δ ≤ − f f. To get regularized solution of the equation Au = f we consider the regularization methods in the general form (see [15]), using the approximation r u () δ f A A A g u A A Ag A I u r r r * * 0 * *) () (+ − =. (2) Here is the initial approximation, r is the regularization parameter, I is the identity operator and the generating function 0 u () λ r g satisfies the conditions (3)-(5). 0 , sup * 0 ≥ ≤ ≤ ≤ r r g r a γ λ λ λ , (3) () 0 0 0 , 0 , 1 sup p p r r g p p r p a ≤ ≤ ≥ ≤ − − ≤ ≤ γ λ λ λ λ , (4) () 0 , sup 0 ≥ ≤ ≤ ≤ r r g r a γ λ λ. (5) Here * 0 , , γ γ p and p γ are positive constants, A A a * ≥ , 1 0 ≤ γ and the greatest value of , for which the inequality (4) holds is called the qualification of method. 0 p The following regularization methods are special cases of general method (2). P1 The Tikhonov method. Here () δ α α f A A A I u * 1 * − + = 1 1 1 0 = = = + = = = − − − γ γ λ λ α p r g r u r () p p p p p − − = 1 1 γ. P2 The iterative variant of the Tikhonov method. Let H u u m N m ∈ = ≥ ∈ α , 0 0 , 1 ,-initial approximation and. Here m n f A u A A I u n n = + + = − − δ α α α α () 1 p m p …