Iterative Methods for Approximate Solution of Inverse Problems

Inexact Newton regularization methods in Hilbert scales

“…We will estimate the norms of T (1) j (z) and T (2) j (z) for z ∈ Γ α j . With the help of Assumption 3, (2.5) and (2.11), similar to the derivation of (2.19) we have…”

“…Combining the above estimates on T (1) j (z) and T (2) j (z) and noting b+s 2(a+s) ≤ 1 2 , it follows for z ∈ Γ α j that…”

“…It is easy to check |ϕ α (z)| α −1 on Γ (1) α and |ϕ α (z)| R −1 on Γ (2) α . Moreover, on Γ (3) α ∪ Γ (4) α there holds…”

“…It is easy to see that |ϕ α (z)| α −1 on Γ (1) α , |ϕ α (z)| R −1 on Γ (2) α and (4) α . Therefore where T n := F (x n ).…”

A mixed Newton-Tikhonov method for nonlinear ill-posed problems

“…In the last decades, the theory and methods to solve nonlinear ill-posed problems have been developed greatly [3][4][5][6][7][8] . Since most inverse problems proposed in practice are nonlinear and ill-posed, so the techniques to solve these problems have critical meaning not only to the theory of inverse problems but also to practical applications.…”