Optimal convergence rates for inexact Newton regularization with CG as inner iteration

We provide a list of errata:

1. Page 3: the bracket before Theorem 2.2 should read as “(see Lemma 7.10, Lemma 7.11, Theorem 7.12, and their proofs in …):”.

2. The estimate in Theorem 2.2 should read

   ∥ T † ⁢ y - x k ⁢ ( δ , y δ ) δ ∥ ≤ c ⁢ ( τ , μ ) ⁢ ( ∥ z ∥ + ∥ v ∥ 1 2 ⁢ μ + 1 ⁢ δ 2 ⁢ μ 2 ⁢ μ + 1 ) .

3. Some changes in the proof of Theorem 4.3:

   1. Page 7: the last seven lines should read as (I only cite the changes and ignore text)

      x † - x 0 = ( T * ⁢ T ) 1 2 ⁢ v = T * ⁢ w = T n * ⁢ R ⁢ ( x † , x n δ ) * ⁢ w + Q ⁢ ( x † , x n δ ) * ⁢ w

      = ( T n * ⁢ T n ) 1 2 ⁢ v n + Q ⁢ ( x † , x n δ ) * ⁢ w

      with ∥ v ∥ = ∥ w ∥ and ∥ v n ∥ ≤ ∥ R ⁢ ( x † , x n δ ) * ⁢ w ∥ ≤ ( 1 + c R ⁢ ρ γ ) ⁢ ∥ v ∥ . Together with …

      P n ⁢ ( x † - x 0 ) = P n ⁢ Q ⁢ ( x † , x n δ ) * ⁢ w ,

      T n † ⁢ y n = ( T n * ⁢ T n ) 1 2 ⁢ v n + ( I - P n ) ⁢ Q ⁢ ( x † , x n δ ) * ⁢ w .

      Now Theorem …

      ∥ x † - x N δ ∥ ≤ ∥ P n ⁢ ( x † - x 0 ) ∥ + ∥ T n † ⁢ y n - h n , k n δ ∥ ≤ c ¯ ⁢ ( ∥ Q ⁢ ( x † , x n δ ) ∥ ⁢ ∥ v ∥ + ∥ v ∥ 1 2 ⁢ δ n 1 2 ) .

   2. Page 8:

      1. in formula (4.16) the constant c should be replaced by c R .

      2. in the formula on line 6, T † ⁢ y n = … should be replaced by T n † ⁢ y n = ( T n * ⁢ T n ) μ ⁢ v n + ( I - P n ) ⁢ z n .

      3. formula (4.17) should be replaced by

         ∥ x † - x N δ ∥ ≤ c ¯ ⁢ ( ∥ z n ∥ + ∥ v n ∥ 1 2 ⁢ μ + 1 ⁢ δ n 2 ⁢ μ 2 ⁢ μ + 1 )

         ≤ c ¯ ⁢ c μ ⁢ ∥ Q ⁢ ( x † , x n δ ) ∥ 2 ⁢ μ ⁢ ∥ v ∥ + c ¯ ⁢ ( 1 + c R ⁢ ρ γ ) 2 ⁢ μ 2 ⁢ μ + 1 ⁢ ∥ v ∥ 1 2 ⁢ μ + 1 ⁢ δ n 2 ⁢ μ 2 ⁢ μ + 1 .

      4. in the last line of the proof (3.6) should be replaced by (3.3), (3.6).

      5. (4.19) should be replaced by F ′ ⁢ ( a ~ ) ⁢ h = R ⁢ ( a ~ , a ) ⁢ F ′ ⁢ ( a ) ⁢ h + Q ⁢ ( a ~ , a ) ⁢ h .