Augmented Arnoldi-Tikhonov Regularization Methods for Solving Large-Scale Linear Ill-Posed Systems

Algorithm 3

Newton's method for 𝜙_𝑚(𝜇)=𝜂²𝜀².

(1) Set 𝜇₀=0 and 𝑖=0.

(2) Solve the least-squares problem

      min_{𝑧∈ℝ𝑚+𝑘+1}𝜇_𝑖^1/2(𝐻𝑅⁻¹)^𝑇𝐼0𝑧−(𝑈,𝑉_𝑚+1)^𝑇𝑏

   to obtain 𝑧_{𝜇𝑖,𝑚}.

(3) Compute 𝜙_𝑚(𝜇_𝑖)=𝑧^𝑇_{𝜇𝑖,𝑚}𝑧_{𝜇𝑖,𝑚}+‖(𝐼−𝑃)𝑏‖².

(4) Solve the least-squares problem

     min_{𝑧∈ℝ𝑚+𝑘+1}𝜇_𝑖^1/2(𝐻𝑅⁻¹)^𝑇𝐼0𝑧−𝐻𝑅⁻¹(𝐻𝑅⁻¹)^𝑇𝑧_{𝜇𝑖,𝑚}

   to obtain 𝑤_{𝜇𝑖,𝑚}.

(5) Compute 𝜙^′_𝑚(𝜇_𝑖)=−2𝑧^𝑇_{𝜇𝑖,𝑚}𝑤_{𝜇𝑖,𝑚}.

(6) Compute the new approximation

        𝜇_𝑖+1=𝜇_𝑖−𝜙_𝑚(𝜇_𝑖)−𝜂²𝜀²𝜙^′_𝑚(𝜇_𝑖).

(7) If |𝜇_𝑖+1−𝜇_𝑖|<10⁻⁶, stop; else 𝜇_𝑖∶=𝜇_𝑖+1, 𝑖∶=𝑖+1, and go to 2.