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

Algorithm 2

Augmented Arnoldi-Tikhonov regularization method.

Input: 𝐴∈ℝ^𝑛×𝑛, 𝑏∈ℝ^𝑛, 𝑊∈ℝ^𝑛×𝑘, 𝜀, 𝜂, 𝑚₀.

Output: 𝑥_𝜇,𝑚, 𝜇_𝑚, 𝑚=𝑚_min+𝑚₀.

 (1) Compute the modified Arnoldi decomposition 𝐴[𝑊,𝑉_𝑚]=[𝑈,𝑉_𝑚+1]𝐻, with 𝑚=𝑚_min+𝑚₀,

   where 𝑚_min is the smallest number such that ‖(𝐼−𝑃)𝑏‖²<𝜂²𝜀².

 (2) Compute the solution 𝜇_𝑚 of the equation 𝜙_𝑚(𝜇)=𝜂²𝜀² by Newton's method.

 (3) Compute the solution 𝑦_𝜇,𝑚 of the least-squares problem (13) and obtain the approximate

   solution 𝑥_𝜇,𝑚=[𝑊,𝑉_𝑚]𝑦_𝜇,𝑚.