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

Algorithm 1

Modified Arnoldi process.

Input: 𝐴∈ℝ^𝑛×𝑛, 𝑏∈ℝ^𝑛, 𝑊∈ℝ^𝑛×𝑘, 𝑚.

Output: 𝑈∈ℝ^𝑛×𝑘, 𝑉_𝑚+1∈ℝ^{𝑛×(𝑚+1)}, 𝐻∈ℝ^{(𝑚+𝑘+1)×(𝑚+𝑘)}.

 (1) Compute QR-factorization 𝐴𝑊=𝑈𝐻.

 (2) Compute 𝑣₁=𝐴𝑏−𝑈(𝑈^𝑇𝐴𝑏), 𝑣₁=𝑣₁/𝑣₁, and set 𝑉₁=[𝑣₁].

 (3) For 𝑗=1,2,…,𝑚 Do:

    𝑣_𝑗+1∶=𝐴𝑣_𝑗;

    For 𝑖=1,2,…,𝑘 Do:

     ℎ_{𝑖,𝑗+𝑘}∶=𝑈(∶,𝑖)^𝑇𝑣_𝑗+1;

     𝑣_𝑗+1∶=𝑣_𝑗+1−ℎ_{𝑖,𝑗+𝑘}𝑈(∶,𝑖);

    End For

    For 𝑖=1,2,…,𝑗 Do:

     ℎ_{𝑖+𝑘,𝑗+𝑘}∶=𝑣^𝑇_𝑖𝑣_𝑗+1;

     𝑣_𝑗+1∶=𝑣_𝑗+1−ℎ_{𝑖+𝑘,𝑗+𝑘}𝑣_𝑖;

    End For

    ℎ_{𝑗+𝑘+1,𝑗+𝑘}∶=𝑣_𝑗+1/𝑣_𝑗+1;

    𝑣_𝑗+1∶=𝑣_𝑗+1/ℎ_{𝑗+𝑘+1,𝑗+𝑘};

    𝑉_𝑗+1∶=[𝑉_𝑗,𝑣_𝑗+1];

     End For