Numerical Solution of Diffusion Models in Biomedical Imaging on Multicore Processors

Algorithm 1

Preconditioned GMRES for solving a linear system 𝐴𝑥=𝑏. Input: 𝐴 (matrix coefficient), 𝑏 (right hand side), 𝑃 (preconditioner). Output: 𝑥_𝑚, approximate solution at the 𝑚th step. For all iteration, a matrix-vector product is required.

Compute 𝑟₀=𝑃(𝑏−𝐴𝑥₀), 𝛽∶=‖𝑟₀‖₂ and 𝑣₁∶=𝑟₀/𝛽

Define the (𝑚+1)×𝑚 𝐻_𝑘={ℎ_𝑖𝑗}_{1≤𝑖≤𝑚+1,1≤𝑗≤𝑚}. Set 𝐻_𝑚=0.

for 𝑘=1 to 𝑚 do

  Compute 𝑤_𝑗∶=𝑃𝐴𝑣_𝑗

  For 𝑖=1 to 𝑘 do:

  ℎ_𝑖𝑗∶=(𝑤_𝑗,𝑣_𝑖)

  𝑤_𝑗∶=𝑤_𝑗−ℎ_𝑖𝑗𝑣_𝑖

end for

  ℎ_𝑗+1,𝑗=‖𝑤_𝑗‖₂. If ℎ_𝑗+1,𝑗=0 set 𝑚∶=𝑗 and go to 12

  𝑣_𝑗+1=𝑤_𝑗/ℎ_𝑗+1,𝑗

end for

Compute 𝑦_𝑚 the minimiser of ‖𝛽𝑒₁−𝐻_𝑚𝑦‖₂

Set 𝑥_𝑚∶=𝑥₀+𝑉_𝑚𝑦_𝑚.