Sparse Regularization-Based Reconstruction for Bioluminescence Tomography Using a Multilevel Adaptive Finite Element Method

Algorithm 1

TNIPM algorithm for BLT reconstruction at the 𝑘th mesh level.

Initialize the parameters: relative tolerance 𝜉>0, 𝑡=1/𝜆, 𝑆^𝑝𝑘=0, 𝑢=𝟏=(1,...,1)∈𝑅^𝑁𝑝;

repeat

1. Compute the search direction (Δ𝑆^𝑝𝑘,Δ𝑢) as an approximate solution to the Newton system

  (14) by preconditioned conjugate gradient method;

2. Compute the step size a by backtracking line search;

3. Update the iterate by (𝑆^𝑝𝑘,𝑢)∶=(𝑆^𝑝𝑘,𝑢)+𝑎(Δ𝑆^𝑝𝑘,Δ𝑢);

4. Construct a dual feasible point 𝜈 from (10);

5. Evaluate the duality gap 𝛾 from (11);

6. quit  if 𝛾/𝐺(𝑣)≤𝜉;

7. Update t: 𝑡{max{𝜇min{2𝑁^𝑝/𝛾,𝑡},𝑡},𝑎≥𝑎_min𝑡,𝑎<𝑎_min, where 𝜇>1 and 𝑎_min∈(0,1].