Initialize the parameters: relative tolerance 𝜉 > 0 , 𝑡 = 1 / 𝜆 , 𝑆 𝑝 𝑘 = 0 , 𝑢 = 𝟏 = ( 1 , . . . , 1 ) 𝑅 𝑁 𝑝 ;
repeat
1. Compute the search direction ( Δ 𝑆 𝑝 𝑘 , Δ 𝑢 ) 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 𝑝 𝑘 , 𝑢 ) = ( 𝑆 𝑝 𝑘 , 𝑢 ) + 𝑎 ( Δ 𝑆 𝑝 𝑘 , Δ 𝑢 ) ;
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 { m a x { 𝜇 m i n { 2 𝑁 𝑝 / 𝛾 , 𝑡 } , 𝑡 } , 𝑎 𝑎 m i n 𝑡 , 𝑎 < 𝑎 m i n , where 𝜇 > 1 and 𝑎 m i n ( 0 , 1 ] .
Algorithm 1: TNIPM algorithm for BLT reconstruction at the 𝑘 th mesh level.