Adjoint Variable Method for the Study of Combined Active and Passive Magnetic Shielding

Algorithm 1

Gradient-adjoint algorithm.

1. 𝑖=0 and estimate the initial value 𝐩₀.

2. Compute 𝐴(𝐩_𝑖) from the direct problem (2.5) setting 𝐩=𝐩_𝑖.

3. Compute 𝜉 from the dual problem (3.7) using 𝐴(𝐩_𝑖) instead of 𝐴(𝐩).

4. Using explicit expression (3.10) for the derivative, compute 𝜕𝐹(𝐩_𝑖)/𝜕𝐩.

5. By a line search algorithm determine the optimal step 𝛼_𝑖.

6. Compute 𝐩_𝑖+1=𝐩_𝑖−𝛼_𝑖(𝜕𝐹(𝐩_𝑖)/𝜕𝐩).

7. If convergence is reached (controlling e.g., the difference 𝐹(𝐩_𝑖)−𝐹(𝐩_𝑖+1))

 then stop, otherwise set 𝑖∶=𝑖+1 and go to step 2 of the algorithm.