Numerical Optimal Control for Problems with Random Forced SPDE Constraints

Algorithm 1

(1) Given 𝑧⁰ and tol>0, set 𝑘=0

(2) Compute 𝐾, 𝑀, 𝐴, 𝑢, and 𝑏

(3) Compute (𝐹_𝑘)_𝑖=𝑏−𝑀𝑢if𝑘=1−𝑀𝑢if𝑘=2,…,𝑃.

(4) Solve 𝐾𝜇=𝐹

(5) Compute 𝐽^′𝑧^𝑘=𝑀(𝛼𝑧^𝑘−𝜇₁).

(6) Compute 𝐺₁=𝑀𝑣 and 𝐺_𝑡=0 for 𝑡=2,…,𝑃

(7) Solve 𝐾𝑤=𝐺=𝐺^𝑇₁,…,𝐺^𝑇_𝑃^𝑇

(8) Compute 𝐻_𝑘=𝑀𝑤_𝑘 for 𝑘=1,…,𝑃

(9) Solve 𝐾𝑞=𝐻=𝐻^𝑇₁,…,𝐻^𝑇_𝑃^𝑇

(10) Compute 𝐽^′′(𝑧)𝑣≈𝑀(𝛼𝑣−𝑞₁)

(11) If ∇𝐽̂𝑧^𝑘<tol, stop.

(12) Solve 𝐽^′′𝑧^𝑘𝑠^𝑘=−𝐽^′𝑧^𝑘 using PCG method with preconditioner 𝑃=𝐺⨂𝐾₀:

 (i) Set 𝑥₀=0.

 (ii) Set 𝑟₀←𝐽^′′𝑧^𝑘𝑥₀+𝐽^′𝑧^𝑘;

 (iii) Solve 𝑃𝑦₀=𝑟₀ for 𝑦₀;

 (iv) Set 𝑝₀=−𝑟₀, 𝑡←0;

 (v) While 𝑟_𝑡>tol

           𝛼_𝑘←𝑟^𝑇_𝑡𝑦_𝑡𝑝^𝑇_𝑡𝐴𝑝_𝑡

           𝑥_𝑡+1←𝑥_𝑡+𝛼_𝑡𝑝_𝑡

           𝑟_𝑡+1←𝑟_𝑡+𝛼_𝑡𝐴𝑝_𝑡

           𝑃𝑦_𝑡+1←𝑟_𝑡+1

          𝛽_𝑡+1←𝑟^𝑇_𝑡+1𝑦_𝑡+1𝑟^𝑇_𝑡𝑦_𝑡

          𝑝_𝑡+1←−𝑦_𝑡+1+𝛽_𝑡+1𝑝_𝑡

            𝑡←𝑡+1

    End (while)

 (vi) 𝑠^𝑘=𝑥_𝑡+1

(13) Perform Armijo line-search.

 (i) Set 𝛼^𝑘=1 and evaluate 𝐽(𝑧^𝑘+𝛼^𝑘𝑆^𝑘)

 (ii) While 𝐽𝑧^𝑘+𝛼^𝑘𝑆^𝑘>𝐽𝑧^𝑘+10⁻⁴𝛼^𝑘𝑆^𝑘∇𝐽̂𝑧^𝑘 do

           (i) Set 𝛼^𝑘=𝛼^𝑘/2 and evaluate 𝐽𝑧^𝑘+𝛼^𝑘𝑆^𝑘.

(14) Set 𝑧^𝑘+1=𝑧^𝑘+𝛼^𝑘𝑆^𝑘, 𝑘←𝑘+1. Go to (2).