Numerical Optimal Control for Problems with Random Forced SPDE Constraints

Figure 2

Illustration of optimal control and state achieved by solving Example 2, taking 𝑢(𝑥,𝑦)=2𝜋²sin(𝜋𝑥)sin(𝜋𝑦), 𝑁=3, 𝐾=4, and 𝛼=0.00001.

(a) Desired values

(b) Mean values

(c) Approximated solution (State)

(d) Control

(e) Adjoint

(f) Expected value-SGS

(g) Standard dev.-SGS

(h) Sparsity of global matrix 𝑛𝑧=3046160