Algorithm 2.

Newton step. We illustrate the solution of the reduced KKT system (4.1) using a PCG method at a given outer iteration k ∈ N. The steps to compute the Hessian matrix vector product are given in lines 4–8.

1:	${\eta }_k\leftarrow min(0.5,{({\Vert {\mathit{g}}_k^h\Vert }_2/{\Vert {\mathit{g}}_0^h\Vert }_2)}^{1/2})$
2:	${\stackrel{\sim }{\mathit{v}}}_0^h\leftarrow 0$, ${\mathit{r}}_0\leftarrow -g_k^h$, ${\mathit{z}}_0\leftarrow {({\mathcal{A}}^h)}^{-1}{\mathit{r}}_0$, s0 ← z0, l ← 0
3:	while l < n do
4:	${\stackrel{\sim }{m}}_l^h(t=0)\leftarrow 0$
5:	${\stackrel{\sim }{m}}_l^h\leftarrow \text{solve}$ (3.9a) forward in time given $m_k^h$, ${\mathit{v}}_k^h$ and ${\stackrel{\sim }{\mathit{v}}}_l^h$	⊲ inc. forward solve
6:	${\stackrel{\sim }{\lambda }}_l^h(t=1)\leftarrow -{\stackrel{\sim }{m}}_l^h(t=1)$	⊲ inc. adjoint solve
7:	${\stackrel{\sim }{\lambda }}_l^h\leftarrow \text{solve}$ (3.9c) backward in time given ${\lambda }_k^h$, ${\mathit{v}}_k^h$ and ${\stackrel{\sim }{\mathit{v}}}_l^h$
8:	${\stackrel{\sim }{\mathit{s}}}_l\leftarrow \text{apply}$ ${\mathscr{H}}_l^h$ to sl as indicated in (3.9e) given ${\lambda }_k^h$, ${\stackrel{\sim }{\lambda }}_l^h$, $m_k^h$ and ${\stackrel{\sim }{m}}_l^h$
9:	${\kappa }_l\leftarrow \langle {\mathit{r}}_l,z_l\rangle /\langle s_l,{\stackrel{\sim }{s}}_l\rangle$
10:	${\stackrel{\sim }{\mathit{v}}}_{l+1}^h\leftarrow {\stackrel{\sim }{\mathit{v}}}_l^h+{\kappa }_l{\mathit{s}}_l$
11:	${\mathit{r}}_{l+1}\leftarrow {\mathit{r}}_l-{\kappa }_l{\stackrel{\sim }{\mathit{s}}}_l$
12:	if ‖rl+1‖2 < ηk break
13:	${\mathit{z}}_{l+1}\leftarrow {({\mathcal{A}}^h)}^{-1}{\mathit{r}}_{l+1}$
14:	μl ← 〈zl+1, rl+1〉/〈zl, rl〉
15:	sl+1 ← zl+1 + μlsl
16:	l ← l + 1
17:	end while