Algorithm 1.

Outer iteration of the designed inexact Newton–Krylov method.

1:	${\mathit{v}}_0^h\leftarrow 0$ compute $m_0^h,{\lambda }_0^h,{\mathcal{J}}^h({\mathit{v}}_0^h)$, and ${\mathit{g}}_0^h$; k ← 0
2:	while true do
3:	stop ← (4.9)
4:	if stop break
5:	sk ← solve (4.4) given $m_k^h,{\lambda }_k^h,{\mathit{v}}_k^h$, and ${\mathit{g}}_k^h$	▷ Newton step
6:	αk ← perform line search on sk
7:	${\mathit{v}}_{k+1}^h\leftarrow {\mathit{v}}_k^h+{\alpha }_k{\mathit{s}}_k$,
8:	$m_{k+1}^h(t=0)\leftarrow m_T^h$
9:	$m_{k+1}^h\leftarrow \text{solve}$ (4.1a) forward in time given ${\mathit{v}}_{k+1}^h$	▷ forward solve
10:	${\lambda }_{k+1}^h(t=1)\leftarrow (m_R^h-m_{k+1}^h(t=1))$
11:	${\lambda }_{k+1}^h\leftarrow \text{solve}$ (4.1b) backward in time given ${\mathit{v}}_{k+1}^h$ and $m_{k+1}^h$	▷ adjoint solve
12:	compute ${\mathcal{J}}^h({\mathit{v}}_{k+1}^h)$ and ${\mathit{g}}_{k+1}^h$ given $m_{k+1}^h,{\lambda }_{k+1}^h$, and ${\mathit{v}}_{k+1}^h$
13:	k ← k + 1
14:	end while