Algorithm 2.

Hessian matrix-vector product of the designed inexact Newton–Krylov algorithm at outer iteration k ∈ N. We illustrate the computational steps required for applying ℋ^h in (4.3d) to the PCG search direction at inner iteration index ι ∈ N.

1:	${\stackrel{\sim }{m}}_{\iota }^h(t=0)\leftarrow 0$
2:	${\stackrel{\sim }{m}}_{\iota }^h\leftarrow \text{solve}$ (4.3a) forward in time given $m_k^h,{\mathit{v}}_k^h$, and ${\stackrel{\sim }{\mathit{v}}}_{\iota }^h$	▷ incremental forward solve
3:	${\stackrel{\sim }{\lambda }}_{\iota }^h(t=1)\leftarrow -{\stackrel{\sim }{m}}_{\iota }^h(t=1)$
4:	${\stackrel{\sim }{\lambda }}_{\iota }^h\leftarrow \text{solve}$ (4.3b) backward in time given ${\lambda }_k^h,{\mathit{v}}_k^h$, and ${\stackrel{\sim }{\mathit{v}}}_{\iota }^h$	▷ incremental adjoint solve
5:	apply ${\mathcal{H}}_{\iota }^h$ in (4.3d) to the PCG search direction given ${\lambda }_k^h,{\stackrel{\sim }{\lambda }}_{\iota }^h,m_k^h$, and ${\stackrel{\sim }{m}}_{\iota }^h$