Algorithm 1. Multiscale Hybrid TP Approximation

1) $U^{*}\left(0\right)=Q^{*}\left(u\left(0\right)\right)$

2) Run the coarse propagator ${\mathcal{C}}_{\left[T\mathrm{,0}\right]}{U}^{\mathrm{<mo>\; *\; </mo>}}\left(0\right)$

3) Derive a spline representation of the coarse solution (in terms of $\theta$ ), $S\left(\theta \right)\approx C_{\left[\theta \mathrm{,0}\right]}\left(U^{\mathrm{<mo>\; *\; </mo>}}\left(0\right)\right)$

4) Compute terms of the TP Sum, Equation (46), at $\theta =g\frac{\text{π}}{4}+\frac{H}{\text{π}}h$ for $g\in \left\{\mathrm{0,1,2,3,4}\right\}$ and $h\in \left\{\mathrm{0,1,2}\right\}$ .

5) Approximate the TP Sum using Equation (48).