almost periodic in $\kappa$. In general, computing (\[eq:discrete approx of Young measure, intro\]) scales linearly in $\varepsilon^{-1}$, but needs to be performed only once for a fine enough energy grid in order for interpolation to be accurate; this is analogous to the pre-computation of Planck-weighted or Rosseland-weighted opacities for use in multigroup transport codes, which also scales linearly in $\varepsilon^{-1}$.

Now define $\kappa_{ij}\left(T\right)=\chi\left(\sigma_{i,j},E_{i},T\right)$ and $$\Psi_{ij}\left(\mathbf{x},\boldsymbol{\Omega}\right)=\int_{E_{i}}^{E_{i+1}}\Psi\left(\mathbf{x},\boldsymbol{\Omega},E,\sigma_{i,j}\right)dE,\,\,\,\,\,\,\,\, S_{i}\left(E,T\right)=\int_{E_{i}}^{E_{i+1}}S\left(E,T\right)dE.$$ Then using the representation (\[eq:discrete approx of Young measure, intro\]) in (\[eq:transport equation on extended phase space, intro\]), we obtain the multigroup-type equations $$\begin{aligned}
\boldsymbol{\Omega}\cdot\nabla_{\mathbf{x}}\Psi_{ij}+\kappa_{ij}\left(T\left(\mathbf{x}\right)\right)\Psi_{ij} & = & \kappa_{ij}\left(T\left(\mathbf{x}\right)\right)S_{i}\left(x\right)+\nonumber \\
 &  & \int_{\mathbb{S}^{2}}\Sigma^{s}\left(\mathbf{x},E\rightarrow E',\boldsymbol{\Omega}\cdot\boldsymbol{\Omega}'\right)\Psi_{ij}d\boldsymbol{\Omega}',\label{eq:transport multiband-multigroup, intro}\end{aligned}$$ This corresponds to a standard multigroup-type approximation of (\[eq:transport equation on extended phase space, intro\]) at each band parameter $\xi=\sigma_{i,j}$. The homogenized solution at $I_{0}\left(\mathbf{x},\boldsymbol{\Omega},E\right)$ in each group interval $\left[E_{i},E_{i+1}\right]$ is then given by weighting against the discrete approximation (\[eq:discrete approx of Young measure, intro\]) of the Young measure $\lambda_{E}$, $$\psi_{0}\left(\mathbf{x},\boldsymbol{\Omega},E\right)\approx\sum_{j=1}^{m}p_{i,j}\Psi_{ij}\left(\mathbf{x},\boldsymbol{\Omega}\right),\,\,\,\, E\in\left[E_{i},E_{i+1}\right].\label{eq:formula for psi_o, discretized Young measures}$$ Here we used the discrete approximation (\[eq:discrete approx of Young measure, intro\]) of the Young measure in (\[eq:psi\_0 as weighting against Young measure, intro\]). We remark that the values of $p_{i,j}$ can be obtained via interpolation from a pre-computed table, as 