Algorithm 1 Offline phase

Require: database of ${\mathit{\theta }}_k$ for $k=1,\dots ,N_c$, truncations $K_{\varrho }^c$, number of snapshots $N_t$.

for $i\leftarrow 1,\dots ,N_t$ do

if $(\varrho =u)$ then

${\mathit{S}}_u^i\leftarrow \left[\begin{array}{c}\left\{\mathit{u}\right\}\left(\left\{\mathit{X}\right\},t^i,{\mathit{\theta }}_1\right),\dots ,\left\{\mathit{u}\right\}\left(\left\{\mathit{X}\right\},t^i,{\mathit{\theta }}_{N_c}\right)\end{array}\right]$; ${\mathcal{S}}_u\leftarrow [{\mathcal{S}}_u;{\mathit{S}}_u^i]$;

else

${\mathit{S}}_v^i\leftarrow \left[\begin{array}{c}\left\{\mathit{v}\right\}\left(\left\{\mathit{X}\right\},t^i,{\mathit{\theta }}_1\right),\dots ,\left\{\mathit{v}\right\}\left(\left\{\mathit{X}\right\},t^i,{\mathit{\theta }}_{N_c}\right)\end{array}\right]$; ${\mathcal{S}}_v\leftarrow [{\mathcal{S}}_v,{\mathit{S}}_v^i]$;

end if

end for

for $j\leftarrow 1,\dots ,N_c$ do

if $(\varrho =u$) then

${\mathit{T}}_u^j\leftarrow \left[\begin{array}{c}\left\{\mathit{u}\right\}\left(\left\{\mathit{X}\right\},t^1,{\mathit{\theta }}_j\right),\dots ,\left\{\mathit{u}\right\}\left(\left\{\mathit{X}\right\},t^{N_t},{\mathit{\theta }}_j\right)\end{array}\right]$; ${\mathcal{T}}_u\leftarrow [{\mathcal{T}}_u,{\mathit{T}}_u^j]$;

else

${\mathit{T}}_v^j\leftarrow \left[\begin{array}{c}\left\{\mathit{v}\right\}\left(\left\{\mathit{X}\right\},t^1,{\mathit{\theta }}_j\right),\dots ,\left\{\mathit{v}\right\}\left(\left\{\mathit{x}\right\},t^{N_t},{\mathit{\theta }}_j\right)\end{array}\right]$; ${\mathcal{T}}_v\leftarrow [{\mathcal{T}}_v,{\mathit{T}}_v^j]$;

end if

end for

${\mathsf{\Phi }}_{\varrho }\leftarrow$ SVD(${\mathcal{S}}_{\varrho }$), ${\mathrm{\Psi }}_{\varrho }\leftarrow$ SVD(${\mathcal{T}}_{\varrho }$), for $\varrho \leftarrow u,v$;

for $i=1,\dots ,N_t$ do

$A\left(t^i\right)\leftarrow {\left({\mathsf{\Phi }}_u^r\right)}^T\left\{\mathit{u}\right\}\left(\left\{\mathit{X}\right\},t^i,[{\mathit{\theta }}_1,\dots ,{\mathit{\theta }}_{N_c}]\right){\mathrm{\Psi }}_u^r$;

$B\left(t^i\right)\leftarrow {\left({\mathsf{\Phi }}_v^r\right)}^T\left\{\mathit{v}\right\}\left(\left\{\mathit{X}\right\},t^i,[{\mathit{\theta }}_1,\dots ,{\mathit{\theta }}_{N_c}]\right){\mathrm{\Psi }}_v^r$;

end for