IRPJAST
CHAMP's Jastrow factor computation using the IRPF90 method
Original equation:
$$ \sum_{i=2}^{Ne} \sum_{j=1}^i \sum_{pkl} \sum_a^{Nn} c_{apkl}\, r_{ij}^k\, ( R_{ia}^l + R_{ja}^l) ( R_{ia} R_{ja})^m $$
Expanding, one obtains:
$$ \sum_{i=2}^{Ne} \sum_{j=1}^i \sum_{pkl} \sum_a^{Nn} c_{apkl} R_{ia}^{p-k-l}\, r_{ij}^k\, R_{ja}^{p-k+l} + c_{apkl} R_{ia}^{p-k+l}\, r_{ij}^k\, R_{ja}^{p-k-l} $$
The equation is symmetric if we exchange $i$ and $j$, so we can rewrite
$$ \sum_{i=1}^{Ne} \sum_{j=1}^{Ne} \sum_{pkl} \sum_a^{Nn} c_{apkl} R_{ia}^{p-k-l}\, r_{ij}^k\, R_{ja}^{p-k+l} $$
If we reshape $R_{ja}^p$ as a matrix $R_{j,al}$ of size $N_e \times N_n(N_c+1)$, for every $k$, we can pre-compute the matrix product
$$ C_{i,al}^{k} = \sum_j r_{ij}^k\, R_{i,al} $$ which can be computed efficiently with BLAS. We can express the total Jastrow as:
$$ \sum_{i=1}^{Ne} \sum_{pkl} \sum_a^{Nn} c_{apkl} R_{ia}^{p-k-l}\, C_{i,a(p-k+l)}^k $$
Rank reduction
Idea
The idea is to use SVD and fast-updates to reduce the rank of the intermediate \(r^k_{ij} = \Gamma_{i,j,k}\) like so.
\[ \Gamma_{i,j,k} = U_{i,d,k} \cdot D_{d,d,k} \cdot V^T_{d,j,k} \]
Where \( D_{d,d,k}\) is a diagonal in indices \(i,j\) and is of rank \(d \ll Min(i,j)\).
Auto-generate derivatives
Idea
The calculation of first and second derivatives of a network of tensors contracted using BLAS can also be expressed as a sum of BLAS contractions. However, there are many pitfalls. Here we shall consider an automated generation of contraction scheme to optimize the order in which the BLAS contraction is performed.