irpjast/README.org

* 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 \(d \ll Min(i,j)\).

Algorithms for fast rank-1 updates of a r-thin SVD has been published in the following papers:

1. https://doi.org/10.1016/j.laa.2005.07.021

2. https://doi.org/10.1002/pamm.200810827

** 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.