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