cschwan
commented
1 year ago Yesterday I realized that this problem is actually one of linear (in)dependence and therefore one can frame this as linear algebra problem.
Imagine we have a grid with only one bin and one order, only consisting of $n$ subgrids $g_j$, where $j \in { 1, 2, \ldots, n }$ denotes the channel (index). We then usually have $m$ (non-zero) PDF combinations labelled with $i \in { 1, 2, \ldots, m }$. To give an example of the PDF combinations we have $b1 = f\mathrm{u} (x1, Q^2) f{\bar{\mathrm{u}}} (x_2, Q^2)$ and $b2 = f\mathrm{c} (x1, Q^2) f{\bar{\mathrm{c}}} (x_2, Q^2)$, for instance. Note that usually we have $m > n$, because we can often merge PDF combinations together when they have similar matrix elements.
A convolution then can be written as
$$ \int_0^1 \mathrm{d} x_1 \int_0^1 \mathrm{d} x2 \int{Q\mathrm{min}^2}^{Q\mathrm{max}^2} \mathrm{d} Q^2 G (x_1, x_2, Q^2) $$
where
$$ G (x_1, x2, Q^2) = \sum{i=1}^m \sum_{j=1}^n bi A{ij} g_j $$
is the important bit. The objects $b = (b_1, b_2, \ldots, b_m)$ and $g = (g_1, g_2, \ldots, gn)$ are vectors, i.e. elements of a vector space. $(A{ij})_{i,j=1}^{m,n}$ is a real and in general non-square $m \times n$ matrix. We can then write the important bit as
$$ G (x_1, x_2, Q^2) = b^T A g $$
In essense, this just transforms the (usually) flavour basis contained expressed by $b$ to a different basis, which is a linear combination of the former.
We can now reformulate the problem in terms of linear algebra: find a decomposition $A = L R$ such that the matrix $R$ has the smallest number of non-zero rows. This will reduce the number subgrids.
I believe this can be done by using a (rank-revealing?) LU decomposition with partial pivoting. This would decompose $A = P^T L U$ so that we have
$$ G = b^T P^T L U g = b^T (P^T L) (U g) = (b')^T A' g'$$
where $g' = U g$ are the new subgrids, which are a linear combination of the old subgrids $g$, and the new lumi definition $A' = P^T L$.
Hawaiian Vrap produces luminosities that
Here's an example for the first bin of
DYE866R_P, all other bins have the same tuples, but with different factors:One strategy to make the luminosity function more readable and to optimize it is to
l = 0into four entries)