I am trying to implement the algorithm contained in this article here. It is about solving a 2 and 2.5D Fredholm integral, focused on bidimensional NMR experiments. I've made significant progress, despite this being my very first time doing any computational science. The output I get is clearly wrong, but seems shifted from what would be ideal. I am trying to stamp out problematic areas, and am wondering if I interpreted this passage wrong:
In principle, the 2-D problem in (2) can be reduced to a 1-D problem $m_r = K_0f_r + \epsilon_r \quad r=1,...,R$. Here, the vectors $m_r$, $f_r$ and $e_r$ are obtained by lexicographically ordering matrices $M_r$, $F_r$, $E_r$.
For reference, eq (2) that they mention is the Fredholm integral we are trying to solve, which relates the measured data $M_r$ and the joint probability distribution $\mathcal{F}_r$ of the relaxation times (x, y):
$$M_r(\tau_1,\tau_2) = \int \int k_1(x, \tau_1)k_2(y, \tau_2)\mathcal{F}_r(x,y)dxdy + \epsilon_r(\tau_1,\tau_2) \quad r=1,...,R$$
According to the text, the shape of $M_r$ is $N_1\times N_2$ and $m_r$ is $(N_1N_2\times 1)$. I thought I could use a simple reshape, but looking at definitions of lexicographic, there seems to be a sort involved. So, which one of these two python snippets is right? Or is it something else entirely?
m_r = np.sort(M_r, axis=None)[:, np.newaxis]orm_r = M_r.reshape((N1*N2, 1))?
Later, they also state that
The matrix $F_r$ is estimated by reordering $f_r$ into the matrix notation.
So, I could do a simple reshape back. But if I have to unsort $F_r$, which has the shape $N_x \times N_y$, I am lost.
np.sorttrynp.lexsort. $\endgroup$ – Alone Programmer Mar 20 '20 at 20:38a = np.array([[1,2,3],[4,5,6]]).np.lexsort(a)returns[1,2,3], which has the wrong shape in my example. And besides, how would I unsort $F_r$ if I usedlexsortonM_r? $\endgroup$ – K.Cl Mar 20 '20 at 20:47