/Physical-Model

Primary LanguagePython

Introduction

The script is trying to solve Bloch equations as described in the pdf attached.

Equations:

The objective is maximize function $J$:

$$ \begin{equation} \tag{3} \displaystyle J = \frac{1}{N} \sum_j Y_{n+1}^T U_n\left(\omega_j, \theta_n\right)\cdots U_k\left(\omega_j, \theta_k\right)\cdots U_1\left(\omega_j, \theta_1\right)X_0. \end{equation} $$

Where $U_k\left(\omega, \theta_k\right)$ is calculated from :

$$ \begin{equation} \tag{2} \displaystyle U_k\left(\omega, \theta_k\right) = exp\left(\partial t\left(\omega\Omega_z+cos\left(\theta\right) \Omega_x + sin\left(\theta\right)\Omega_y\right)\right). \end{equation} $$

$X,Y$ is caluclated from equation 4:

$$ \begin{equation} \tag{4} \displaystyle Y_{k+1}^T = Y_{n+1}^T U_n\left(\omega_j, \theta_n\right) \cdots U_{k+1}\left(\omega_j, \theta_{k+1}\right), \end{equation} $$

$$ \begin{equation} \tag{4} \displaystyle X_{k-1}^T = U_n \left(\omega_j, \theta_n\right)\cdots U_{k+1}\left(\omega_j, \theta_k+1\right) X_0. \end{equation} $$

Then, we can calculate $e$ from equation 8:

$$ \begin{equation} \tag{8} \displaystyle e = \sum_j X_{k-1} \times Y_{k+1}, \end{equation} $$

where $X$ refers to $cross\ product$.

Updating $\theta$ from equ:

$$ \displaystyle tan\left(\theta\right) := \frac{e_2}{e_1}, $$

as:

$$e = \begin{pmatrix}e_1 \cr e_2 \cr e_3 \cr \end{pmatrix}.$$

Alogrithms

The optimization algorithm is as follows:

  1. Assume $θ_i = 0$ (from $\theta_1$ to $\theta_n$).
  2. We calculate initial $J$ from $equation (3)$.
  3. Calculate all $Y_k$ and $X_k$ as defined in $equation (4)$.
  4. Using $equation (8)$ we calculate $e$ which should be a three dimensions vector.
  5. Calculate a new set of $\theta_k$ using $tan(\theta_k) = \frac{e_2}{e_1}$.
  6. Do this for all $\theta$.
  7. We recalculate new $J$ from $equation (3)$.
  8. Repeat steps from 3 to 7.
  9. We need to get J to reach 1. Only then we can stop the iteration.

This method is implemented in git branch approach A
The method was slow and iteration time is over 50 seconds for trial.
Another approach was tried in the Master branch

  1. Assume $θ_i = 0$ (from $θ_1$ to $θ_n$).
  2. Calculate a tensor with (N,n,3,3) dimensions where N: number of random $\omega$ and n is the number of selected $\theta$.
  3. Calculate Y, X, and e from the tensor slicing method.
  4. Updating $\theta$ and recalculate. This approach reduced the time significantly to less than 3 seconds per trial.