Add coil limits to the inverse solver

[timothy-nunn]

Adds coil limits to the inverse solver and cleans up some of the delta_current calculations.

Both of these changes have been tested on a MAST-U example.

Change delta_current calculation

Changed the calculation of delta_current from

freegsnke/freegsnke/inverse.py

Lines 409 to 410 in d8e17ed

	mat = np.linalg.inv(np.matmul(self.A.T, self.A) + reg_matrix)
	delta_current = np.dot(mat, np.dot(self.A.T, self.b))

to use np.linalg.solve. This ensures we get the most accurate solution and is possibly more numerically stable.

There is negligible difference caused by this change for the MAST-U example.

Coil limits

Adds support for specifying coil limits. When limits are specified, a convex solver is used to solve the least-squares problem; this is because you can specify additional constraints on the convex solver, such as box constraints for the coil limits. These additional constraints have an added slack to their bound which allows the coil limits to be violated for a penalty.

The problem that is being solved is slightly different due to the coil limits and slack.

$$ \min_{x} | A\vec{x} - \vec{b}|^2 + | \gamma \vec{x} |^2 + \mu| \vec{S}_{\text{upper}} |^2 + \mu| \vec{S}_{\text{lower}} |^2 $$

subject to

$$ \vec{x} + \vec{I}^c \leq \vec{I}^c_{\text{max}} + \vec{S}_{\text{upper}} $$

$$ \vec{x} + \vec{I}^c \geq\vec{I}^c_{\text{min}} - \vec{S}_{\text{lower}} $$

where

• $\vec{S}$ is the slack for the upper/lower limit. A slack of $0$ implies that the limit is satisfied. Slack must be positive.
• $\vec{I}^c$ is the current coil currents
• $\vec{x}$ is the step to be taken in the coil currents
• $\vec{I}^c_{\text{max}}$ and $\vec{I}^c_{\text{min}}$ are the maximum and minimum coil currents, respectively.
• $\mu$ scales the penalty in the objective function from a non-zero slack (coil limit violation).

As can be seen, using the convex solver with no coil limits yields near-identical results to above:

When we add limits, the solution changes because the inverse solver attempts to ensure the coil currents are always within limits:

[kpentland]

Hi Tim,

This looks awesome at first glance! I'm just going to stress test it a bit on some real equilibria then will add some more comments.

I've updated some of the text in the Example01 notebook and have added a plot that shows the coil currents within their limits.

[theo-brown]

Please can the seed be exposed to the user as an optional input?

[kpentland]

Done!

[theo-brown]

Nice, thanks guys!