Pade approximant based spectral extrapolation

[Arjun-Khurana]

After too much delay, this is an initial PR for Pade-approximant based spectral extrapolation, hopefully closing #2192. It essentially works in the same way as Harminv (thanks @Alex-Kaylor for help with the initial implementation) by aggregating field and computing an analytic frequency response using the rational Pade Approximant.

Like Harminv, Pade is constructed as a class that holds as a member variable (data) the aggregated field data, parameters for sampling the data, and the computed analytic function.

To initialize, one need only provide a field component c and a point pt at which to perform the computation.

Optional arguments include sample_rate, start_time and stop_time, which sample the collected field data according to samples = self.data[self.start_time : self.stop_time : self.sample_rate].

Additionally, there are m and n, which respectively set the orders of the numerator and denominator. Alternatively, one can specify m_frac or n_frac, which specify m and n as fractions of the length of the collected time series. If none of these are specified, the default behavior is to use half the length of the time series as the order of the numerator m, and N-m-2 as the order of the denominator.

Speaking of the algorithm, the code is currently implemented by calling the builtin scipy pade routine. Separately, I have investigated the behavior of the Trefethen SVD-based algorithm and found it to be much harder to get it to converge (though in theory it is more robust to noise in the input data). Thus, the existing implementation only uses the builtin scipy algorithm, but it would be nice to at least have the option of using either algorithm. The way I have implemented the Trefethen algorithm here provides straightforward potential interchange of Trefethen and scipy.

I have also written a test case that compares the center frequency of a peak in the Pade frequency response to the first Harminv mode, included in test_ring.py. This case passes when I run it locally with python3 -m pytest python/tests/test_ring.py but fails on my fork's CI workflow, I'm sure I'm doing something incorrectly there.

[Arjun-Khurana]

Things to do before I would consider this PR "complete":

• Understand and correctly implement the Trefethen algorithm as an option
• Create a stop_when_pade_converged style step function that computes the Pade frequency response every some number of time steps and stops when it converges within some tolerance

Suggestions and improvements are welcome.

[smartalecH]

I would punt on the SVD approach. You can always add more PRs with additional functionality later.

For now, focus on the simplest working routine that's easy to augment later.

[smartalecH]

This is why the CI dies (you can download the log as an artifact):

ERROR: test_pade (__main__.TestRing)
----------------------------------------------------------------------
Traceback (most recent call last):
  File "/home/runner/work/meep/meep/build/meep-1.27.0-beta/_build/sub/python/../../../python/tests/test_ring.py", line 77, in test_pade
    self.init()
  File "/home/runner/work/meep/meep/build/meep-1.27.0-beta/_build/sub/python/../../../python/tests/test_ring.py", line 49, in init
    self.p = mp.Pade(mp.Ez, mp.Vector3(r + 0.1), sample_rate=4)
AttributeError: module 'meep' has no attribute 'Pade'

Be sure to include your new module/function in meep.i (The swig file):

meep/python/meep.i

Line 1718 in c376694

from .simulation import (

[oskooi]

Good work. We should add a tutorial for this feature after this is merged.

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[oskooi]

Getting there. Some more suggested changes.

[oskooi]

You will also need to create a new entry for the Pade class in doc/docs/Python_User_Interface.md.in (similar to Harminv):

meep/doc/docs/Python_User_Interface.md.in

Lines 651 to 655 in c376694

	#### Harminv Step Function
	The following step function collects field data from a given point and runs [Harminv](https://github.com/NanoComp/harminv) on that data to extract the frequencies, decay rates, and other information.
	* [Harminv class](#Harminv)

This is necessary for converting its docstrings into Markdown for the readthedocs manual.

[stevengj]

In a separate PR, it would be good to show how to use this to calculate transmission and reflection spectra in problems with single-mode ports and sharp resonances. For example, to compute the transmission/reflection spectrum through N layers of a 2d photonic crystal at normal incidence.

[oskooi]

Just two final comments/suggestions.

[oskooi]

LGTM.