In order for LIGO to detect gravitational waves and do astrophysics, it
is crucial to have an accurate model of the gravitational waves.
Numerical relativity (NR) is the only method that accurately models the
merger part of a binary black hole signal. However, these
simulations are prohibitively expensive for most direct applications.
Therefore, several approximate waveform models have been developed over
the years. The two main approaches have been dubbed "Phenomenological"
and "Effective-one-body" waveforms. These models typically make some
assumptions about the phenomenology of the waveforms, based on good
physical motivations. Then, any remaining free parameters are set by
fitting against NR simulations. These models are also quite fast and
can be directly used for GW applications.
Surrogate modeling is a data driven approach to waveform modeling that
has come about in the last few years. Here, we do not make any
assumptions about the underlying phenomenology but rather use the NR
waveforms themselves to implicitly reconstruct the phenomenology. This
is achieved by first building an accurate basis using the NR waveforms
themselves and then using some fancy interpolation methods to construct
a waveform model.
This website shows a video demonstration
of the surrogate modelling procedure. We construct a surrogate model
for nonspinning binary black holes. The total mass scales out of the
system and the only free parameter is the mass ratio (q). We use the
SEOBNRv4 waveform model for this demonstration, but the same can be
done with NR waveforms. These movies can be found
here. Note that this is a very high-level
description of the process, where I skip most of the details. Refer to
1308.3565 for the technical details. Some recent surrogate models are described in 1812.07865 and 1905.09300.
Surrogate modeling demo
Let's break it down
To build a surrogate model, we begin with
a dataset of waveforms. In this case, this is nonspinning waveforms
with q between 1 and 10. The first step is to construct an accurate
basis that represents our space of waveforms. We do this using the very
waveforms we are trying to model. The basis functions are picked in an
iterative manner where in each step the waveform that has the highest
projection error onto the current basis, gets added to the basis for
the next iteration. We proceed until the projection error goes below a
chosen threshold. In this manner we pick out the most representative
waveforms to form our basis, and reduce the dataset in the q direction
to a small set of basis functions.
In the movie below, we show what the
selected waveforms might look like. We first show the real part of the
(2,2) mode, which is highly oscillatory. It is much easier to build a
model for a slowly varying function, therefore we instead work with the
amplitude and phase of the (2,2) mode. At the end of the movie we show
the amplitudes of the selected waveforms, which are much simpler. In
this example, we will construct a surrogate model for the amplitude.
Having constructed our basis, the next
step is to reduce the data in the time direction. This is done using
the empirical interpolation method, which iteratively picks out the
time values that are most representative. Only these time values,
called the empirical time nodes, are used in constructing an
empirical interpolant in time. This is shown in the movie below.
At this point we have condensed our dataset
in both q and time directions. Given our basis functions, all we
need to evaluate the waveform is the basis coefficients so that
we can project the basis functions. Thanks to the empirical
interpolant method, these coefficients only need to be evaluated
at the empirical time nodes. However, what if we want to evaluate
the waveform at a random point in the parameter space where we
don't have a basis function?
To do this, we first construct fits
across parameter space, at each of the empirical time nodes, as
Finally, to evaluate the waveform at
a random parameter space point: We first evaluate the fits at that
point. This gives usthe basis coefficients, which we use to
project the basis functions to get the amplitude evaluation.
Downloading these movies
You can download these movies by right-clicking on
them. On Chrome choose "Save Video As..", and similar for other browsers.
You can also get them directly from my Github repo as follows. Don't forget to credit me.
git clone firstname.lastname@example.org:vijayvarma392/SurrogateMovie.git
List of visualizations
waves.mp4: Reduced basis construction.
ei.mp4: Empirical interpolation.
fits.mp4: Fits across parameter space.
eval.mp4: Evalution of surrogate.
full_movie.mp4: The whole shebang.
full_movie.gif: gif version of above file. Lower quality and
higher file size, so use the mp4 version unless you need a
Each of the above file names, except full_movie.mp4 also has a
companion .png file that has the first frame of the video. Also,
waveamp.png shows the last frame of waves.mp4.
These movies were made by me,
Please credit me, and cite this website, if you use these visualizations in
your work, presentations or outreach.