Astronomical Radiative Transfer and Video Games:

A Match Made in the Sun

Chris Osborne

University of Glasgow

Who recognises this?

Cornell University

And how does it connect to this?

VLT (ESO)

Physics!

\begin{align*} (\hat{\omega} \cdot \nabla) L(\vec{p}, \hat{\omega}) = &-\chi(\vec{p}, \hat{\omega}) L(\vec{p}, \hat{\omega}) \\ &+ \eta(\vec{p}, \hat{\omega}) \\ &+ \sigma_s(\vec{p}, \hat{\omega}) \oint_{\mathbb{S}^2} p(\vec{p}, \hat{\omega}^\prime, \hat{\omega}) L(\vec{p}, \hat{\omega}^\prime)\, d\hat{\omega}^\prime \end{align*}

This hybrid notation will probably make everyone mad!

L radiance,
\eta emission coefficient,
\chi absorption coefficient,
\sigma_s scattering coefficient,
p phase function.

Key takeaway: this is a recursive integro-differential equation!

Non-LTE Radiative Transfer

Coupling between radiation field and atomic transitions.
Extremely computationally demanding, but necessary for many spectral lines.
Also, ionisation.

HINODE BFI

Most Important Quantities

Radiative rates depend on intensity.
Intensity depends on radiative rates.

\begin{align*} J_\nu(\vec{p}) &= \frac{1}{4\pi}\oint_\mathbb{S^2} I_\nu(\vec{p},\hat{\omega}) \mathop{d \hat{\omega} } \\ &\vec{p} \in \mathbb{R}^2, \hat{\omega} \in \mathbb{S}^2\\ \end{align*}

Specific intensity and its first moment.
3 and 5 dimensional functions! (+ wavelength…)
Recursive component solved by fixed-point iteration in the atomic populations.

Monte Carlo

Ray Effects

Desired Result

Tricky Radiation

In the steady-state solution of radiative transfer coupling is global.
This hinders parallelisation (and SC imposes its own additional dependencies even within a patch).
Monte Carlo approaches have poor convergence rates.
Spectroscopy has very high anisotropy requirements.
Is there some middle ground?

Video games are now solving an adjacent problem at reasonable resolutions hundreds of times a second on single workstations…

Radiance Cascades

Radiance Cascades: Observations

Linear light source and blocker:

\begin{cases} A < B,\\ \alpha > \beta. \end{cases}

with some small angles approximations…

\begin{cases} \Delta_s < F(D) \propto D\quad&\mathrm{(spatial)}\\ \Delta_\omega < G(1/D) \propto 1/D\quad&\mathrm{(angular)} \end{cases}

We can exploit this!
Represent contributions from different shells separately.

How do we use this?

Encode radiance field as a set of cascades, each describing I_\nu in annuli around \vec{p}.
If penumbra criterion is satisfied, cascade is linearly interpolateable!
- Serves as Nyquist criterion
The cascades sparsely encode the radiation field throughout the domain…
But not in a form we can use directly.
Solution:

Interpolation. But with minimal error.

Interpolation leads to reuse of rays in upper cascades and the effective construction of exponentially more rays than computed directly.

Further Optimisations

GPU-first design (has achieved 40\times perf/Watt vs CPU).
Use a performance-portability library – no ifs, no buts – especially for small teams.
- We use Kokkos with some additional layers.
- Python pre/postprocessing.
First solar non-LTE code to exploit atmospheric sparsity.

Mipmaps & Ray Acceleration

Lower resolution proxies for distant rays: mipmaps (recursive spatial averaging) with directionality and adaptive error bounds (variance of log emissivity and opacity)

In practice, upper cascades (longer rays) sample upper MIPs (fewer texels) when variance is low enough.
Store data in Morton order as much as possible.
Inspired by OpenVDB.

Outlook

Performance Portability libraries are a force multiplier for small teams.
Radiance cascades provide the first meaningful change to solar non-LTE RT in 30 years and it came from outside the field.
There are equivalent/adjacent problems in other fields, and it’s worth our time to seek them out.
- e.g. radiance cascades can be repurposed for cosmological self-gravity

Thanks!

Christopher.Osborne@glasgow.ac.uk

Paper