accuracy.md

Accuracy

What error a given setting buys, measured. Every number here is a measurement on this machine at the stated configuration, so treat it as a scale rather than a constant, and re-measure on your own problem with pyimr.resolution when the answer matters.

Tolerance and the observable you fit

Observables do not converge at the same rate, so the right tolerance depends on which one you fit. Relative error against rtol=1e-11, coupled thermal:

rtol	radius	wall velocity	internal pressure	stress integral	temperature
1e-9	1.5e-10	2.1e-09	1.0e-08	2.2e-09	1.7e-09
1e-7	1.3e-08	2.5e-07	1.6e-06	2.8e-07	2.1e-07
1e-6	3.4e-07	4.4e-06	2.8e-05	4.8e-06	3.7e-06

Internal pressure is roughly 80x less accurate than radius at the same setting, so fitting it needs about two orders tighter for the same relative accuracy.

For likelihood evaluation against experimental radius data, rtol=1e-6, atol=1e-8 is ample: 3.4e-07 against measurement noise nearer 2e-02, and model selection on real data is unchanged from 1e-9 -- same winners, same chi-squared per sample. Keep 1e-10, 1e-12 for sensitivities, where derivatives amplify error, and 1e-9 or tighter when validating against reference trajectories.

Gradient accuracy

Tangent accuracy is not uniform across configurations, and the difference is large enough to matter for anything built on the gradients. Measured against centered differences on the production RHS:

configuration	relative error	limited by
mechanical, every operator	~7e-07	the finite-difference check
coupled heat/mass transfer, thermal = "spectral"	~5e-05	time integration
coupled heat/mass transfer, thermal = "fd"	~8e-05	time integration

The thermal tangents are correct, not defective: their error is flat in the finite-difference step across a 16x range, which rules out truncation in the check, and it responds to the integrator tolerance, which an error in the tangent equations would not. What bounds them is the accuracy to which the augmented state/tangent system is integrated. On the coupled fd case at h = 0.05:

rtol / atol	relative error
1e-9 / 1e-11 (default)	8.43e-05
1e-12 / 1e-14	1.53e-06

So three orders of tolerance buys a factor of about 55, at a large cost in runtime -- the coupled tangent solve is already the slowest operation in the package. Tightening further stops helping, because the centered-difference reference becomes round-off limited before the tangent does.

The mechanical tangents show clean h^2 convergence before reaching their noise floor; the thermal ones do not, because they are already at it.

Gradient-based optimizers and Laplace/EIG calculations on coupled thermal models are therefore working with about four to five significant digits, not the seven the mechanical path gives.

Cost is not bounded by tolerance

A parameter set whose bubble collapses to a fraction of a percent of its maximum and then creeps rather than rebounds takes hundreds of thousands of steps at any tolerance, while the healthy points of the same sweep finish in under 7e3. SimulationConfig.max_steps (default 1_000_000) turns that into a SimulationError at a point of your choosing, which is what makes a grid sweep affordable.