Method-C applied convergence demo¶
A docs-side mirror of the within-stratum optimisation result that
already lives in
tests/benchmarks/test_optimization_trajectory.py.
The numbers and plot on this page are the bundled benchmark output —
see
benchmarks/method_c_demo/README.md
for the regeneration command.
What this page does not claim¶
- This is not a new result. The within-stratum precision gap between Method C and Method A is documented in ADR-0009 / ADR-0011 and exercised by the test cited above. This page just surfaces it alongside the rest of the documentation.
- No claim that Method A is "wrong". The residual it leaves is a
documented bias property of the sigmoid-temperature smoothed
objective at finite
beta.
The report¶
Method-C applied convergence demo¶
Two disks, C1 = (0, 0), R1 optimised toward the analytical ground truth r1* = 1.5, C2 = (1, 0), R2 = 1. The objective is loss(r1) = (union_area(a, b) - target)^2 with the target computed in closed form by brepax.analytical.disk_disk.disk_disk_union_area. The same gradient-descent loop runs twice, once with method="stratum" (Method C) and once with method="smoothing", k=0.1, beta=0.1, resolution=128 (Method A). The numbers below are the output of one command.
Settings: lr=0.01, max_steps=200, init_r1=0.8, target_area=7.880057 (closed form).
Final-step numbers¶
| Method | final r1 | pos err | final loss | steps |
|---|---|---|---|---|
| Method C (stratum) | 1.501310 | 1.31e-03 | 3.62e-06 | 200 |
| Method A (smoothing, beta=0.1) | 1.478751 | 2.12e-02 | 1.75e-15 | 200 |
Method C converges to the grid-discretisation floor of its stratum-aware integrator. Method A's residual is the bias introduced by the sigmoid temperature beta; it does not shrink with more gradient-descent steps because the bias is inherent to the smoothed objective, not a transient of the optimiser. This is the documented behaviour from tests/benchmarks/test_optimization_trajectory.py; the purpose of this page is to make it visible alongside the rest of the documentation rather than to claim it as a new result.
Convergence plot¶
Regenerate with uv run python -m benchmarks.method_c_demo.run.