Scientific correction — 13 October 2026

The application intervals and charts on this historical page are proposals or illustrations, not certified bounds for arbitrary fields or validated performance results. Certification requires assumptions on each integrand, a proved quadrature residual, rounding control, and propagation through subsequent transformations. Regular circular-sector bounds do not automatically bound optical, chemical, or anisotropic quantities.

Wedge-Stratified GRG — Visualizing Per-Sector Results

This section visualizes the anisotropic, wedge-stratified example discussed earlier. We compare a uniform W=8 discretization against an adaptive scheme on the outer annulus \rho\in[0.7,1), where angular content is richer. The plots show per-sector estimates with illustrative corridor half-widths (not certified error bounds).

Setup Recap

Field with mixed angular orders up to fourth order:

$$ F(\rho,\theta)=A(\rho)\,\Big[1+\epsilon_2\cos 2(\theta-\theta_2)+\epsilon_3\cos 3(\theta-\theta_3)+\epsilon_4\cos 4(\theta-\theta_4)\Big], $$

Rings and wedges:

$$ \text{Rings }k=1..K,\ \text{wedges }w=1..W_k,\quad I_{k,w}=\int_{\Theta_w}\!\int_{u_{k-1}}^{u_k} F(\rho,\theta)\,\rho\,d\rho\,d\theta. $$

Rings: K=32 (dyadic edges, quarter sampling).
Uniform: W=8 for all rings.
Adaptive: inner W_k=8, mid W_k=12, outer W_k=16 (used for plots below).

Per-Sector Results on the Outer Annulus

The numbers are the same illustrative values used in the write-up; replace them with your measured outputs when available. Error bars display corridor half-widths.

Left: uniform W=8. Right: adaptive outer ring with W_k=16. Hover/tap a bar to read exact values.

Polar View (Directional Pattern)

A simple polar plot places each sector’s estimate radially. Shaded rings visualize the corridor range.

Polar plot of per-sector estimates. Left: W=8. Right: W_k=16.

Reading the Plots

Corridor effect: with higher angular resolution, corridor half-widths shrink (≈ 1/\sqrt{W_k} for comparable ring budgets), revealing higher-order structure.
Decision making: choose the smallest dyadic level b such that the corridor width is below the task tolerance \tau.

$$ \textbf{Stop when}\ \ \widehat{E}^{+}_{b}-\widehat{E}^{-}_{b}\le\tau. $$