GRG — Wedge Visualization

GRG — Wedge Visualization

Sep 21, 2025

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 certified corridor half-widths.

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