Residual Critiques v2

Residual Critiques v2

Sep 20, 2025

GRG — Residual Critiques: Demo Status, Benchmarks-in-Progress, Complex Anisotropy, and Cost Comparison

This page acknowledges what is still pending (public demo and published benchmarks) and adds a more demanding wedge-stratified example (freeform-like anisotropy). We also outline when GRG is preferable versus angular bases (e.g., Zernike) as W grows.

1) Mini-App Demo & Screencast — Current Status

  • Badge: Demo forthcoming — the public link and 2–3 min screencast will appear in the site header Community → Mini-App and in this page’s hero as a primary CTA.
  • Presets: Optics (MTF/PSF), Chemistry (RDF). Each preset provides CSV with estimate and certified corridors.
  • Transparency: until the link is live, this page keeps the badge above the fold, so the gap is explicit.

2) Benchmarks — What Will Be Published

Side-by-side, corridor-aware comparisons vs Zernike, polar B-splines, Fourier–Bessel, and pixel discretizations. Datasets and scripts are public; winner is declared only when corridor half-width is < task tolerance \tau and intervals do not overlap.

$$ \text{Report}\quad \|T_{\text{GRG}} - T_{\text{ref}}\|_2,\ \text{runtime},\ \text{memory}\quad\text{vs}\quad b,\ N=2^b,\ \text{problem size}. $$

3) Complex Anisotropy — Freeform-Like Example (Concrete K, Adaptive W)

Scenario. A weakly freeform optical surface induces an intensity field with mixed angular content up to fourth order. We integrate over dyadic rings and adaptive wedges to capture anisotropy where gradients are steep.

$$ F(\rho,\theta) \;=\; A(\rho)\,\bigl[\,1+\epsilon_2\cos 2(\theta-\theta_2)+\epsilon_3\cos 3(\theta-\theta_3)+\epsilon_4\cos 4(\theta-\theta_4)\,\bigr] $$
  • Rings: K=32 (dyadic edges; quarter sampling uses \Delta u=2^{-b-2} with b=5).
  • Adaptive wedges: inner rings (\rho<0.4) use W_k=8; mid (0.4\le \rho<0.7) use W_k=12; outer (\rho\ge 0.7) use W_k=16.
  • Parameters: A(\rho)=\rho^{0.6}, \epsilon_2=0.14, \epsilon_3=0.07, \epsilon_4=0.05, phases \theta_2=15^\circ, \theta_3=50^\circ, \theta_4=100^\circ.

Per-sector results (outer annulus, \rho\in[0.7,1))

Values below are illustrative and show how corridors tighten with higher W_k where angular content is richer.

Uniform W = 8

Sector wEstimateCorridor ±
10.121±0.0070
20.128±0.0072
30.136±0.0074
40.142±0.0075
50.141±0.0075
60.134±0.0073
70.126±0.0071
80.121±0.0070

Adaptive Wk (8→12→16)

Sector wEstimateCorridor ±
10.121±0.0046
20.129±0.0048
30.138±0.0049
40.144±0.0050
50.143±0.0050
60.135±0.0048
70.127±0.0047
80.121±0.0046

Takeaway. Adaptive wedges localize work where anisotropy is strongest; corridor half-widths fall ≈ proportional to 1/\sqrt{W_k} for comparable ring budgets, exposing higher-order angular content without oversampling isotropic regions.

4) Cost for Large W — When GRG Beats/Matches Angular Bases

With K rings and W wedges, work and memory scale as:

$$ \text{GRG (wedge)}:\ \ \text{work}\sim O(K\cdot W),\quad \text{memory}\sim O(K\cdot W). $$

Rule of thumb. If anisotropy is low (energy mostly in orders ≤ 2), a short Zernike truncation can be competitive. As angular richness grows or is localized, GRG with adaptive W_k becomes preferable—no global orthogonalization, easy corridor control, and targeted refinement.

Scenario Angular content Suggested method Rationale
Near-radial optics Dominant 2-fold Zernike (low order) or GRG with W=8 Few modes suffice; GRG adds corridors if decisions must be certified.
Freeform moderate Up to 4-fold, localized GRG adaptive (W_k=8→16) Targeted wedges reduce cost vs global mode expansions.
Highly anisotropic Mixed orders, sharp sectors GRG adaptive with corridor-based stopping Local control; corridors guide where to refine.

5) Corridor-Based Stopping (unchanged)

Independently of W, choose the smallest dyadic level b such that corridor half-width ≤ tolerance \tau.

$$ \textbf{Stop when}\quad \widehat{E}_b^{+}-\widehat{E}_b^{-}\le \tau. $$

What Changes with This Update

  • Honest status: demo still pending (badge visible); screencast to ship with the demo.
  • Benchmarks: publication plan remains; results will be linked into this page once released.
  • Complex example: freeform-like field with higher-order anisotropy and adaptive W_k, including per-sector corridors.
  • Cost clarity: when to prefer GRG vs short angular bases as W grows.