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.
Unified Update

Unified Update

Sep 20, 2025

GRG — Unified Update: Examples, Validation, Mini-App, and Anisotropic Extension

This unified page consolidates the latest GRG status: cross-domain numerical examples (optics and chemistry), an external validation plan against standard bases (e.g., Zernike), public Mini-App notes, and a practical recipe for anisotropic, non-radial systems via wedge-stratified GRG. All formulas use turns-first dyadic refinement with certified corridors for auditable error control.

Key Takeaways

  • π-free numerics. GRG avoids \pi in accumulation, improving digital precision and reproducibility.
  • Certified corridors. Every estimate comes with upper/lower bounds; stopping rules are explicit.
  • Interdisciplinary reach. Geometry, optics, chemistry, and beyond—all under a single corridor framework.
  • Open path to validation. Public scripts, datasets, and side-by-side benchmarks with standard methods.

Numerical Example A — Optics (MTF at a Target Frequency)

Goal. Maximize on-axis contrast at target spatial frequency using corridor-aware sampling. We set unit pupil and dyadic refinement with b \in \mathbb{N}_{\ge 0}, N = 2^b, and \Delta u = 2^{-b-2}.

$$ \mathrm{MTF}(f) \;=\; \frac{1}{\pi}\! \int_{0}^{2\pi}\!\!\int_{0}^{1} P(\rho,\theta)\,\cos\!\bigl(2\pi f\,\rho\cos\theta\bigr)\,\rho\,d\rho\,d\theta $$

Illustrative setup. Clear pupil (P=1). Apodizer family A_\alpha(\rho)=\rho^\alpha. Target frequency f^\star is scaled to the unit pupil. For each b, GRG produces an estimate and a certified corridor.

MTF at f^\star with GRG corridors (illustrative values).
bN=2^bΔuMTF est.Corridor ±
4162^{-6}=0.0156250.412±0.020
5322^{-7}=0.00781250.417±0.012
6642^{-8}=0.003906250.419±0.007
71282^{-9}=0.0019531250.420±0.004

Practical stop. Choose the smallest b such that the corridor half-width is below a task tolerance \tau.

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

Numerical Example B — Chemistry (Radial Distribution Function)

Goal. Estimate a smooth radial distribution in a dense cluster (e.g., ligand shell) using dyadic radii and corridor-aware accumulation. After mild smoothing with a compact kernel, corridors certify stability across small samples.

$$ g(r) \;=\; \frac{1}{4\pi r^2 \Delta r}\,\frac{1}{N} \sum_{i=1}^{N} \#\bigl\{\, j\neq i:\ r \le \|x_j-x_i\| < r+\Delta r \,\bigr\} $$
Radius rΔug(r) estimateCorridor ±
0.202^{-8}1.38±0.06
0.402^{-8}0.97±0.05
0.602^{-8}0.81±0.04
0.802^{-8}0.76±0.04

Interpretation. Corridor-aware RDF avoids over-reading small oscillations and makes cross-dataset comparisons fair and reproducible.

Validation Plan — Public, Scriptable, Comparable

We will release side-by-side benchmarks with open scripts and datasets. Figures and tables will include corridors so decisions are auditable.

$$ \text{Compare}\quad \bigl\|\,T_{\text{GRG}} - T_{\text{ref}}\,\bigr\|_{2} \ \ \text{vs.}\ \ b,\; \text{runtime},\; \text{memory}, $$
  • Tasks: PSF/MTF fitting; surface reconstruction; denoising with corridor stops; radial distributions.
  • Baselines: Zernike, polar B-splines, Fourier-Bessel, and pixel discretizations.
  • Artifacts: One-command scripts; dataset DOIs; fixed seeds/parameters; CSV outputs for independent plotting.

Mini-App — Visibility & Usability

The public demo exposes corridor overlays, a b slider, CSV/PNG export, and presets (optics/chemistry). A short screencast walks through an optimization and a corridor-based stopping rule.

Anisotropy — Wedge-Stratified GRG for Non-Radial Systems

Idea. Partition each ring into W angular wedges. Corridors apply per wedge; certified aggregation preserves guarantees. Increase W where angular gradients are steep.

$$ \text{For rings }k=1..K,\ \text{wedges }w=1..W:\quad I_{k,w}\;=\;\int_{\theta\in\Theta_w}\!\int_{\rho\in[u_{k-1},u_k)} F(\rho,\theta)\,\rho\,d\rho\,d\theta. $$
Wedge-stratified corridors Annulus split into angular wedges; thick arcs indicate certified corridor bounds aggregated per wedge.
Per-wedge certification enables anisotropic handling without losing corridor guarantees.

FAQ — What’s New Here?

  • Concrete cross-domain demos: optics (MTF) and chemistry (RDF) with corridor-aware tables/plots.
  • External validation: transparent plan, open artifacts, standard baselines.
  • Mini-App: public demo with overlays, dyadic control, CSV/PNG export, and a short screencast.
  • Non-radial systems: wedge-stratified recipe with simple heuristics for W allocation.