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:
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.
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.
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.
Values below are illustrative and show how corridors tighten with higher W_k where angular content is richer.
Uniform W = 8
Sector w
Estimate
Corridor ±
1
0.121
±0.0070
2
0.128
±0.0072
3
0.136
±0.0074
4
0.142
±0.0075
5
0.141
±0.0075
6
0.134
±0.0073
7
0.126
±0.0071
8
0.121
±0.0070
Adaptive Wk (8→12→16)
Sector w
Estimate
Corridor ±
1
0.121
±0.0046
2
0.129
±0.0048
3
0.138
±0.0049
4
0.144
±0.0050
5
0.143
±0.0050
6
0.135
±0.0048
7
0.127
±0.0047
8
0.121
±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:
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 adaptiveW_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.
GRG — Addressing Residual Critiques: Demo Access, Benchmarks, and a Concrete Wedge-Stratified Example
This page responds to the latest external feedback by: (i) exposing how the public Mini-App demo and screencast will be surfaced, (ii) detailing the publication plan for side-by-side benchmarks, and (iii) adding a fully worked anisotropic, wedge-stratified example with concrete W, K, per-sector corridor bounds, and a clear complexity model.
1) Mini-App Demo & Screencast — Access Plan
Public demo URL: will be placed in the site header under Community → Mini-App and reiterated in the hero of all demo-related posts (single click from the homepage).
Screencast: a short walkthrough (≈2–3 min) showing corridor overlays, the b slider, CSV/PNG export, and a corridor-based stopping rule.
Repro artifacts: each preset (Optics/MTF, Chemistry/RDF) includes a “Download CSV” button containing the estimate and its certified corridor.
Status marker: until the public URL is visible, the hero of this page shows a “Demo forthcoming” badge so the absence is obvious—not hidden.
2) Benchmarks — What Will Be Published
We will release side-by-side comparisons versus Zernike, polar B-splines, Fourier–Bessel, and pixel discretizations. Each figure carries certified corridors so decisions are auditable.
Decision rule: declare a winner only when corridor half-width < task tolerance \tau and confidence intervals do not overlap.
3) Concrete Anisotropic Example — Wedge-Stratified GRG
Scenario. A weakly freeform pupil with sagittal/meridional asymmetry (e.g., elliptical sag). We integrate a radial-angular field F(\rho,\theta) over dyadic rings and angular wedges, with certified corridors per sector.
K=32 dyadic rings (b=5 for ring edges; quarter sampling uses \Delta u=2^{-b-2}).
Two angular granularities: W=8 (coarse) and W=16 (refined).
Field: F(\rho,\theta)=A(\rho)\,\bigl(1+\epsilon\cos 2(\theta-\theta_0)\bigr) with A(\rho)=\rho^{0.6}, \epsilon=0.15, \theta_0=20^\circ.
Per-sector corridors (illustrative values)
We report the aggregate over outer half of the pupil \rho\in[0.5,1). Numbers show the estimate and the symmetric corridor half-width.
W = 8
Sector w
Estimate
Corridor ±
1
0.124
±0.0060
2
0.131
±0.0063
3
0.137
±0.0065
4
0.141
±0.0066
5
0.139
±0.0066
6
0.133
±0.0064
7
0.127
±0.0061
8
0.123
±0.0060
W = 16
Sector w
Estimate
Corridor ±
1
0.062
±0.0035
2
0.066
±0.0036
3
0.070
±0.0037
4
0.072
±0.0038
…
…
…
16
0.061
±0.0035
Observation. Doubling W tightens per-sector corridors roughly as 1/\sqrt{W} (for comparable sampling budgets), while revealing the expected \cos 2(\theta-\theta_0) anisotropy.
With K=32 and W=\{8,16\}, the refined case doubles angular bins, hence ~2× arithmetic and storage. In practice, ring accumulation is cache-friendly; wall-clock growth is close to linear until compute becomes memory-bound. Corridor contraction with b remains unchanged; W mainly dictates angular resolution.
Visualization — Wedge-Stratified Layout
Per-sector certification on the outer half of the pupil; increase W where angular gradients are steep.
4) Practical Stopping with Corridors
Independently of W, choose the smallest dyadic level b such that corridor half-width is below the tolerance relevant to the task.
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}.
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).
b
N=2^b
Δu
MTF est.
Corridor ±
4
16
2^{-6}=0.015625
0.412
±0.020
5
32
2^{-7}=0.0078125
0.417
±0.012
6
64
2^{-8}=0.00390625
0.419
±0.007
7
128
2^{-9}=0.001953125
0.420
±0.004
Practical stop. Choose the smallest b such that the corridor half-width is below a task tolerance \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 r
Δu
g(r) estimate
Corridor ±
0.20
2^{-8}
1.38
±0.06
0.40
2^{-8}
0.97
±0.05
0.60
2^{-8}
0.81
±0.04
0.80
2^{-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.
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.
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