Add notebook demonstrating equivalence of equiangular samplings #278
There are no files selected for viewing
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| Original file line number | Diff line number | Diff line change | ||||
|---|---|---|---|---|---|---|
| @@ -0,0 +1,375 @@ | ||||||
| { | ||||||
| "cells": [ | ||||||
| { | ||||||
| "cell_type": "markdown", | ||||||
| "id": "76430a19-e222-4c2c-9e4e-743641a5e12e", | ||||||
| "metadata": {}, | ||||||
| "source": [ | ||||||
| "# Equivalence of various equiangular samplings\n", | ||||||
| "\n", | ||||||
| "In this demo we show the equivalence between Fejer/Clenshaw-Curtis sampling and other equiangular sampling schemes, including MW, MWSS and DH." | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 1, | ||||||
| "id": "be9189dc-7827-416a-9340-fb66fbc390e1", | ||||||
| "metadata": {}, | ||||||
| "outputs": [ | ||||||
| { | ||||||
| "name": "stderr", | ||||||
| "output_type": "stream", | ||||||
| "text": [ | ||||||
| "JAX is not using 64-bit precision. This will dramatically affect numerical precision at even moderate L.\n" | ||||||
| ] | ||||||
| } | ||||||
| ], | ||||||
| "source": [ | ||||||
| "import s2fft \n", | ||||||
| "import numpy as np" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "markdown", | ||||||
| "id": "81b92ac3-36e8-40fc-93f6-45394eb673dd", | ||||||
| "metadata": {}, | ||||||
| "source": [ | ||||||
| "Set band-limit for tests." | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 2, | ||||||
| "id": "463a7044-99c3-45e6-bce2-3b3c7dd0e259", | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "L = 4" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "markdown", | ||||||
| "id": "f59ae250-b286-4a9a-8a45-32098fa134a1", | ||||||
| "metadata": {}, | ||||||
| "source": [ | ||||||
| "## Equivalence of MW and Fejer type 1 sampling" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 3, | ||||||
| "id": "8559d609-67d5-4561-bb43-2de124f49e25", | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "thetas_mw = s2fft.sampling.s2_samples.thetas(L, sampling=\"mw\")" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 4, | ||||||
| "id": "bed1f19f-d015-4284-ade6-eeb8954d34b9", | ||||||
| "metadata": {}, | ||||||
| "outputs": [ | ||||||
| { | ||||||
| "data": { | ||||||
| "text/plain": [ | ||||||
| "array([0.44879895, 1.34639685, 2.24399475, 3.14159265])" | ||||||
| ] | ||||||
| }, | ||||||
| "execution_count": 4, | ||||||
| "metadata": {}, | ||||||
| "output_type": "execute_result" | ||||||
| } | ||||||
| ], | ||||||
| "source": [ | ||||||
| "thetas_mw" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 5, | ||||||
| "id": "ad22a641-3940-489e-98f5-315f0ca4c5e6", | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "n = L - 1/2" | ||||||
|
There was a problem hiding this comment. It's unclear what |
||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 6, | ||||||
| "id": "de96933f-6b89-4796-bdcd-1eb4959a1a55", | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "thetas_mw_fejer1 = (np.arange(0, n) + 0.5) * np.pi / n" | ||||||
|
There was a problem hiding this comment. It would be useful to have a reference for this formula or at least some explanation of how it arises. |
||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 7, | ||||||
| "id": "974e4e4d-0753-45da-8eab-ecf0cf21f501", | ||||||
| "metadata": {}, | ||||||
| "outputs": [ | ||||||
| { | ||||||
| "data": { | ||||||
| "text/plain": [ | ||||||
| "array([0.44879895, 1.34639685, 2.24399475, 3.14159265])" | ||||||
| ] | ||||||
| }, | ||||||
| "execution_count": 7, | ||||||
| "metadata": {}, | ||||||
| "output_type": "execute_result" | ||||||
| } | ||||||
| ], | ||||||
| "source": [ | ||||||
| "thetas_mw_fejer1" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 8, | ||||||
| "id": "01fd0a9d-180d-4275-b4bb-4e269aa1d466", | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "assert np.allclose(thetas_mw, thetas_mw_fejer1)" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "markdown", | ||||||
| "id": "0f13b08a-3df9-4999-a1a0-f022b36c7aa8", | ||||||
| "metadata": {}, | ||||||
| "source": [ | ||||||
| "## Equivalence of MWSS and Clenshaw-Curtis (Fejer type 2 with end-points) sampling " | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 9, | ||||||
| "id": "3df7cacf-90c1-44c5-9d84-4c0164b5df4e", | ||||||
| "metadata": {}, | ||||||
| "outputs": [ | ||||||
| { | ||||||
| "data": { | ||||||
| "text/plain": [ | ||||||
| "array([0. , 0.78539816, 1.57079633, 2.35619449, 3.14159265])" | ||||||
| ] | ||||||
| }, | ||||||
| "execution_count": 9, | ||||||
| "metadata": {}, | ||||||
| "output_type": "execute_result" | ||||||
| } | ||||||
| ], | ||||||
| "source": [ | ||||||
| "thetas_mwss = s2fft.sampling.s2_samples.thetas(L, sampling=\"mwss\")\n", | ||||||
| "thetas_mwss" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 10, | ||||||
| "id": "c7e7af1f-d1dd-4d32-bbaf-3bf61f6026b1", | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "n = L" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 11, | ||||||
| "id": "5334dba6-f247-45ea-8340-9e13d1321ab9", | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "thetas_mwss_fejer2 = np.arange(1, n) * np.pi / n" | ||||||
|
There was a problem hiding this comment. Again without context unclear where this formula comes from / what is being shown here. |
||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 12, | ||||||
| "id": "c8dbc543-71a7-4b4d-b8d0-3ca1af2fb360", | ||||||
| "metadata": {}, | ||||||
| "outputs": [ | ||||||
| { | ||||||
| "data": { | ||||||
| "text/plain": [ | ||||||
| "array([0.78539816, 1.57079633, 2.35619449])" | ||||||
| ] | ||||||
| }, | ||||||
| "execution_count": 12, | ||||||
| "metadata": {}, | ||||||
| "output_type": "execute_result" | ||||||
| } | ||||||
| ], | ||||||
| "source": [ | ||||||
| "thetas_mwss_fejer2" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "markdown", | ||||||
| "id": "ae69b19f-c6cb-4f94-a591-946ab9c235fa", | ||||||
| "metadata": {}, | ||||||
| "source": [ | ||||||
| "Note that Fejer tyle 2 sampling does not include end-points." | ||||||
|
There was a problem hiding this comment.
Suggested change
|
||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 13, | ||||||
| "id": "6484fd92-7c6c-45dd-87c6-170c2cd94699", | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "assert np.allclose(thetas_mwss[1:-1], thetas_mwss_fejer2)" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 14, | ||||||
| "id": "c80ef1dd-1b41-401c-a506-95da1e842bc1", | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "thetas_mwss_cc = np.arange(0, n + 1) * np.pi / n" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 15, | ||||||
| "id": "2c4b38b9-8e40-4050-8942-77cc9dbc2b52", | ||||||
| "metadata": {}, | ||||||
| "outputs": [ | ||||||
| { | ||||||
| "data": { | ||||||
| "text/plain": [ | ||||||
| "array([0. , 0.78539816, 1.57079633, 2.35619449, 3.14159265])" | ||||||
| ] | ||||||
| }, | ||||||
| "execution_count": 15, | ||||||
| "metadata": {}, | ||||||
| "output_type": "execute_result" | ||||||
| } | ||||||
| ], | ||||||
| "source": [ | ||||||
| "thetas_mwss_cc" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 16, | ||||||
| "id": "c5818c01-bcb5-4df6-9a94-097af75c504f", | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "assert np.allclose(thetas_mwss, thetas_mwss_cc)" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "markdown", | ||||||
| "id": "303b49fa-ceff-4f36-99fd-adc714d93140", | ||||||
| "metadata": {}, | ||||||
| "source": [ | ||||||
| "## Equivalence of DH and Fejer type 1 sampling" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 17, | ||||||
| "id": "d412cec5-47e1-4146-955b-3aed62318647", | ||||||
| "metadata": {}, | ||||||
| "outputs": [ | ||||||
| { | ||||||
| "data": { | ||||||
| "text/plain": [ | ||||||
| "array([0.19634954, 0.58904862, 0.9817477 , 1.37444679, 1.76714587,\n", | ||||||
| " 2.15984495, 2.55254403, 2.94524311])" | ||||||
| ] | ||||||
| }, | ||||||
| "execution_count": 17, | ||||||
| "metadata": {}, | ||||||
| "output_type": "execute_result" | ||||||
| } | ||||||
| ], | ||||||
| "source": [ | ||||||
| "thetas_dh = s2fft.sampling.s2_samples.thetas(L, sampling=\"dh\")\n", | ||||||
| "thetas_dh" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 18, | ||||||
| "id": "19336652-2ccb-48d5-9bbe-1de0b352d5d7", | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "n = 2 * L" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 19, | ||||||
| "id": "c58e6bbe-4892-46d3-8478-747d282630a0", | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "thetas_dh_fejer1 = (np.arange(0, n) + 0.5) * np.pi / n" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 20, | ||||||
| "id": "383a53ad-0fbe-40e2-8c4f-fdb3d630596c", | ||||||
| "metadata": {}, | ||||||
| "outputs": [ | ||||||
| { | ||||||
| "data": { | ||||||
| "text/plain": [ | ||||||
| "array([0.19634954, 0.58904862, 0.9817477 , 1.37444679, 1.76714587,\n", | ||||||
| " 2.15984495, 2.55254403, 2.94524311])" | ||||||
| ] | ||||||
| }, | ||||||
| "execution_count": 20, | ||||||
| "metadata": {}, | ||||||
| "output_type": "execute_result" | ||||||
| } | ||||||
| ], | ||||||
| "source": [ | ||||||
| "thetas_dh_fejer1" | ||||||
| ] | ||||||
| }, | ||||||
| { | ||||||
| "cell_type": "code", | ||||||
| "execution_count": 21, | ||||||
| "id": "ed2f9425-bf04-45b6-bb94-1c5546dd1417", | ||||||
| "metadata": {}, | ||||||
| "outputs": [], | ||||||
| "source": [ | ||||||
| "assert np.allclose(thetas_dh, thetas_dh_fejer1)" | ||||||
| ] | ||||||
| } | ||||||
| ], | ||||||
| "metadata": { | ||||||
| "kernelspec": { | ||||||
| "display_name": "Python 3 (ipykernel)", | ||||||
| "language": "python", | ||||||
| "name": "python3" | ||||||
| }, | ||||||
| "language_info": { | ||||||
| "codemirror_mode": { | ||||||
| "name": "ipython", | ||||||
| "version": 3 | ||||||
| }, | ||||||
| "file_extension": ".py", | ||||||
| "mimetype": "text/x-python", | ||||||
| "name": "python", | ||||||
| "nbconvert_exporter": "python", | ||||||
| "pygments_lexer": "ipython3", | ||||||
| "version": "3.11.11" | ||||||
| } | ||||||
| }, | ||||||
| "nbformat": 4, | ||||||
| "nbformat_minor": 5 | ||||||
| } | ||||||
Oops, something went wrong.
Add this suggestion to a batch that can be applied as a single commit.
This suggestion is invalid because no changes were made to the code.
Suggestions cannot be applied while the pull request is closed.
Suggestions cannot be applied while viewing a subset of changes.
Only one suggestion per line can be applied in a batch.
Add this suggestion to a batch that can be applied as a single commit.
Applying suggestions on deleted lines is not supported.
You must change the existing code in this line in order to create a valid suggestion.
Outdated suggestions cannot be applied.
This suggestion has been applied or marked resolved.
Suggestions cannot be applied from pending reviews.
Suggestions cannot be applied on multi-line comments.
Suggestions cannot be applied while the pull request is queued to merge.
Suggestion cannot be applied right now. Please check back later.
There was a problem hiding this comment.
Choose a reason for hiding this comment
The reason will be displayed to describe this comment to others. Learn more.
I think tutorial is better than demo for consistency with terminology we use in documentation and that it's worth expanding the sampling scheme acronyms as it may not be clear what these refer to .