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Add notebook demonstrating equivalence of equiangular samplings
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‎notebooks/equivalence_equiangular_sampling.ipynb‎

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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "76430a19-e222-4c2c-9e4e-743641a5e12e",
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"metadata": {},
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"source": [
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"# Equivalence of various equiangular samplings\n",
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"\n",
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"In this demo we show the equivalence between Fejer/Clenshaw-Curtis sampling and other equiangular sampling schemes, including MW, MWSS and DH."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "be9189dc-7827-416a-9340-fb66fbc390e1",
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"metadata": {},
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"outputs": [
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"JAX is not using 64-bit precision. This will dramatically affect numerical precision at even moderate L.\n"
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]
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}
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],
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"source": [
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"import s2fft \n",
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"import numpy as np"
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]
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},
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{
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"cell_type": "markdown",
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"id": "81b92ac3-36e8-40fc-93f6-45394eb673dd",
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"metadata": {},
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"source": [
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"Set band-limit for tests."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "463a7044-99c3-45e6-bce2-3b3c7dd0e259",
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"metadata": {},
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"outputs": [],
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"source": [
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"L = 4"
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]
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},
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{
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"cell_type": "markdown",
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"id": "f59ae250-b286-4a9a-8a45-32098fa134a1",
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"metadata": {},
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"source": [
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"## Equivalence of MW and Fejer type 1 sampling"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "8559d609-67d5-4561-bb43-2de124f49e25",
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"metadata": {},
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"outputs": [],
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"source": [
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"thetas_mw = s2fft.sampling.s2_samples.thetas(L, sampling=\"mw\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "bed1f19f-d015-4284-ade6-eeb8954d34b9",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([0.44879895, 1.34639685, 2.24399475, 3.14159265])"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"thetas_mw"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "ad22a641-3940-489e-98f5-315f0ca4c5e6",
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"metadata": {},
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"outputs": [],
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"source": [
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"n = L - 1/2"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"id": "de96933f-6b89-4796-bdcd-1eb4959a1a55",
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"metadata": {},
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"outputs": [],
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"source": [
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"thetas_mw_fejer1 = (np.arange(0, n) + 0.5) * np.pi / n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"id": "974e4e4d-0753-45da-8eab-ecf0cf21f501",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([0.44879895, 1.34639685, 2.24399475, 3.14159265])"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"thetas_mw_fejer1"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"id": "01fd0a9d-180d-4275-b4bb-4e269aa1d466",
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"metadata": {},
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"outputs": [],
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"source": [
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"assert np.allclose(thetas_mw, thetas_mw_fejer1)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "0f13b08a-3df9-4999-a1a0-f022b36c7aa8",
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"metadata": {},
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"source": [
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"## Equivalence of MWSS and Clenshaw-Curtis (Fejer type 2 with end-points) sampling "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"id": "3df7cacf-90c1-44c5-9d84-4c0164b5df4e",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([0. , 0.78539816, 1.57079633, 2.35619449, 3.14159265])"
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]
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},
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"execution_count": 9,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"thetas_mwss = s2fft.sampling.s2_samples.thetas(L, sampling=\"mwss\")\n",
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"thetas_mwss"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"id": "c7e7af1f-d1dd-4d32-bbaf-3bf61f6026b1",
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"metadata": {},
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"outputs": [],
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"source": [
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"n = L"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"id": "5334dba6-f247-45ea-8340-9e13d1321ab9",
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"metadata": {},
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"outputs": [],
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"source": [
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"thetas_mwss_fejer2 = np.arange(1, n) * np.pi / n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"id": "c8dbc543-71a7-4b4d-b8d0-3ca1af2fb360",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([0.78539816, 1.57079633, 2.35619449])"
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]
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},
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"execution_count": 12,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"thetas_mwss_fejer2"
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]
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},
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{
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"cell_type": "markdown",
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"id": "ae69b19f-c6cb-4f94-a591-946ab9c235fa",
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"metadata": {},
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"source": [
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"Note that Fejer tyle 2 sampling does not include end-points."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"id": "6484fd92-7c6c-45dd-87c6-170c2cd94699",
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"metadata": {},
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"outputs": [],
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"source": [
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"assert np.allclose(thetas_mwss[1:-1], thetas_mwss_fejer2)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 14,
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"id": "c80ef1dd-1b41-401c-a506-95da1e842bc1",
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"metadata": {},
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"outputs": [],
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"source": [
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"thetas_mwss_cc = np.arange(0, n + 1) * np.pi / n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"id": "2c4b38b9-8e40-4050-8942-77cc9dbc2b52",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([0. , 0.78539816, 1.57079633, 2.35619449, 3.14159265])"
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]
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},
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"execution_count": 15,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"thetas_mwss_cc"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 16,
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"id": "c5818c01-bcb5-4df6-9a94-097af75c504f",
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"metadata": {},
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"outputs": [],
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"source": [
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"assert np.allclose(thetas_mwss, thetas_mwss_cc)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "303b49fa-ceff-4f36-99fd-adc714d93140",
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"metadata": {},
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"source": [
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"## Equivalence of DH and Fejer type 1 sampling"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 17,
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"id": "d412cec5-47e1-4146-955b-3aed62318647",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([0.19634954, 0.58904862, 0.9817477 , 1.37444679, 1.76714587,\n",
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" 2.15984495, 2.55254403, 2.94524311])"
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]
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},
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"execution_count": 17,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"thetas_dh = s2fft.sampling.s2_samples.thetas(L, sampling=\"dh\")\n",
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"thetas_dh"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 18,
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"id": "19336652-2ccb-48d5-9bbe-1de0b352d5d7",
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"metadata": {},
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"outputs": [],
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"source": [
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"n = 2 * L"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 19,
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"id": "c58e6bbe-4892-46d3-8478-747d282630a0",
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"metadata": {},
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"outputs": [],
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"source": [
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"thetas_dh_fejer1 = (np.arange(0, n) + 0.5) * np.pi / n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 20,
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"id": "383a53ad-0fbe-40e2-8c4f-fdb3d630596c",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([0.19634954, 0.58904862, 0.9817477 , 1.37444679, 1.76714587,\n",
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" 2.15984495, 2.55254403, 2.94524311])"
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]
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},
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"execution_count": 20,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"thetas_dh_fejer1"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 21,
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"id": "ed2f9425-bf04-45b6-bb94-1c5546dd1417",
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"metadata": {},
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"outputs": [],
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"source": [
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"assert np.allclose(thetas_dh, thetas_dh_fejer1)"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.11"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}

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