{ "cells": [ { "cell_type": "markdown", "id": "afc07c2f-f818-45a3-adc1-4ccc177fb49e", "metadata": {}, "source": [ "# Performance of `evolution`\n", "## Cutoff in the coefficients\n", "\n", "The time evolution operator for a Hamiltonian $H$\n", "$$ O[H, \\Delta t] = \\text{exp}\\left(-i H \\Delta t\\right) $$\n", "uses a KPM expansion as a function of time $\\Delta t$, instead of the usual Fermi energy $\\varepsilon$ of spectral functions.\n", "\n", "In the expansion \n", "$$ O[H, \\Delta t] = \\sum_{m=0}^{\\infty} T_m(H) c_m(\\Delta t)$$\n", "the coefficients $c_m$ are proportional to the Bessel functions of the first kind of order $m$, so they decay exponentially fast after a certain threshold $m^*$.\n", "\n", "Thus, is it extremely efficient to cutoff the expansion for a certain target accuracy\n", "$$ O[H, \\Delta t] \\simeq \\sum_{m=0}^{m^*} T_m(H) c_m(\\Delta t)$$\n", "\n", "The exact value of $m^*$ depends on the time step times the bounds of the spectrum $\\Delta t\\times\\Delta E$, and on the desired accuracy.\n", "To find $m^*$ for all values of $\\Delta t\\times\\Delta E$, we define a few typical target accuracies: $10^{-8}, 10^{-16}, 10^{-32}$, and numerically obtain the answer. \n", "Then, we will use a numerical estimate of the function that fits $m^*$." ] }, { "cell_type": "code", "execution_count": 1, "id": "678ee99d-938d-4560-8008-48eedefd3194", "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "(function(root) {\n", " function now() {\n", " return new Date();\n", " }\n", "\n", " var force = true;\n", " var py_version = '3.3.2'.replace('rc', '-rc.').replace('.dev', '-dev.');\n", " var reloading = false;\n", " var Bokeh = root.Bokeh;\n", "\n", " if (typeof (root._bokeh_timeout) === \"undefined\" || force) {\n", " root._bokeh_timeout = Date.now() + 5000;\n", " root._bokeh_failed_load = false;\n", " }\n", "\n", " function run_callbacks() {\n", " try {\n", " root._bokeh_onload_callbacks.forEach(function(callback) {\n", " if (callback != null)\n", " callback();\n", " });\n", " } finally {\n", " delete root._bokeh_onload_callbacks;\n", " }\n", " console.debug(\"Bokeh: all callbacks have finished\");\n", " }\n", "\n", " function load_libs(css_urls, js_urls, js_modules, js_exports, callback) {\n", " if (css_urls == null) css_urls = [];\n", " if (js_urls == null) js_urls = [];\n", " if (js_modules == null) js_modules = [];\n", " if (js_exports == null) js_exports = {};\n", "\n", " root._bokeh_onload_callbacks.push(callback);\n", "\n", " if (root._bokeh_is_loading > 0) {\n", " console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n", " return null;\n", " }\n", " if (js_urls.length === 0 && js_modules.length === 0 && Object.keys(js_exports).length === 0) {\n", " run_callbacks();\n", " return null;\n", " }\n", " if (!reloading) {\n", " console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n", " }\n", "\n", " function on_load() {\n", " root._bokeh_is_loading--;\n", " if (root._bokeh_is_loading === 0) {\n", " console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n", " run_callbacks()\n", " }\n", " }\n", " window._bokeh_on_load = on_load\n", "\n", " function on_error() {\n", " console.error(\"failed to load \" + url);\n", " }\n", "\n", " var skip = [];\n", " if (window.requirejs) {\n", " window.requirejs.config({'packages': {}, 'paths': {'jspanel': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/jspanel', 'jspanel-modal': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/modal/jspanel.modal', 'jspanel-tooltip': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/tooltip/jspanel.tooltip', 'jspanel-hint': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/hint/jspanel.hint', 'jspanel-layout': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/layout/jspanel.layout', 'jspanel-contextmenu': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/contextmenu/jspanel.contextmenu', 'jspanel-dock': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/dock/jspanel.dock', 'gridstack': 'https://cdn.jsdelivr.net/npm/gridstack@7.2.3/dist/gridstack-all', 'notyf': 'https://cdn.jsdelivr.net/npm/notyf@3/notyf.min'}, 'shim': {'jspanel': {'exports': 'jsPanel'}, 'gridstack': {'exports': 'GridStack'}}});\n", " require([\"jspanel\"], function(jsPanel) {\n", "\twindow.jsPanel = jsPanel\n", "\ton_load()\n", " })\n", " require([\"jspanel-modal\"], function() {\n", "\ton_load()\n", " })\n", " require([\"jspanel-tooltip\"], function() {\n", "\ton_load()\n", " })\n", " require([\"jspanel-hint\"], function() {\n", "\ton_load()\n", " })\n", " require([\"jspanel-layout\"], function() {\n", "\ton_load()\n", " })\n", " require([\"jspanel-contextmenu\"], function() {\n", "\ton_load()\n", " })\n", " require([\"jspanel-dock\"], function() {\n", "\ton_load()\n", " })\n", " require([\"gridstack\"], function(GridStack) {\n", "\twindow.GridStack = GridStack\n", "\ton_load()\n", " })\n", " require([\"notyf\"], function() {\n", "\ton_load()\n", " })\n", " root._bokeh_is_loading = css_urls.length + 9;\n", " } else {\n", " root._bokeh_is_loading = css_urls.length + js_urls.length + js_modules.length + Object.keys(js_exports).length;\n", " }\n", "\n", " var existing_stylesheets = []\n", " var links = document.getElementsByTagName('link')\n", " for (var i = 0; i < links.length; i++) {\n", " var link = links[i]\n", " if (link.href != null) {\n", "\texisting_stylesheets.push(link.href)\n", " }\n", " }\n", " for (var i = 0; i < css_urls.length; i++) {\n", " var url = css_urls[i];\n", " if (existing_stylesheets.indexOf(url) !== -1) {\n", "\ton_load()\n", "\tcontinue;\n", " }\n", " const element = document.createElement(\"link\");\n", " element.onload = on_load;\n", " element.onerror = on_error;\n", " element.rel = \"stylesheet\";\n", " element.type = \"text/css\";\n", " element.href = url;\n", " console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n", " document.body.appendChild(element);\n", " } if (((window['jsPanel'] !== undefined) && (!(window['jsPanel'] instanceof HTMLElement))) || window.requirejs) {\n", " var urls = ['https://cdn.holoviz.org/panel/1.3.6/dist/bundled/floatpanel/jspanel4@4.12.0/dist/jspanel.js', 'https://cdn.holoviz.org/panel/1.3.6/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/modal/jspanel.modal.js', 'https://cdn.holoviz.org/panel/1.3.6/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/tooltip/jspanel.tooltip.js', 'https://cdn.holoviz.org/panel/1.3.6/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/hint/jspanel.hint.js', 'https://cdn.holoviz.org/panel/1.3.6/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/layout/jspanel.layout.js', 'https://cdn.holoviz.org/panel/1.3.6/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/contextmenu/jspanel.contextmenu.js', 'https://cdn.holoviz.org/panel/1.3.6/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/dock/jspanel.dock.js'];\n", " 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js_urls.length; i++) {\n", " var url = js_urls[i];\n", " if (skip.indexOf(url) !== -1 || existing_scripts.indexOf(url) !== -1) {\n", "\tif (!window.requirejs) {\n", "\t on_load();\n", "\t}\n", "\tcontinue;\n", " }\n", " var element = document.createElement('script');\n", " element.onload = on_load;\n", " element.onerror = on_error;\n", " element.async = false;\n", " element.src = url;\n", " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", " document.head.appendChild(element);\n", " }\n", " for (var i = 0; i < js_modules.length; i++) {\n", " var url = js_modules[i];\n", " if (skip.indexOf(url) !== -1 || existing_scripts.indexOf(url) !== -1) {\n", "\tif (!window.requirejs) {\n", "\t on_load();\n", "\t}\n", "\tcontinue;\n", " }\n", " var element = document.createElement('script');\n", " element.onload = on_load;\n", " element.onerror = on_error;\n", " element.async = false;\n", " element.src = url;\n", " element.type = \"module\";\n", " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", " document.head.appendChild(element);\n", " }\n", " for (const name in js_exports) {\n", " var url = js_exports[name];\n", " if (skip.indexOf(url) >= 0 || root[name] != null) {\n", "\tif (!window.requirejs) {\n", "\t on_load();\n", "\t}\n", "\tcontinue;\n", " }\n", " var element = document.createElement('script');\n", " element.onerror = on_error;\n", " element.async = false;\n", " element.type = \"module\";\n", " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", " element.textContent = `\n", " import ${name} from \"${url}\"\n", " window.${name} = ${name}\n", " window._bokeh_on_load()\n", " `\n", " document.head.appendChild(element);\n", " }\n", " if (!js_urls.length && !js_modules.length) {\n", " on_load()\n", " }\n", " };\n", "\n", " function inject_raw_css(css) {\n", " const element = document.createElement(\"style\");\n", " element.appendChild(document.createTextNode(css));\n", " document.body.appendChild(element);\n", " }\n", "\n", " var js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-3.3.2.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.3.2.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.3.2.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.3.2.min.js\", \"https://cdn.holoviz.org/panel/1.3.6/dist/panel.min.js\"];\n", " var js_modules = [];\n", " var js_exports = {};\n", " var css_urls = [];\n", " var inline_js = [ function(Bokeh) {\n", " Bokeh.set_log_level(\"info\");\n", " },\n", "function(Bokeh) {} // ensure no trailing comma for IE\n", " ];\n", "\n", " function run_inline_js() {\n", " if ((root.Bokeh !== undefined) || (force === true)) {\n", " for (var i = 0; i < inline_js.length; i++) {\n", "\ttry {\n", " inline_js[i].call(root, root.Bokeh);\n", "\t} catch(e) {\n", "\t if (!reloading) {\n", "\t throw e;\n", "\t }\n", "\t}\n", " }\n", " // Cache old bokeh versions\n", " if (Bokeh != undefined && !reloading) {\n", "\tvar NewBokeh = root.Bokeh;\n", "\tif (Bokeh.versions === undefined) {\n", "\t Bokeh.versions = new Map();\n", "\t}\n", "\tif (NewBokeh.version !== Bokeh.version) {\n", "\t Bokeh.versions.set(NewBokeh.version, NewBokeh)\n", "\t}\n", "\troot.Bokeh = Bokeh;\n", " }} else if (Date.now() < root._bokeh_timeout) {\n", " setTimeout(run_inline_js, 100);\n", " } else if (!root._bokeh_failed_load) {\n", " console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n", " root._bokeh_failed_load = true;\n", " }\n", " root._bokeh_is_initializing = false\n", " }\n", "\n", " function load_or_wait() {\n", " // Implement a backoff loop that tries to ensure we do not load multiple\n", " // versions of Bokeh and its dependencies at the same time.\n", " // In recent versions we use the root._bokeh_is_initializing flag\n", " // to determine whether there is an ongoing attempt to initialize\n", " // bokeh, however for backward compatibility we also try to ensure\n", " // that we do not start loading a newer (Panel>=1.0 and Bokeh>3) version\n", " // before older versions are fully initialized.\n", " if (root._bokeh_is_initializing && Date.now() > root._bokeh_timeout) {\n", " root._bokeh_is_initializing = false;\n", " root._bokeh_onload_callbacks = undefined;\n", " console.log(\"Bokeh: BokehJS was loaded multiple times but one version failed to initialize.\");\n", " load_or_wait();\n", " } else if (root._bokeh_is_initializing || (typeof root._bokeh_is_initializing === \"undefined\" && root._bokeh_onload_callbacks !== undefined)) {\n", " setTimeout(load_or_wait, 100);\n", " } else {\n", " root._bokeh_is_initializing = true\n", " root._bokeh_onload_callbacks = []\n", " var bokeh_loaded = Bokeh != null && (Bokeh.version === py_version || (Bokeh.versions !== undefined && Bokeh.versions.has(py_version)));\n", " if (!reloading && !bokeh_loaded) {\n", "\troot.Bokeh = undefined;\n", " }\n", " load_libs(css_urls, js_urls, js_modules, js_exports, function() {\n", "\tconsole.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n", "\trun_inline_js();\n", " });\n", " }\n", " }\n", " // Give older versions of the autoload script a head-start to ensure\n", " // they initialize before we start loading newer version.\n", " setTimeout(load_or_wait, 100)\n", "}(window));" ], "application/vnd.holoviews_load.v0+json": "(function(root) {\n function now() {\n return new Date();\n }\n\n var force = true;\n var py_version = '3.3.2'.replace('rc', '-rc.').replace('.dev', '-dev.');\n var reloading = false;\n var Bokeh = root.Bokeh;\n\n if (typeof (root._bokeh_timeout) === \"undefined\" || force) {\n root._bokeh_timeout = Date.now() + 5000;\n root._bokeh_failed_load = false;\n }\n\n function run_callbacks() {\n try {\n root._bokeh_onload_callbacks.forEach(function(callback) {\n if (callback != null)\n callback();\n });\n } finally {\n delete root._bokeh_onload_callbacks;\n }\n console.debug(\"Bokeh: all callbacks have finished\");\n }\n\n function load_libs(css_urls, js_urls, js_modules, js_exports, callback) {\n if (css_urls == null) css_urls = [];\n if (js_urls == null) js_urls = [];\n if (js_modules == null) js_modules = [];\n if (js_exports == null) js_exports = {};\n\n root._bokeh_onload_callbacks.push(callback);\n\n if (root._bokeh_is_loading > 0) {\n console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n return null;\n }\n if (js_urls.length === 0 && js_modules.length === 0 && Object.keys(js_exports).length === 0) {\n run_callbacks();\n return null;\n }\n if (!reloading) {\n console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n }\n\n function on_load() {\n root._bokeh_is_loading--;\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n run_callbacks()\n }\n }\n window._bokeh_on_load = on_load\n\n function on_error() {\n console.error(\"failed to load \" + url);\n }\n\n var skip = [];\n if (window.requirejs) {\n window.requirejs.config({'packages': {}, 'paths': {'jspanel': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/jspanel', 'jspanel-modal': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/modal/jspanel.modal', 'jspanel-tooltip': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/tooltip/jspanel.tooltip', 'jspanel-hint': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/hint/jspanel.hint', 'jspanel-layout': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/layout/jspanel.layout', 'jspanel-contextmenu': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/contextmenu/jspanel.contextmenu', 'jspanel-dock': 'https://cdn.jsdelivr.net/npm/jspanel4@4.12.0/dist/extensions/dock/jspanel.dock', 'gridstack': 'https://cdn.jsdelivr.net/npm/gridstack@7.2.3/dist/gridstack-all', 'notyf': 'https://cdn.jsdelivr.net/npm/notyf@3/notyf.min'}, 'shim': {'jspanel': {'exports': 'jsPanel'}, 'gridstack': {'exports': 'GridStack'}}});\n require([\"jspanel\"], function(jsPanel) {\n\twindow.jsPanel = jsPanel\n\ton_load()\n })\n require([\"jspanel-modal\"], function() {\n\ton_load()\n })\n require([\"jspanel-tooltip\"], function() {\n\ton_load()\n })\n require([\"jspanel-hint\"], function() {\n\ton_load()\n })\n require([\"jspanel-layout\"], function() {\n\ton_load()\n })\n require([\"jspanel-contextmenu\"], function() {\n\ton_load()\n })\n require([\"jspanel-dock\"], function() {\n\ton_load()\n })\n require([\"gridstack\"], function(GridStack) {\n\twindow.GridStack = GridStack\n\ton_load()\n })\n require([\"notyf\"], function() {\n\ton_load()\n })\n root._bokeh_is_loading = css_urls.length + 9;\n } else {\n root._bokeh_is_loading = css_urls.length + js_urls.length + js_modules.length + Object.keys(js_exports).length;\n }\n\n var existing_stylesheets = []\n var links = document.getElementsByTagName('link')\n for (var i = 0; i < links.length; i++) {\n var link = links[i]\n if (link.href != null) {\n\texisting_stylesheets.push(link.href)\n }\n }\n for (var i = 0; i < css_urls.length; i++) {\n var url = css_urls[i];\n if (existing_stylesheets.indexOf(url) !== -1) {\n\ton_load()\n\tcontinue;\n }\n const element = document.createElement(\"link\");\n element.onload = on_load;\n element.onerror = on_error;\n element.rel = \"stylesheet\";\n element.type = \"text/css\";\n element.href = url;\n console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n document.body.appendChild(element);\n } if (((window['jsPanel'] !== undefined) && (!(window['jsPanel'] instanceof HTMLElement))) || window.requirejs) {\n var urls = ['https://cdn.holoviz.org/panel/1.3.6/dist/bundled/floatpanel/jspanel4@4.12.0/dist/jspanel.js', 'https://cdn.holoviz.org/panel/1.3.6/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/modal/jspanel.modal.js', 'https://cdn.holoviz.org/panel/1.3.6/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/tooltip/jspanel.tooltip.js', 'https://cdn.holoviz.org/panel/1.3.6/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/hint/jspanel.hint.js', 'https://cdn.holoviz.org/panel/1.3.6/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/layout/jspanel.layout.js', 'https://cdn.holoviz.org/panel/1.3.6/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/contextmenu/jspanel.contextmenu.js', 'https://cdn.holoviz.org/panel/1.3.6/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/dock/jspanel.dock.js'];\n for (var i = 0; i < urls.length; i++) {\n skip.push(urls[i])\n }\n } if (((window['GridStack'] !== undefined) && (!(window['GridStack'] instanceof HTMLElement))) || window.requirejs) {\n var urls = ['https://cdn.holoviz.org/panel/1.3.6/dist/bundled/gridstack/gridstack@7.2.3/dist/gridstack-all.js'];\n for (var i = 0; i < urls.length; i++) {\n skip.push(urls[i])\n }\n } if (((window['Notyf'] !== undefined) && (!(window['Notyf'] instanceof HTMLElement))) || window.requirejs) {\n var urls = 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\n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import holoviews as hv\n", "hv.extension()\n" ] }, { "cell_type": "code", "execution_count": 2, "id": "b0c2c1c9-47cd-466e-8f46-fad06283fb36", "metadata": {}, "outputs": [], "source": [ "import scipy\n", "import scipy.optimize\n", "from scipy.special import jv\n" ] }, { "cell_type": "code", "execution_count": 3, "id": "21504e7c-9efb-409a-8a14-d3f86655054a", "metadata": {}, "outputs": [], "source": [ "from kpm_tools.evolution import coef_evo\n" ] }, { "cell_type": "markdown", "id": "ae3eabbb-6c96-4499-9551-acab82c3afd7", "metadata": {}, "source": [ "\n", "## numerical evaluation of the coefficients\n", "\n", "First, we obtain an `ndarray` of the coefficients vs the time delta $\\Delta t$ and the number of moments `num_moments`.\n", "\n", "Here, we use $\\Delta E = 1$, such that `dt` is the rescaled time delta $\\Delta t \\times \\Delta E$. " ] }, { "cell_type": "code", "execution_count": 4, "id": "16305d37-d864-4b26-bada-3720a18d41fb", "metadata": {}, "outputs": [], "source": [ "dt_array = np.logspace(-3,2,1000)\n", "num_moments = 500\n" ] }, { "cell_type": "code", "execution_count": 5, "id": "e58fe285-fbb2-4c39-b58b-a098e606df11", "metadata": {}, "outputs": [], "source": [ "coefs_array = np.array([\n", " coef_evo(num_moments, 0, ab=(2,0), delta_time=dt)\n", " for dt in dt_array\n", "])\n", "coefs_array = coefs_array[:,1:,0]\n" ] }, { "cell_type": "code", "execution_count": 6, "id": "63d2e9dc-13af-4f30-b742-f1c50974647a", "metadata": {}, "outputs": [], "source": [ "num_moments_array = np.arange(1, num_moments)\n" ] }, { "cell_type": "markdown", "id": "bce1b2d9-5c48-4a9a-9818-fde110efeff3", "metadata": {}, "source": [ "## make a fit\n", "\n", "### Numerical estimation of the best `num_moments`\n", "Then we obtain the best `num_moments` for a given `dt` and `accuracy`." ] }, { "cell_type": "code", "execution_count": 7, "id": "9c772263-2eb8-4d49-b038-8842972f7872", "metadata": {}, "outputs": [], "source": [ "accuracy = 1e-16\n" ] }, { "cell_type": "code", "execution_count": 8, "id": "45b5e0fe-4df1-44b5-ae6f-1b17ada7e64b", "metadata": {}, "outputs": [], "source": [ "# get the transition point in `m` such that coefs are smaller than accuracy\n", "\n", "def max_m_array(coefs_array, accuracy):\n", " return np.count_nonzero(np.abs(coefs_array) > accuracy, axis=1)\n" ] }, { "cell_type": "code", "execution_count": 9, "id": "7d89d044-e66d-4298-a437-2bf2e0742af0", "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/tmp/ipykernel_5521/785646435.py:5: RuntimeWarning: divide by zero encountered in log\n", " np.log(np.abs(coefs_array))/np.log(10)\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ ":Overlay\n", " .QuadMesh.I :QuadMesh [M,$dt$] (z)\n", " .Curve.I :Curve [x] (y)" ] }, "execution_count": 9, "metadata": { "application/vnd.holoviews_exec.v0+json": {} }, "output_type": "execute_result" } ], "source": [ "(\n", " hv.QuadMesh((\n", " num_moments_array,\n", " dt_array,\n", " np.log(np.abs(coefs_array))/np.log(10)\n", " ),\n", " kdims=['M', '$dt$'],\n", " vdims=hv.Dimension('z', label='${log}_{10}|c_n|$', range=(np.log(accuracy)/np.log(10), 0))\n", " ).opts(colorbar=True, fig_size=200)\n", " *\n", " hv.Curve((\n", " max_m_array(coefs_array, accuracy),\n", " dt_array,\n", " ))\n", ")[:30,:4]\n" ] }, { "cell_type": "markdown", "id": "e60af675-2128-4ec0-812e-7a63bfc2453c", "metadata": {}, "source": [ "We can already see that only a handful of moments is needed for a fairly large vale of the time step `dt`." ] }, { "cell_type": "markdown", "id": "8713f990-6eb3-463d-8692-b18acd93725a", "metadata": {}, "source": [ "\n", "### Analitical fit of the best `num_moments`\n", "\n", "Since we don't want to store arrays of best `num_moments`, we make\n", "an analytical approximation to the best `num_moments` for a given accuray and as a function of `dt`.\n" ] }, { "cell_type": "code", "execution_count": 10, "id": "bcacfa28-2ef0-4c02-b49a-20f5b5b60835", "metadata": {}, "outputs": [], "source": [ "def func(dt, A, B, C, D):\n", " return (A * np.log(1/accuracy) * (dt) ** B) + C * dt + D * np.log(1/accuracy)\n", "\n", "p_acc16, _ = scipy.optimize.curve_fit(func,\n", " dt_array,\n", " max_m_array(coefs_array, accuracy),\n", " p0=(0.5, 0.25, 2, 0),\n", "\n", "# bounds=([0,0,0,2], [1,1,1,2+1e-8])\n", " )\n" ] }, { "cell_type": "code", "execution_count": 11, "id": "da947bd6-9ee7-485a-b5c0-1331c07f10ab", "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ ":Overlay\n", " .QuadMesh.I :QuadMesh [M,$dt$] (z)\n", " .Curve.I :Curve [x] (y)" ] }, "execution_count": 11, "metadata": { "application/vnd.holoviews_exec.v0+json": {} }, "output_type": "execute_result" } ], "source": [ "hv.QuadMesh((\n", " num_moments_array,\n", " dt_array,\n", " np.log(np.abs(coefs_array+np.finfo(float).eps))/np.log(10)\n", "),\n", " kdims=['M', '$dt$'],\n", " vdims=hv.Dimension('z', label='$|c_n|$', range=(np.log(accuracy)/np.log(10), 0))\n", ").opts(logy=True, logz=False, colorbar=True, fig_size=200) * \\\n", "hv.Curve((np.array([func(dt, *p_acc16) for dt in dt_array]),\n", " dt_array)).opts(logy=True, logx=True)\n" ] }, { "cell_type": "markdown", "id": "63ad6320-1f42-4556-a77e-2eb18314fd0e", "metadata": {}, "source": [ "## Result\n", "\n", "Finally, these parameters give the best choice of `num_moments`\n", "as a function of `dt` for\n", "typical accuracy goals: `1e-8`, `1e-16`, `1e-32`." ] }, { "cell_type": "code", "execution_count": 12, "id": "a6c802ad", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'p_acc32 = [0.28132266, 0.32958046, 2.04428473, 0.09146841]'" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "'p_acc8 = [0.42464678, 0.33500138, 2.00044861, 0.07265139]'\n", "'p_acc16 = [0.34310197, 0.33722835, 2.01232381, 0.08868359]'\n", "'p_acc32 = [0.28132266, 0.32958046, 2.04428473, 0.09146841]'\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "548000c4-04cc-4f1a-8977-aadeef099537", "metadata": {}, "source": [ "From the scaling analysis we can see that large time steps are possible, and require a proportionally larger amount of moments in the KPM expansion.\n", "\n", "Smaller timesteps have a slight overhead in the number of moments:\n", "$m^*=4$ for very small time steps of $10^{-3}$ but only $m^*=6$ for a ten times larger time step $10^{-2}$.\n", "\n", "Thus, it is more convenient -in terms of performance- to make one big time step, rather than many subsequent smaller time steps.\n", "\n", "Nevertheless, it is still fairly efficient to make many time steps for visualizing the time evolution." ] } ], "metadata": { "kernelspec": { "display_name": "kt", "language": "python", "name": "kt" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.13" } }, "nbformat": 4, "nbformat_minor": 5 }