31:[] 2f:{"title":"Monte-Carlo Integration","cells":[{"id":0,"value":"# Monte-Carlo Integration","pinned":false,"hidden":false,"mode":"md"},{"id":275,"value":"An example of the numerical Monte-Carlo integration technique for the function ${tex` f(x)=4\\cdot \\sqrt{1-x^2}`}.\n\nThe analytical solution of the following integral is ${tex`\\int_0 ^1 f(x)=\\pi\\approx 3.1416`}","pinned":false,"hidden":false,"mode":"md"},{"id":286,"value":"### Monte-Carlo simulation","pinned":false,"hidden":false,"mode":"md"},{"id":89,"value":"The integral has the approximate value of ${`${result.integral}`} for ${`${iterations}`} iterations.","pinned":false,"hidden":false,"mode":"md"},{"id":104,"value":"viewof reload = Inputs.button(\"reload integral\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":14,"value":"Plot.plot({\n grid: true,\n marks: [\n Plot.ruleY([0]),\n Plot.line(curve, {x: (d) => d[0], y: (d) => d[1]}),\n Plot.dot(result.pointsInside, {x: \"x\", y: \"y\", stroke: \"orange\"}),\n Plot.dot(result.pointsOutside, {x: \"x\", y: \"y\", stroke: \"#76b7b3\"}),\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":119,"value":"viewof histoSamples = Inputs.range([1, 100], {value: 10, label: \"How many samples should be calculated?\", step: 1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":138,"value":"For ${`${histoSamples}`} samples, with ${`${iterations}`} iterations respectively, the mean value of the integral is ${`${Math.round(d3.mean(histoSameIter.map((d) => d.integral))*10000)/10000}`}, with a variance of ${`${Math.round(d3.variance(histoSameIter.map((d) => d.integral))*10000)/10000}`}.","pinned":false,"hidden":false,"mode":"md"},{"id":123,"value":"Plot.plot({\n y: {grid: true},\n marks: [\n Plot.rectY(histoSameIter, {x1: histoSameIter.map((d, ind) => ind), x2: histoSameIter.map((d, ind) => ind+1), y: \"integral\", insetRight: 1}),\n Plot.ruleY([0]),\n Plot.ruleY([3.14]),\n Plot.axisX({anchor: \"bottom\", label: \"sample\"}),\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":228,"value":"The variance of ${tex`N_{MC}=10, 100, 1000, 10000, 100000, 1000000`} iterations is ${`${Math.round(d3.deviation(histoDifferentIterFixed.map((d) => d.integral))*1000)/1000}`}","pinned":false,"hidden":false,"mode":"md"},{"id":216,"value":"Plot.plot({\n y: {grid: true},\n x: {type: \"log\"},\n marks: [\n Plot.rectY(histoDifferentIterFixed, {x1: histoDifferentIterFixed.map((d, ind) => d.iterations), x2: histoDifferentIterFixed.map((d, ind) => d.iterations+10**ind), y: \"integral\", insetRight: 1}),\n Plot.ruleY([0]),\n Plot.ruleY([3.14]),\n Plot.axisX({anchor: \"bottom\", label: \"Iterationen\"}),\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":158,"value":"viewof maxIterations = Inputs.range([100, 50000], {value: 1000, step: 100, label: \"Maximum iterations\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":198,"value":"The mean value of ${maxIterations}/100=${maxIterations/100} samples, with different number of iterations, is ${`${Math.round(d3.mean(histoDifferentIter.map((d) => d.integral))*10000)/10000} ± ${Math.round(d3.variance(histoDifferentIter.map((d) => d.integral))*10000)/10000}`}.","pinned":false,"hidden":false,"mode":"md"},{"id":170,"value":"Plot.plot({\n y: {grid: true},\n marks: [\n Plot.rectY(histoDifferentIter, {x1: histoDifferentIter.map((d, ind) => d.iterations), x2: histoDifferentIter.map((d, ind) => d.iterations+100), y: \"integral\", insetRight: 1, label: \"Iterationen\"}),\n Plot.ruleY([0]),\n Plot.ruleY([3.14]),\n Plot.axisX({anchor: \"bottom\", label: \"Iterationen\"}),\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":210,"value":"## Functions","pinned":false,"hidden":false,"mode":"md"},{"id":5,"value":"f = (x) => {\n return 4*Math.sqrt(1-x**2)\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":213,"value":"histoDifferentIterFixed = {\n const iterations = [10, 100, 1000, 10000, 100000, 1000000];\n const results = [];\n return iterations.map((iter) => {\n let result = integral(iter);\n result[\"iterations\"] = iter;\n return result\n })\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":181,"value":"histoDifferentIter = {\n const results = [];\n for (let i=100; i {\n reload;\n const pointsOutside = [];\n const pointsInside = [];\n const yRange = 4;\n for (let i =0; i< maxIter; i++) {\n const randPointX = Math.random();\n const randPointY = Math.random()*yRange;\n if (f(randPointX) >= randPointY) {\n pointsInside.push({\n x: randPointX,\n y:randPointY\n })\n } else pointsOutside.push({\n x: randPointX,\n y: randPointY\n })\n }\n return {\n integral: pointsInside.length/maxIter*yRange,\n pointsInside: pointsInside,\n pointsOutside: pointsOutside\n }\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":97,"value":"iterations = 1000","pinned":false,"hidden":false,"mode":"ojs"},{"id":70,"value":"result = integral(iterations)","pinned":false,"hidden":false,"mode":"ojs"},{"id":12,"value":"curve = {\n const pointsPerUnit = 1000;\n const curve = [];\n for(let i=0; i<=1.1; i+=1/pointsPerUnit) {\n curve.push([i, isNaN(f(i)) ? 0:f(i)])\n }\n return curve\n}","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W31","stdlib":"1","titling":"auto"}