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![The first thoughts and attempts I made to practice [the Monte Carlo Method]
were suggested by a question which occurred to me in 1946 as I was
convalescing from an illness and playing solitaires. The question was what are
the chances that a Canfield solitaire laid out with 52 cards will come out
successfully? After spending a lot of time trying to estimate them by pure
combinatorial calculations, I wondered whether a more practical method than
"abstract thinking" might not be to lay it out say one hundred times and simply
observe and count the number of successful plays. This was already possible to
envisage with the beginning of the new era of fast computers, and I immediately
thought of problems of neutron diffusion and other questions of mathematical
physics, and more generally how to change processes described by certain
differential equations into an equivalent form interpretable as a succession of
random operations. Later [in 1946], I described the idea to John von Neumann,
and we began to plan actual calculations.
–- Stanislaw Ulam
https://en.wikipedia.org/wiki/Monte_Carlo_method](https://image.slidesharecdn.com/ychao20160820-160821054423/85/slide-15-320.jpg)
![import random
import numpy
import ROOT
m_total = 10**5
m_radius = 1
def in_circle(point):
x = point[0]
y = point[1]
return (x**2 + y**2) < m_radius**2
m_count = m_inside = 0
c1 = ROOT.TCanvas("pi", "#pi", 600, 600)
m_x = numpy.linspace(0, 1.,101)
m_y = (1.**2 - m_x**2)**0.5
c1.cd()
g = ROOT.TGraph(len(m_x), m_x, m_y)
g.SetTitle("#pi")
g.GetXaxis().SetTitle("x")
g.GetYaxis().SetTitle("y")
hist1 = ROOT.TH2F("hist1", "#pi outside", 100, 0, 1, 100, 0, 1)
hist2 = ROOT.TH2F("hist2", "#pi inside", 100, 0, 1, 100, 0, 1)
for i in range(m_total):
point = random.random()*m_radius, random.random()*m_radius
if in_circle(point):
hist2.Fill(point[0], point[1])
m_inside += 1
else:
hist1.Fill(point[0], point[1])
m_count += 1
pi = (m_inside * 1.0 / m_count) * 4
print(pi)
hist1.SetMarkerColor(4)
hist1.Draw("same")
hist2.SetMarkerColor(2)
hist2.Draw("same")
g.SetLineWidth(3)
g.Draw("same")
c1.SaveAs("pi.png")
$ python pyroot_pi.py
3.14108
git@github.com:yuanchao/MCnVC.git](https://image.slidesharecdn.com/ychao20160820-160821054423/85/slide-23-320.jpg)
Uploaded byYuan CHAO
蒙地卡羅模擬與志願運算
本文件介绍了数值模拟和志愿计算，特别是蒙特卡洛方法在粒子物理中的应用，例如计算圆周率和布丰投针问题。还讨论了参与大型强子对撞机实验的志愿计算项目，并且介绍了开放数据的概念和获取方式。最后，提供了关于如何参与相关计算的指导。
![[ZigBee 嵌入式系統] ZigBee 應用實作 - 使用 TI Z-Stack Firmware](https://cdn.slidesharecdn.com/ss_thumbnails/zigbeeappimplementation-150613072040-lva1-app6891-thumbnail.jpg?width=640&height=640&fit=bounds)
蒙地卡羅模擬與志願運算
- 1. 數值模擬與志願運算數值模擬與志願運算 ---- 輕鬆上手參加輕鬆上手參加 LHCLHC實驗實驗 Yuan CHAO ( 趙元 ) (National Taiwan University, Taipei, Taiwan) COSCUP 2016/08/20-21
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- 4. 4 簡介簡介 數值計算模擬數值計算模擬 蒙地卡羅方法蒙地卡羅方法 面積法計算面積法計算 ππ 布豐投針問題布豐投針問題 參與參與 LHCLHC實驗實驗 志願運算志願運算 公開數據公開數據
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- 7. 7 標準模型標準模型 Standard ModelStandardModel ~10-18 m宇宙的尺度 http://htwins.net/scale2/ ~10-1 m 膠子 光子 W/Z 子 重力子 強作用力 電磁力 弱作用力 重力 夸 克 輕 子 奈米 =10-9 m
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- 13. 蒙地卡羅法 ( 近似解) 拉斯維加斯法 ( 正解 )
- 14. 蒙地卡羅法 ( 給定時間) 拉斯維加斯法 ( 未知時間 )
- 15. The first thoughtsand attempts I made to practice [the Monte Carlo Method] were suggested by a question which occurred to me in 1946 as I was convalescing from an illness and playing solitaires. The question was what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count the number of successful plays. This was already possible to envisage with the beginning of the new era of fast computers, and I immediately thought of problems of neutron diffusion and other questions of mathematical physics, and more generally how to change processes described by certain differential equations into an equivalent form interpretable as a succession of random operations. Later [in 1946], I described the idea to John von Neumann, and we began to plan actual calculations. –- Stanislaw Ulam https://en.wikipedia.org/wiki/Monte_Carlo_method
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- 19. Pie are squareπr2 r
- 20. Pie are squareπr2 對稱 r
- 21. Pie are squareπr2 對稱 r
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- 23. import random import numpy importROOT m_total = 10**5 m_radius = 1 def in_circle(point): x = point[0] y = point[1] return (x**2 + y**2) < m_radius**2 m_count = m_inside = 0 c1 = ROOT.TCanvas("pi", "#pi", 600, 600) m_x = numpy.linspace(0, 1.,101) m_y = (1.**2 - m_x**2)**0.5 c1.cd() g = ROOT.TGraph(len(m_x), m_x, m_y) g.SetTitle("#pi") g.GetXaxis().SetTitle("x") g.GetYaxis().SetTitle("y") hist1 = ROOT.TH2F("hist1", "#pi outside", 100, 0, 1, 100, 0, 1) hist2 = ROOT.TH2F("hist2", "#pi inside", 100, 0, 1, 100, 0, 1) for i in range(m_total): point = random.random()*m_radius, random.random()*m_radius if in_circle(point): hist2.Fill(point[0], point[1]) m_inside += 1 else: hist1.Fill(point[0], point[1]) m_count += 1 pi = (m_inside * 1.0 / m_count) * 4 print(pi) hist1.SetMarkerColor(4) hist1.Draw("same") hist2.SetMarkerColor(2) hist2.Draw("same") g.SetLineWidth(3) g.Draw("same") c1.SaveAs("pi.png") $ python pyroot_pi.py 3.14108 git@github.com:yuanchao/MCnVC.git
- 24. 布豐投針問題 George Louis, Comtede Buffon in 18th https://en.wikipedia.org/wiki/Georges-Louis_Leclerc,_Comte_de_Buffon
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- 26. 布豐投針法求 π Mario Lazzariniin 1901 https://en.wikipedia.org/wiki/Buffon%27s_needle 拋了3408次針，得到π的近似值為355/113。
- 27. 布豐投針法求 π 對稱性 考慮針的中心點 x落在 的機率都相同，針的方向也是 與線條相交時的距離 所以相交的機率為 https://en.wikipedia.org/wiki/Buffon%27s_needle a l b t http://blog.linux.org.tw/~jserv/archives/002004.html
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- 36. CMS 實驗組 ~3000 科學家 182研究單位 來自 42 個國家 ~130 電算中心 200k – 250k jobs 同時執行 使用 WLCG 運算網格
- 37. 37 Atlas@homeAtlas@home 的經驗的經驗 也執行實際的模擬工作也執行實際的模擬工作 每個批次每個批次 2525個事例個事例 (( 正常為正常為 10001000 事例事例 )) 執行時間執行時間 1 – 41 – 4 小時小時 (( 正常為正常為 88 核心核心 1212 小時小時 )) 虛擬環境映象檔虛擬環境映象檔 ~500 MB (~500 MB ( 展開約展開約 1GB )1GB ) 下載數據下載數據 1 – 100 MB1 – 100 MB 回傳數據回傳數據 ~100MB~100MB 虛擬環境記憶體需求虛擬環境記憶體需求 2GB2GB 如何參與？如何參與？ 下載下載 BOINCBOINC 用戶端軟體用戶端軟體 (Mac, Linux, Windows)(Mac, Linux, Windows) 選擇選擇 Atlas@homeAtlas@home 或或 CMS@homeCMS@home 並建立帳號並建立帳號 開始計算！開始計算！ BOINCBOINC 軟體可以排程，在閒置時執行軟體可以排程，在閒置時執行 http://boinc.berkeley.edu/
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