The Monte Carlo Solution to Endless Iterated Integrals
Transcript of The Monte Carlo Solution to Endless Iterated Integrals
BY: PAULANA HALLFORT VALLEY STATE UNIVERSITY
From Impossible to a Split Second:
The Monte Carlo Solution To Endless Iterated Integrals
ABOUT• Atlanta, GA• Third year
student, The Fort Valley State University
• Double major: Mathematics and Computer Science
• Avid Programmer• Mother and
Engaged
PROBABILITY
Theory of Sometimes
Likelihood of an outcome between 0 and 1
Probability of an event happening
=
Number of ways it can happen
Total number of outcomes
LAW OF LARGE NUMBERS
•Perform a test a large number of times•Average of results are close to expected value•“Guarantees” stable results over long-term experiments•Cannot be applied to small number of trials
MONTE CARLO SIMULATION
•Used in ’30s and ‘40s by atomic physicists•Used to predict the outcome of chemical reactions•Physicists were fans of gambling•Hence name “Monte Carlo”
MODERN UTILITY
•Entities using MC:•General Motors•Procter and Gamble•Eli Lilly•Sears•National Labs (ORNL, Los Alms, etc.)•Major Wall Street Firms•Financial Planners•Engineering and Research Organizations
MODERN UTILITY
•Uncertainty Estimations•Complex Scientific Calculations•Reliability Engineering•Estimating average return•Riskiness of new products•Projecting net income•Predicting costs•Optimal capacities
THE IDEA• Box off the region of integration
• Calculate the area of the box
• Randomly place points in the box
• Count number of points in the box, Nb
• Count number of points under the function, Nf
• Area of f(x) = Area of box * (Nf/Nb)
SIMPLE INTEGRALS
The Monte Carlo Technique
Mean Value Theorem
Sample f(x) uniformly xi = a + (b - a)rand()
Large number of Samples 1,000 – 10,000 – 50,000
Estimation error𝜎
√𝑁
Random Numbers
xi = a + (b - a)rand()
MONTE CARLO METHOD
1.Generate random numbers for each variable
2.Evaluate them at the function
3.Average the calculated values
4.Multiply by the difference in the limits
WHY?
Pros•Diversify curriculum
•Alternate methods•Increase understanding
•Approach more difficult problems
Cons•Accessibility•Application•Ease of use/manipulation
EXCEL IMPLEMENTATION
xi = 1 +(3 - 1)rand()f(xi) = Xi
2 for each xi
f(xi)/N * (3-1) = final estimation
EXCEL IMPLEMENTATION
•Integral converges as samples increases
•For N = 1000: 8.76027356 vs. 8.6666 estimated actual•Error: 1.080079613 %•Not efficient for single integrals
•f(x) = x^2 + y^2•X (-2,2) Y (-2,2)
•N = 1000•Estimate: 42.725187
•Actual: 42.66666667
•Error: 0.137157187%
EXCEL IMPLEMENTATION
MONTE CARLO INTEGRATION
Upper Lim it 1 f(x) = (x+3)/sqrt(4+(x̂ 2)) estim ate errorLower Lim it 0 actual
Error. Check your internet connection and try again. Speed: 100 Values: 1000 (M ax: 30000)x f(x) avg final error
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PURPOSE
•Calculate complex, multi-dimensional integrals•
•Reduce calculations from traditional methods• Discretization of integrals• For 100 points in each integration there are 10012 =1024 calculations
• Assuming 1 Giga evaluations/sec, it would take over 107 years!
POWER OF MC
For N = 106 , there are ~106 calculations
At 1 Giga evaluations/sec it would take ~10-3 sec
CONCLUSION
•Powerful method to estimate complex integrals
•Little computational or coding effort•Does not provide exact value•Approximate values good enough for practical application
•Larger dimensional integrals can be easily calculated
•Other methods may make calculations impossible
ACKNOWLEDGEMENTS
•Fort Valley State University•FVSU Chapter of Association for Computing Machinery•Dr. Masoud Naghedolfeizi – Project Advisor•Dr. Sanjeev Arora - Grant Provider"Establishing an Undergraduate STEM Teaching and Research Laboratory at FVSU“•Dr. Dawit Aberra – FVSU Dean of Mathematics and Computer Science