Exploring Monte Carlo Integration Methods for Advanced Applications
Uncover the fascinating history and applications of Monte Carlo integration, from its origins in nuclear physics to its diverse uses in computational science, statistics, and engineering. Dive into the specifics of crude Monte Carlo integration, advanced convergence techniques, interesting discoveries, and practical coding examples, highlighting its slow but dimension-independent convergence nature. Discover the power and potential of Monte Carlo integration for high-dimensional evaluations.
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Presentation Transcript
Monte Carlo Integration Madison Redman MAT360 Project #2
History of Monte carlo methods o Developed and formalized by notable Math and Science figures Fermi, Ulam, von Neumann, Metropolis, and others o The Manhattan Project and the Atomic Bomb o Used to study the probabilistic behavior of neutron transport in fissile materials o Started in nuclear physics o Now used in many different disciplines o Computational science o Statistics o Computer graphics o Engineering
Overview of Monte carlo methods o Random or stochastic sampling o Law of Large Numbers o Monte Carlo Simulation o Hit or Miss Monte Carlo Method o Monte Carlo Integration o Technique of estimating or averaging using random samples
Specifically: crude Monte carlo Integration ? ? ?(?) ? ? ? ?(??) ? ?=0 ? for some n = number of points (nodes) And points are randomly selected using uniform distribution
More on Mc integration o Converges with order O( ?) o Accelerated Convergence o Quasi-MC Int. converges with order O((log(?))?1 ?) o Importance Sampling, cuts error in half o Actually, fairly slow to converge o But, independent of dimension, so preferable for higher-dimension problems
Interesting discoveries ??(?) ? ? ? Two points, end points: ? 2[? ? + ?(?)] 2[? ?0 + ?(?0+ )] Very big generalization, chances are very slim that a and b are randomly selected BUT, looks like the Trapezoid Rule!! ?0+2 ?(?) ? And, look at Simpson s Rule: ?0 3[? ?0 + 4? ?0+ + ?(?0+ 2 )] Again, looks very similar!
A little bit of code o Uniform distribution function o Random Sampling o Different result each time
So, very slow to converge BUT, preferable for high-dimension evaluations An overlook on mc integration Used in many different fields Especially: Computer graphics Nuclear physics Engineering
Bibliography https://cs.dartmouth.edu/~wjarosz/publications/dissertation/appendixA.pdf https://towardsdatascience.com/the-basics-of-monte-carlo-integration-5fe16b40482d https://www.scratchapixel.com/lessons/mathematics-physics-for-computer-graphics/monte-carlo- methods-mathematical-foundations/quick-introduction-to-monte-carlo-methods.html https://www.math.pku.edu.cn/teachers/litj/notes/numer_anal/MCQMC_Caflisch.pdf https://towardsdatascience.com/the-basics-of-monte-carlo-integration-5fe16b40482d https://docs.octave.org/v4.2.2/Random-Number-Generation.html
