Monte Carlo Computations notes

# Monte Carlo Computations notes - AMSC/CMSC 660 Scientific...

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Unformatted text preview: AMSC/CMSC 660 Scientific Computing I Fall 2006 UNIT 4: Monte Carlo Methods Dianne P. O’Leary c ° 2002,2004,2006 What is a Monte-Carlo method? In a Monte-Carlo method , the desired answer is formulated as a quantity in a stochastic model and estimated by random sampling of the model. Example: • If we have a cube, with the sides numbered 1 to 6, we might toss the cube 120 times, observe which side comes up on top each time, and study whether the sides occur with approximately equal frequency. • If we have a “black box” that takes a number between 0 and 1 as input and emits a number between 0 and 1 as an output, we could feed the box m numbers and observe the m outputs of the box and use the average of the observations as an estimate of the statistical mean of the process defined by the black box. Two basic principles • There is an important difference between – Monte Carlo methods, which estimate quantities by random sampling, and – pseudo-Monte Carlo methods, which use samples that are more systematically chosen. In some sense, all practical computational methods are pseudo-Monte Carlo , since random number generators implemented on machines are generally not truly random. So the distinction between the methods is a bit fuzzy. But we’ll use the term Monte Carlo for samples that are generated using pseudo-random numbers generated by a computer program. • Monte Carlo methods are (at least in some sense) methods of last resort . They are generally quite expensive and only applied to problems that are too difficult to handle by deterministic (non-stochastic) methods. The Plan: • Basic statistics: Random and pseudo-random numbers and their generation 1 • Monte Carlo methods for numerical integration • Monte Carlo methods for optimization • An example of Monte Carlo methods for counting Basic statistics: Random and pseudo-random numbers and their generation • What is a random number? • What are the mean and variance of a random sample? • What is a distribution? What are its mean and variance? • How are pseudorandom numbers generated? Examples of how to generate random numbers • Take n papers and number them 1 to n . Put them in a box, and draw one at random. After you record the resulting number, put the paper back in the box. You are taking random numbers that are uniformly distributed among the values { 1 , 2 ,... ,n } . • Make a spinner by anchoring a needle at the center of a circle. Draw a radius line on the circle. Spin the needle, and measure the angle it forms with the radius line. You obtain random numbers that are uniformly distributed on the interval [0 , 2 π ) . • The First Six Million Prime Numbers , C.L Baker and F. J. Gruenberger, The Microcard Foundation, Madison, WI, 1959 • If, on average, a radioactive substance emits α-particles every μ seconds, then the time between two successive emissions has the exponential distribution with mean μ . ( Note: This is a special case of the Gamma distribution...
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Monte Carlo Computations notes - AMSC/CMSC 660 Scientific...

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