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Unformatted text preview: AMSC/CMSC 660 Scientific Computing I Fall 2006 UNIT 4: Monte Carlo Methods Dianne P. OLeary c 2002,2004,2006 What is a MonteCarlo method? In a MonteCarlo 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 pseudoMonte Carlo methods, which use samples that are more systematically chosen. In some sense, all practical computational methods are pseudoMonte Carlo , since random number generators implemented on machines are generally not truly random. So the distinction between the methods is a bit fuzzy. But well use the term Monte Carlo for samples that are generated using pseudorandom 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 (nonstochastic) methods. The Plan: Basic statistics: Random and pseudorandom 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 pseudorandom 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 ) . 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 particles every seconds, then the time between two successive emissions has the exponential distribution with mean . ( Note: This is a special case of the Gamma distribution...
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