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lec28_11202006 - 10.34 Numerical Methods Applied to...

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10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #28: Guest lecture on Monte Carlo / MD. Intro to MC Methods random points stochastic – element of randomness contrast with standard integration algorithms when is MC useful? trapezoid point in curve = 1 From Friday point out curve = 0 >= < dx x p x f f ) ( ) ( integral of p(q)f(q)dq where p(q) is probability distribution could do by sampling Comparison of Accuracy MC – accuracy ~ N -0.5 effect of dimension Other methods – accuracy ~ N ~1/d on accuracy Figure 1. Trapezoidal rule versus Monte Carlo integration. Random States calculation of area of hyper-sphere for calculation of Pi o chance of hit Æ 0 Importance sampling – concentrates sampling in regions of higher probability dx x p x p x p x f dx x p x f ) ( ) ( ) ( ) ( ) ( ) ( Ergodicity MC often used to simulate time-dependent processes, although there is no “time” in MC simulations
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