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monte_carlo_sampling - The Monte Carlo sampling What is...

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The Monte Carlo sampling
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What is Monte Carlo? A method to evaluate integrals Popular in all areas of science, economics, sociology, etc. Involves: Random sampling of domain Evaluation of function Acceptance/rejection of points
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Why Monte Carlo? The domain of integration is complex: Complex environment lighting The integrals are very high-dimensional: Solution of the radiative transfer equation
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Basic Monte Carlo Algorithm Suppose we want to approximate in a high-dimensional space For i = 1 to n Pick a point x i at random Accept or reject the point based on criterion If accepted, then add f(x i ) to total sum Error typically decays as ( 29 Z f d = x x 1 N
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Example: multiple dimensions What is the average of a variable for a N - dimensional probability distribution? Two approaches: Quadrature Discretize each dimension into a set of n points Possibly use adaptivity to guide discretization For a reasonably smooth function, error decreases as n -N/2 Monte Carlo Sample m points from the space Possibly weight sampling based on reference function Error decreases as m -1/2
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Problems: sampling tails of distributions We want to Integrate a sharply-peaked function Use Monte Carlo with uniformly-distributed random numbers What happens? Very few points contribute to the integral (~9%) Poor computational efficiency/convergence Can we ignore the tails? NO! Solution: use a different distribution -1 -0.5 0.5 1 0.2 0.4 0.6 0.8 1 ( 29 ( 29 ( 29 12 exp 100 f x x = - 0.2 0.4 0.6 0.8 1 200 400 600 800
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Improved sampling: change of variables One way to improve sampling is to change variables: New distribution is flatter Uniform variates more useful Advantages: Simplicity Very useful for generating distributions of non-uniform variates (coming up) Disadvantages Most useful for invertible functions
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Change of variables: method 1. Given an integral 1. Transform variables 1. Choose variables to give (nearly) flat distribution 1. Integrate ( 29 1.
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