Lecture 008

Numerical Integration

Trapezoid Rule

Trapezoid Rule

Curse of Dimensionality in Trapezoid Rule with Multidimensional Integrals and Fubini's Theorem

Curse of Dimensionality in Trapezoid Rule with Multidimensional Integrals and Fubini's Theorem

The basic idea is: as dimension increase, you need more sample points. So only Monte Carlo can save us.

Desired Properties of Uniform Random:

Modern example is PCG32 (pcg-random.org)

But estimating with average of random sample will give us wrong estimation.

Wrong Estimation of klzzwxh:0002 with naive summation and average

Wrong Estimation of f(X) with naive summation and average

Solution: Approximate f by averaging realizations of a random variable X, and explicitly accounting for its PDF:

F \simeq \frac{1}{N} \sum_{i = 1}^N \frac{f(x_i)}{p(x_i)}

Correct, on everage, Monte Carlo Integration. With assumption that klzzwxh:0006 when klzzwxh:0007 (ie. assuming unbiased klzzwxh:0008)

Correct, on everage, Monte Carlo Integration. With assumption that p(X_i) \neq 0 when f(X_i) \neq 0 (ie. assuming unbiased p)

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Derivation of: For an unbiased estimator, its average error is equal to its variance!

Derivation of: For an unbiased estimator, its average error is equal to its variance!

Variance and STD of the Monte Carlo Error

Variance and STD of the Monte Carlo Error

Pros

Cons

Monte Carlo Sampling by Choosing p(x)

Rejection Sampling:

Change of variable in transforming between distributions

Change of variable in transforming between distributions

Example: Polar Coordinate Parameterization

Example: Polar Coordinate Parameterization

Derivation of Spherical Parameterization

Derivation of Spherical Parameterization

Result

Result

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