Part 2: Monte Carlo Integration

I experimented with different numbers of samples to examine the convergence rate. I was actually surprised by how good the results were with only 10 samples. Interestingly, the stratified uniform sampling seemed to converge about as rapidly as the stratified combined importance sampling. It also looks like it's far better to simply not use importance sampling than use a poorly suited PDF.

  10 100 1000 10000 100000 1000000
Unstratified Uniform 1.298770760 1.316952491 1.329541091 1.335195991 1.333040189 1.333390281
Stratified Uniform 1.336215529 1.333698606 1.333337135 1.333332810 1.333333338 1.333333333
Stratified x2 1.133689474 1.137735518 1.767145275 1.309771723 1.312213917 1.325302278
Stratified (π/2-x)2 1.589574899 1.127640444 1.239445876 1.318168193 1.327149472 1.323831119
Stratified Combined 1.344254876 1.334577371 1.333194932 1.333333752 1.333333330 1.333333332
Actual 1.333333333

Last modified: Mon May 23 16:37:50 MDT 2005