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Monte Carlo Crash Course: Quasi-Monte Carlo
Probability - Exponentially Better Integration - Sampling - Case Study: Rendering - Quasi-Monte Carlo - Coming Soon… Quasi-Monte Carlo We’ve learned how to define and apply Monte Carlo integration—fundamentally, it’s the only tool we need. In the remaining chapters, we’ll explore ways to reduce variance and successfully sample difficult distributions.
Instead of attempting to distribute samples across an exponential number of regions, a Latin hypercube sampler stratifies each dimension independently. Taking $$\Omega$$ to be the unit square, $$D^*_N$$ is defined as the worst-case difference between the ratio of samples falling inside a rectangle $$\mathcal{R}$$ and the volume of $$\mathcal{R}$$. While this convergence rate is always asymptotically faster than $$\frac{1}{\sqrt{N}}$$, in high dimensions, the associated constant factor (which depends on $$d$$) can still make the approach impractical.
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