What are the advantages and disadvantages of quasi-Monte Carlo methods over random sampling?

1 What is random sampling?

Random sampling is a method of numerical integration that uses randomly generated points to approximate the value of an integral. By sampling the integrand function at these points and then averaging the results, you can get an estimate of the integral. However, this approach can be inefficient, especially for high-dimensional integrals where the points may be sparse and miss important regions of the integrand. Additionally, random sampling can be inaccurate due to the variability and uncertainty of the random points, which may introduce errors or biases in the estimate. Furthermore, it can be difficult to assess the quality and convergence of the estimate as there is no clear criterion or error bound for random sampling.

2 What is quasi-Monte Carlo?

Quasi-Monte Carlo is a method of numerical integration that uses deterministic sequences of points to estimate the value of an integral. The idea is to use points that are more evenly distributed and cover the integration domain better than random points. Quasi-Monte Carlo sequences are designed to have low discrepancy, which means that they avoid clustering and gaps, and mimic the properties of uniform random points. Some examples of quasi-Monte Carlo sequences are the Halton, Sobol, and Niederreiter sequences.

3 What are the advantages of quasi-Monte Carlo?

Quasi-Monte Carlo methods offer a number of advantages over random sampling methods, such as increased efficiency, accuracy, and reliability. These methods require fewer points than random sampling to achieve the same or better accuracy, particularly for high-dimensional integrals. Furthermore, they reduce the variance and error of the estimate, and have deterministic error bounds that depend on the discrepancy of the sequence and the smoothness of the integrand. In addition, they are not reliant on the quality or randomness of the random number generator, and can be reproduced and verified easily.

4 What are the disadvantages of quasi-Monte Carlo?

Quasi-Monte Carlo methods, while advantageous in certain scenarios, do have some drawbacks. These methods require more knowledge and expertise to select and implement the right sequence and algorithm for the problem. Additionally, they assume that the integrand is smooth and periodic, which may not be ideal for integrands with singularities, discontinuities, or irregular boundaries. Furthermore, these methods can be sensitive to aliasing or cancellation effects, which occur when the sequence and integrand have similar patterns or frequencies that interfere with one another.

5 How to choose the best method?

When it comes to numerical integration, there is no one-size-fits-all solution. It depends on the problem, resources, and desired accuracy. Generally speaking, random sampling is best for simple or low-dimensional integrals that require a quick estimate with minimal effort. For complex or high-dimensional integrals, quasi-Monte Carlo is more precise and robust. Additionally, it’s important to compare and test different methods and sequences to evaluate their performance and quality.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?