Abstract: Estimating an expectation or integral is important in high energy physics, Bayesian inference, image rendering, quantitative finance, and uncertainty quantification. Monte Carlo type methods are commonly used. The numerical error can be expressed as a product of three quantities: one measuring the deficit in the sampling scheme, a second measuring the roughness of the function defining the expectation or integral, and a third representing the confounding between that function and the sampling deficit.  

We explain how low discrepancy sampling, also known as the quasi-Monte Carlo method, can substantially improve the efficiency of these calculations. We discuss how to improve efficiency via transformations of the integral. Our data-driven error bounds advise the user when to stop simulating. We illustrate low discrepancy sampling via our QMCPy software library.