A Measure-Theoretic Interpretation of Sample Based Numerical Integration with Applications to Inverse and Prediction Problems under Uncertainty

The integration of functions over measurable sets is a fundamental problem in computational science. When the measurable sets belong to high-dimensional spaces or the function is computationally complex, it may only be practical to estimate integrals based on weighted sums of function values from a finite collection of samples. Monte Carlo, quasi--Monte Carlo, and other (pseudo-)random schemes are common choices for determining a set of samples. These schemes are appealing for their conceptual ease and ability to circumvent, with various degrees of success, the so-called curse of dimensionality. However, convergence is often slow and described in terms of probability. We consider a general measure-theoretic interpretation of any sample based algorithm for numerically approximating an integral. A priori error bounds are proven that provide insight into defining adaptive sampling algorithms solving error optimization problems. We use these bounds to improve integral approximations for both forward and inver...