A MEASURE-THEORETIC INTERPRETATION OF SAMPLE BASED NUMERICAL INTEGRATION WITH APPLICATIONS TO INVERSE AND PREDICTION PROBLEMS UNDER UNCERTAINTY

摘要

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 inverse problems.