Monte Carlo simulation of the Young measures — Comparison of random-number generators

"Young measure" is an abstract notion from mathematical measure theory growing up from the analysis of some variational problems. From the formal point of view it may be regarded as a continuous linear functional defined on the space of Carathéodory integrands. However, calculating an explicit form of specific Young measure based on its formal theoretical definition involves so-called "weak" limits of the sequences of functions satisfying some general conditions. From a strictly mathematical standpoint, it is quite difficult task which can be solved only in the most simple cases. However in many real-world applications it would be enough to learn only some of the most important probabilistic characteristics of the Young distribution. In such a case the possible solution is to adopt Monte Carlo techniques. In the presentation we propose three different algorithms designed for simulating random variables distributed according to the Young measures associated with piecewise functions. Next with the help of computer simulations we compare their statistical performance via some benchmarking problems. In this study we focus on the accurateness of the distribution of the generated sample.