On Simulation of the Young Measures – Comparison of Random-Number Generators

"Young measure" is an abstract notion from mathematical measure theory. Originally, the notion appeared in the context of some variational problems related to the analysis of sequences of “fast” oscillating of functions. From the formal point of view the Young measure may be treated as a continuous linear functional defined on the space of Carathéodory integrands satisfying certain regularity conditions. Calculating an explicit form of specific Young measure is a very important task. However, from a strictly mathematical standpoint it is a very difficult problem not solved as yet in general. Even more difficult would be the problem of calculating Lebasque's integrals with respect to such measures. Based on known formal results it can be done only in the most simple cases. On the other hand in many real-world applications it would be enough to learn only some of the most important probabilistic characteristics of the Young distribution or learn only approximate values of the appropriate integrals. In such a case a 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.