Monte Carlo Methods and the Koksma-Hlawka Inequality

• Main Authors: Sergey Ermakov, Svetlana Leora
• Format: Article
• Language: English
• Published: MDPI AG 2019-08-01
• Series: Mathematics
• Subjects: Monte Carlo method; quasi-Monte Carlo method; Koksma-Hlawka inequality; quasi-random sequences; stochastic processes
• Online Access: https://www.mdpi.com/2227-7390/7/8/725

The solution of a wide class of applied problems can be represented as an integral over the trajectories of a random process. The process is usually modeled with the Monte Carlo method and the integral is estimated as the average value of a certain function on the trajectories of this process. Solving this problem with acceptable accuracy usually requires modeling a very large number of trajectories; therefore development of methods to improve the accuracy of such algorithms is extremely important. The paper discusses Monte Carlo method modifications that use some classical results of the theory of cubature formulas (quasi-random methods). A new approach to the derivation of the well known Koksma-Hlawka inequality is pointed out. It is shown that for high (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>s</mi> <mo>&gt;</mo> <mn>5</mn> </mrow> </semantics> </math> </inline-formula>) dimensions of the integral, the asymptotic decrease of the error comparable to the asymptotic behavior of the Monte Carlo method, can be achieved only for a very large number of nodes <i>N</i>. It is shown that a special criterion can serve as a correct characteristic of the error decrease (average order of the error decrease). Using this criterion, it is possible to analyze the error for reasonable values of <i>N</i> and to compare various quasi-random sequences. Several numerical examples are given. Obtained results make it possible to formulate recommendations on the correct use of the quasi-random numbers when calculating integrals over the trajectories of random processes.