In finance, statistics, physics etc. many problems arise where we are required to calculate or approximate integral which dimension is in hundreds or even thousands. In the diploma seminar we examine some methods that can solve such integrals relatively efficiently.
First, we discuss some integration rules in one dimension from the classical theory of numerical integration (quadrature rules) and comment why their generalization to higher dimensions is not effective. Then we study Monte Carlo method, which successfully eliminate these problems. We derive the error of the method and state the main reason for introducing quasi-Monte Carlo (QMC) methods. Further, we define notions of star discrepancy and variation in the sense of Hardy and Krause that are needed for the Koksma-Hlawka inequality, which is the main result of QMC methods. Moreover we present two main families of QMC methods, lattice rules and digital nets, and describe constructions of some of the most important examples. Finally, we take a look at validity of some of the results we have learned before on one practical example.
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