delo diplomskega seminarja
Timotej Vesel (Author), Marjetka Krajnc (Mentor)

Abstract

V financah, statistiki, fiziki itd. se velikokrat pojavijo problemi, pri katerih je potrebno izračunati oziroma aproksimirati integral dimenzije več sto ali celo več tisoč. V delu diplomskega seminarja si ogledamo nekaj metod, s katerimi lahko takšne integrale relativno učinkovito rešimo. Najprej obravnavamo nekaj pravil za integracijo v eni dimenziji iz klasične teorije numerične integracije (kvadraturna pravila) ter ugotovimo, zakaj njihova posplošitev v več dimenzij ni učinkovita. Nato obravnavamo metodo Monte Carlo, ki uspešno odpravlja te probleme, izpeljemo napako metode in navedemo glavni razlog za vpeljavo kvazi-Monte Carlo (QMC) metod. Za tem definiramo pojma zvezdne diskrepance in variacije v smislu Hardya in Krausa, ki ju potrebujemo za neenakost Koksma-Hlawka, ki je glavni rezultat pri QMC metodah. Potem predstavimo glavni družini QMC metod, mrežna pravila in številske mreže, ter opišemo konstrukcije nekaj najpomembnejših primerov. Nazadnje si na praktičnem primeru ogledamo veljavnost nekaterih rezultatov, ki smo jih spoznali pred tem.

Keywords

matematika;numerična integracija;metoda Monte Carlo;kvazi-Monte Carlo metode;mrežna pravila;številske mreže;

Data

Language: Slovenian
Year of publishing:
Typology: 2.11 - Undergraduate Thesis
Organization: UL FMF - Faculty of Mathematics and Physics
Publisher: [T. Vesel]
UDC: 519.6
COBISS: 18720345 Link will open in a new window
Views: 1166
Downloads: 243
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Other data

Secondary language: English
Secondary title: Lattice rules and quasi-Monte Carlo methods for integration of functions
Secondary abstract: 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.
Secondary keywords: mathematics;numerical integration;Monte Carlo method;quasi-Monte Carlo methods;lattice rules;digital nets;
Type (COBISS): Final seminar paper
Study programme: 0
Embargo end date (OpenAIRE): 1970-01-01
Thesis comment: Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Finančna matematika - 1. stopnja
Pages: 34 str.
ID: 11217812