Kvazi Monte Carlo metoda za numeričku integraciju

master's thesis

• Author: Luka Turčić
• Mentor: Vjekoslav Kovač (mentor)
• Committee member: Vjekoslav Kovač (predsjednik povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Committee member: Miljenko Huzak (član povjerenstva)
• Committee member: Robert Manger (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-07-17, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Kvazi Monte Carlo metoda je determinističko poboljšanje Monte Carlo metode, vjerojatnosne metode koja se u rješavanju determinističkih problema oslanja na generiranje slučajnih uzoraka. Slučajni uzorci Monte Carlo metode su u kvazi Monte Carlo metodi zamijenjeni dobro odabranim determinističkim uzorcima, a idealan problem gdje se to može ilustrirati je problem numeričke integracije. Kriterij odabira determinističkih točaka je diskrepancija - kvantitativna mjera odstupanja od uniformne distribuiranosti niza. Glavni rezultati ovog rada dokazuju da je diskrepancija kriterij odabira točaka kvazi Monte Carlo metode, odnosno pokazuju da nizovi niske diskrepancije garantiraju manju grešku numeričke integracije. Potom rad daje neke konstrukcije takvih nizova, zajedno s dokazima da tako definirani nizovi jesu nizovi niske diskrepancije.

Abstract (english)

Quasi Monte Carlo method is a deterministic refinement of the Monte Carlo method, a probabilistic method which uses random samples for solving deterministic problems. Random samples from the Monte Carlo method are replaced with well-chosen deterministic samples and an ideal problem where this can be shown is the problem of numerical integration. The criterion for choosing deterministic points is discrepancy – a quantitative measure of deviation from the uniform distribution of a sequence. The main results of this thesis prove that discrepancy is the criterion for choosing quasi Monte Carlo samples, that is, they show that sequences of low discrepancy guarantee smaller error of numerical integration. Moreover, the thesis gives several constructions of such sequences with proofs that the constructed sequences are sequences of low discrepancy.

• Keywords: kvazi Monte Carlo metoda; diskrepancija
• Keywords (english): quasi Monte Carlo method; discrepancy
• Language: croatian
• URN:NBN: urn:nbn:hr:217:724450
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-07-02 12:03:11