Aproksimacija razpršenih podatkov z metodo najmanjših kvadratov nad triangulacijami : delo diplomskega seminarja

Jagodnik, Lara (author), Grošelj, Jan (mentor)

V diplomskem delu obravnavamo problem aproksimacije razpršenih podatkov z metodo najmanjših kvadratov nad triangulacijami. Definiramo končno dimenzionalni prostor $S_1^0(\triangle)$ zveznih odsekoma linearnih funkcij nad triangulacijo $\triangle$ in ga opremimo z bazo. Baza prostora je sestavljena iz funkcij z lokalnimi nosilci in grafi piramidaste oblike. Podatke aproksimiramo s funkcijo $f \in S_1^0(\triangle)$, ki jo predstavimo kot linearno kombinacijo baznih funkcij. Koeficiente določimo z metodo najmanjših kvadratov. V delu izpeljemo, da lahko koeficiente $f$ izračunamo z reševanjem predoločenega sistema enačb. Predoločen sistem prevedemo v normalni sistem, ki je določen s simetrično in razpršeno matriko. Njena analiza nam zagotovi obstoj in enoličnost aproksimacijske funkcije.

• Language: Slovenian
• Keywords: triangulacije, metoda najmanjših kvadratov, predoločeni sistemi
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2021
• PID: 20.500.12556/RUL-129789
• UDC: 519.6
• COBISS.SI-ID: 75593475
• Publication date in RUL: 08.09.2021

JAGODNIK, Lara, 2021, Aproksimacija razpršenih podatkov z metodo najmanjših kvadratov nad triangulacijami : delo diplomskega seminarja [online]. Bachelor’s thesis. [Accessed 5 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=129789

Secondary language

• Language: English
• Title: Least squares approximation of scattered data over triangulations

Abstract

In this paper we consider the problem of least squares approximation of scattered data over triangulations. We define finite dimensional space $S_1^0(\triangle)$ of continuous piecewise linear functions over a triangulation $\triangle$ and equip it with a basis. The basis consists of functions with local supports and pyramid-shaped graphs. Data are approximated by a function $f \in S_1^0(\triangle)$, which is represented as a linear combination of basis functions. The coefficients of the function are determined using the least squares method. We derive that coefficients of a function $f$ can be computed with solving an overdetermined system. The overdetermined system can be solved using the corresponding normal system determined by a symmetric sparse matrix. Its analysis ensures the existence and uniqueness of the approximation function.

• Keywords: triangulations, least squares method, overdetermined systems