Aproksimacija razpršenih podatkov z metodo najmanjših kvadratov nad triangulacijami

delo diplomskega seminarja

Lara Jagodnik (Author), Jan Grošelj (Mentor)

Abstract

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.

Keywords

matematika;triangulacije;metoda najmanjših kvadratov;predoločeni sistemi;

Data

Language:	Slovenian
Year of publishing:	2021
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[L. Jagodnik]
UDC:	519.6
COBISS:	75593475
Views:	1032
Downloads:	75
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Least squares approximation of scattered data over triangulations
Secondary 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.
Secondary keywords:	mathematics;triangulations;least squares method;overdetermined systems;
Type (COBISS):	Final seminar paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Finančna matematika - 1. stopnja
Pages:	29 str.
ID:	13335674