Numerical Approximation
Numerisk approximation

NUMN19F, 7.5 högskolepoäng

Gäller från och med: Spring 2021
Beslutad av: Professor Thomas Johansson
Datum för fastställande: 2020-09-24

Allmänna uppgifter

Avdelning: Numerical Analysis
Kurstyp: Gemensam kurs, avancerad nivå och forskarnivå
Kursen ges även på avancerad nivå med kurskod: NUMN19
Undervisningsspråk: English

Syfte

The overall goal of the course is to provide an introduction to classical results and numerical algorithms within approximation theory and prepare the participants for further studies in mathematics and computationally oriented subjects. The purpose is further to develop the participants' ability to solve problems, communicate mathematical reasoning, assess mathematical algorithms and translate them into effective code.

Mål

Kunskap och förståelse

För godkänd kurs skall doktoranden

• be able to motivate and exemplify the need for approximations of functions, both from the theoretical and the computational point of view
• be able to describe how to find good approximations with respect to different norms, in particular the 1-, 2- and supremum-norms, and give an account of the difficulties in each of these cases,
• be able to give an account of the relation between the topology of the approximation space and the existence and uniqueness of best approximations,
• be able to formulate the main theorems of approximation theory, especially the characterisation theorems and the Weierstrass theorem, and outline their proofs.

Färdighet och förmåga

För godkänd kurs skall doktoranden

• be able to identify the relevant approximation algorithm for a given situation, and write a computer program which implements it
• be able to present solutions and numerical results for problems such as the above ones in written and oral form,
• be able to, with adequate terminology and in a logical and well-structured manner, explain the design of the numerical methods and algorithms included in the course.

Värderingsförmåga och förhållningssätt

För godkänd kurs skall doktoranden be able to argue for the importance of approximation theory as a tool in mathematics,computational technology and related subjects.

Kursinnehåll

The course treats: • The approximation problem: Norms, approximation spaces, the Weierstrass theorem. • Theory of best approximation in Euclidean spaces: Existence, uniqueness, characterisation theorems, duals. • Construction of best approximations: Orthogonality, Chebyshev polynomials, Haar spaces, the exchange algorithm.

Kurslitteratur

Iske, A.: Approximation Theory and Algorithms for Data Analysis. Springer, 2019. ISBN 9783030052270.

Kursens undervisningsformer

Undervisningsformer: Föreläsningar, övrigt. Theoretical and practical assignments.

Kursens examination

Examinationsformer: Muntlig tentamen, inlämningsuppgifter
Betygsskala: Underkänd, godkänd
Examinator:

Antagningsuppgifter

Kurstillfällesinformation

Kontaktinformation och övrigt

Kursansvariga:
Hemsida: http://www.ctr.maths.lu.se/course/NewNumApprox/