Sparse representation of multivariate functions based on discrete point evaluations
Sparse representation of multivariate functions based on discrete point evaluations
Author
Byrenheid, Glenn
Type of Scholarly Publication
DissertationDate of Exam
09.11.2018Date of Publication
22.01.2019Advisor
Ullrich, TinoCo-Referee
Griebel, MichaelInvolved Institutions
Rheinische Friedrich-Wilhelms-Universität BonnMetadata
Show full item recordCitable Links
Abstract
Functions provide one of the most important building blocks for model descriptions of reality. Central point of this thesis is the approximation of multivariate functions using Faber-Schauder hat functions. In the first part we describe mixed smoothness Sobolev-Besov-Triebel-Lizorkin spaces by decreasing properties of Faber-Schauder coefficients. This allows us to provide equivalent norm representations based on discrete function values. In the second part we apply this theory to study sparse grid sampling or more generally the problem of sampling recovery for Sobolev classes (especially with integrability $pneq 2$). We provide new convergence estimates for a Faber-Schauder based sparse grid method measuring errors in $L_{q}([0,1]^d)$ with $p
Subjects
Abtastalgorithmen, Dünngitterapproximation, Funktionenräume, Faber-Schauder-Basen, Sobolev-Räume, Besov-Räume, Triebel-Lizorkin-Räume, Sampling, Nichtlineare Approximation, Beste m-Term-Approximation, sampling representations, sampling, sparse grid approximation, function spaces, Faber-Schauder bases, Sobolev spaces, Besov spaces, Triebel-Lizorkin spaces, nonlinear approximation, best m-term approximation, greedy methods
Classification (DDC)
510 MathematikByrenheid, Glenn: Sparse representation of multivariate functions based on discrete point evaluations. - Bonn, 2019. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5n-53130
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5n-53130
@phdthesis{handle:20.500.11811/7838,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5n-53130,
author = {{Glenn Byrenheid}},
title = {Sparse representation of multivariate functions based on discrete point evaluations},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2019,
month = jan,
note = {Functions provide one of the most important building blocks for model descriptions of reality. Central point of this thesis is the approximation of multivariate functions using Faber-Schauder hat functions. In the first part we describe mixed smoothness Sobolev-Besov-Triebel-Lizorkin spaces by decreasing properties of Faber-Schauder coefficients. This allows us to provide equivalent norm representations based on discrete function values. In the second part we apply this theory to study sparse grid sampling or more generally the problem of sampling recovery for Sobolev classes (especially with integrability $pneq 2$). We provide new convergence estimates for a Faber-Schauder based sparse grid method measuring errors in $L_{q}([0,1]^d)$ with $p
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5n-53130,
author = {{Glenn Byrenheid}},
title = {Sparse representation of multivariate functions based on discrete point evaluations},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2019,
month = jan,
note = {Functions provide one of the most important building blocks for model descriptions of reality. Central point of this thesis is the approximation of multivariate functions using Faber-Schauder hat functions. In the first part we describe mixed smoothness Sobolev-Besov-Triebel-Lizorkin spaces by decreasing properties of Faber-Schauder coefficients. This allows us to provide equivalent norm representations based on discrete function values. In the second part we apply this theory to study sparse grid sampling or more generally the problem of sampling recovery for Sobolev classes (especially with integrability $pneq 2$). We provide new convergence estimates for a Faber-Schauder based sparse grid method measuring errors in $L_{q}([0,1]^d)$ with $p
url = {https://hdl.handle.net/20.500.11811/7838}
}
E-Dissertationen: Related items
Showing items related by title, author, creator and subject.
-
On Approximability of Bounded Degree Instances of Selected Optimization Problems
Schmied, Richard (2013-08-07)In order to cope with the approximation hardness of an underlying optimization problem, it is advantageous to consider specific families of instances with properties that can be exploited to obtain efficient approximation ... -
On Discrete and Geometric Firefighting
Schwarzwald, Barbara Anna (2021-08-19)Wildfires ravaging forests around the globe cost lives, homes and billions in damages every year, which motivates the study of effective firefighting. In the area of theoretical computer science, several different models ... -
An Application of Kolmogorov's Superposition Theorem to Function Reconstruction in Higher Dimensions
Braun, Jürgen (2009-12-01)In this thesis we present a Regularization Network approach to reconstruct a continuous function ƒ:[0,1]<sup>n</sup>→<b>R</b> from its function values ƒ(<b>x</b><sub>j</sub>) on discrete data points <b>x</b><sub>j</sub>, ...