Multiscale analysis for ill-posed problems with semi-discrete Tikhonov regularization
type: Item
instance of: scholarly article
title: Multiscale analysis for ill-posed problems with semi-discrete Tikhonov regularization
zbMATH Open document ID: 1254.65071
DOI: 10.1088/0266-5611/28/6/065019
author: Min Zhong
author: Shuai Lu
author: Jin Cheng
published in: Inverse Problems
publication date: 2012-07-11T00:00:00Z
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 47A52
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 65J22
zbMATH DE Number: 6054457
zbMATH Keywords: linear ill-posed problems
zbMATH Keywords: Tikhonov regularization
zbMATH Keywords: Sobolev functions
zbMATH Keywords: multiscale analysis
zbMATH Keywords: moderate ill-posedness
zbMATH Keywords: discrete noisy data
zbMATH Keywords: ill-posed linear operator equations
zbMATH Keywords: Hilbert space
zbMATH Keywords: Morozov's discrepancy principle
zbMATH Keywords: error estimates
zbMATH Keywords: linear inverse problem
zbMATH Keywords: integral operator
zbMATH Keywords: algorithm
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1088/0266-5611/28/6/065019
OpenAlex ID: W2024524509
review text: \textit{H. Wendland}'s results [Numer. Math. 116, No. 3, 493--517 (2010; Zbl 1208.65067)] on multiscale analysis in Sobolev spaces on bounded domains can be applied to the interpretation of indirect measurements when the available data are discrete and noisy. In this context, compactly supported radial basis functions of varying radii are exploited. The present paper deals with the application of the theory from Wendland's paper to the stable approximate solution of ill-posed linear operator equations \(Af=g\) in a Hilbert space setting by a semi-discrete version of Tikhonov's regularization method. In particular, an a posteriori choice of the regularization parameter based on Morozov's discrepancy principle with corresponding error estimates for noisy data is included in the discussion. The authors have in mind linear inverse problems, where the (mostly compact) forward operator \(A\) has the structure of an integral operator NEWLINE\[NEWLINE[Af](x)=\int \limits _\Omega k(x,t) f(t) dt NEWLINE\]NEWLINE over a bounded domain \(\Omega \subset \mathbb{R}^k,\;k=1,2,\dots\). The developed theory and algorithm suppose that the problem is moderately ill-posed, which means here that there are constants \(0<\underline c \leq \overline c<\infty\) such that NEWLINE\[NEWLINE\underline c\,\|f\|_{H^\theta(\Omega)} \leq \|Af\|_{H^{\theta+a}(\Omega)} \leq \overline c\,\|f\|_{H^\theta(\Omega)} NEWLINE\]NEWLINE is valid for all \(f \in H^\theta(\Omega)\) and some \(\theta \in \mathbb{R}\). The constant \(a>0\) represents the degree of ill-posedness of the problem and is finite. Some numerical examples with the simple integration operator as \(A\) on \(\Omega=[0,2] \subset \mathbb{R}\) and two different exact solutions \(f\) illustrate the approach and associated ingredients like reproducing kernel Hilbert spaces, discretization schemes and error behavior.
reviewed by: Bernd Hofmann
description: scientific article; zbMATH DE number 6054457