Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise
type: Item
instance of: scholarly article
title: Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise
zbMATH Open document ID: 1160.65023
author: Thorsten Hohage
author: Mihaela Pricop
published in: Inverse Problems and Imaging
publication date: 2009-01-30T00:00:00Z
review text: The authors are concerned with the parameter identification problems where \(a\) is an unknown parameter in an elliptic differential equation (assumed to belong to a separable Hilbert space \(\mathcal{X}\) which is not directly observable) and \(u\) is the solution to a boundary value problem associated with the given equation. It is only possible to access \(u,\) which belongs to another Hilbert space \(\mathcal{Y},\) and the relation between \(a\) and \(u\) is described in terms of so-called parameter-to-solution (nonlinear) operator \(F:D( F) \to\mathcal{Y},\) \(F(a)=u,\) where \(D( F) \subset\mathcal{X}.\) The exact solution to the problem is \(a_{\dag }\in D( F) \) and the exact data is \(u_{\dag}=F( a_{\dag }) .\) It is assumed that \(u_{\dag}\) is not given explicitly, but through a random variable \(\hat{u}\in\mathcal{Y}\) satisfying \[ \sqrt{\mathbf{E}\| \hat{u}-u_{\dag}\| _{\mathcal{Y}}^{2}} \leq\delta. \] The estimators are defined by means of the Tikhonov functional \[ \hat{a}=\arg\min_{a\in D(F)}( \| F(a)-\hat{u}\| _{\mathcal{Y}}^{2}+\alpha\| a-a_{0}\| _{\mathcal{X}} ^{2}), \] where \(\alpha>0\) is a regularization parameter and \(a_{0}\in\mathcal{X}\) stands for some initial guess of \(a_{\dag}.\) The authors derive estimates on the expected squared error under certain assumptions on the forward operator. They study reconstruction of a reaction coefficient and a diffusion coefficient in two inverse elliptic boundary value problems and demonstrate that assumptions of their convergence result are met in both cases. Numerical experiments illustrating the effect of the smoothness of the parameter \(a_{\dag}\) on the convergence rate of \(\hat{u}\) are provided.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 65N21
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 65L09
Mathematics Subject Classification ID: 34A55
Mathematics Subject Classification ID: 65J22
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 47J06
zbMATH DE Number: 5500621
zbMATH Keywords: statistical inverse problem
zbMATH Keywords: parameter identification
zbMATH Keywords: inverse elliptic boundary value problems
zbMATH Keywords: Tikhonov regularization
zbMATH Keywords: Hilbert scales
zbMATH Keywords: random noise
zbMATH Keywords: reconstruction
zbMATH Keywords: reaction coefficient
zbMATH Keywords: diffusion coefficient
zbMATH Keywords: convergence
zbMATH Keywords: numerical experiments
reviewed by: Yuriy V. Rogovchenko
MaRDI profile type: Publication
full work available at URL: ipi.2008.2.271
OpenAlex ID: W2327825075
DOI: 10.3934/IPI.2008.2.271
description: scientific article; zbMATH DE number 5500621