Journal of Inverse and Ill-posed Problems

Published by De Gruyter

Volume 22 Issue 6

Issue of Journal of Inverse and Ill-posed Problems

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Contents
Journal Overview

Publicly Available December 1, 2014

Frontmatter

Page range: i-iv

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December 13, 2013

An inverse problem for the recovery of the vascularization of a tumor

Thierry Colin, Angelo Iollo, Jean-Baptiste Lagaert, Olivier Saut

Page range: 759-786

Abstract

In this paper we present a simplified model of tumor growth and an associated inverse problem. Our model consists in describing the evolution of a population of cancer cells and the density of oxygen. The growth rate depends on the concentration of oxygen. The oxygen distribution is computed by means of a diffusion equation, the source term being localized on the blood vessels. We consider the inverse problem that consists in recovering the position of the blood vessels assuming the distribution of tumor cells. We use an adjoint method. Results relative to idealized clinical cases are discussed.

November 13, 2013

Estimating the ice thickness of mountain glaciers with a shape optimization algorithm using surface topography and mass-balance

Laurent Michel, Marco Picasso, Daniel Farinotti, Andreas Bauder, Martin Funk, Heinz Blatter

Page range: 787-818

Abstract

We present a shape optimization algorithm to estimate the ice thickness distribution within a two-dimensional, non-sliding mountain glacier, given a transient surface geometry and a mass-balance distribution. The approach is based on the minimization of the surface topography misfit at the end of the glacier's evolution in the shallow ice approximation of ice flow. Neither filtering of the surface topography where its gradient vanishes nor interpolation of the basal shear stress is involved. Novelty of the presented shape optimization algorithm is the use of surface topography and mass-balance only within a time-dependent Lagrangian approach for moving-boundary glaciers. On real-world inspired geometries, it is shown to produce estimations of even better quality in smaller time than the recently proposed steady and transient inverse methods. A sensitivity analysis completes the study and evinces the method's higher susceptibility to perturbations in the surface topography than in surface mass-balance or rate factor.

November 13, 2013

On the determination of the principal coefficient from boundary measurements in a KdV equation

Lucie Baudouin, Eduardo Cerpa, Emmanuelle Crépeau, Alberto Mercado

Page range: 819-845

Abstract

This paper concerns the inverse problem of retrieving the principal coefficient in a Korteweg–de Vries (KdV) equation from boundary measurements of a single solution. The Lipschitz stability of this inverse problem is obtained using a new global Carleman estimate for the linearized KdV equation. The proof is based on the Bukhgeĭm–Klibanov method.

February 15, 2014

Reconstructing conductivities with boundary corrected D-bar method

Samuli Siltanen, Janne P. Tamminen

Page range: 847-870

Abstract

The aim of electrical impedance tomography is to form an image of the conductivity distribution inside an unknown body using electric boundary measurements. The computation of the image from measurement data is a non-linear ill-posed inverse problem and calls for a special regularized algorithm. One such algorithm, the so-called D-bar method, is improved in this work by introducing new computational steps that remove the so far necessary requirement that the conductivity should be constant near the boundary. The numerical experiments presented suggest two conclusions. First, for most conductivities arising in medical imaging, it seems the previous approach of using a best possible constant near the boundary is sufficient. Second, for conductivities that have high contrast features at the boundary, the new approach produces reconstructions with smaller quantitative error and with better visual quality.

February 26, 2014

Regularization of linear inverse problems with total generalized variation

Kristian Bredies, Martin Holler

Page range: 871-913

Abstract

The regularization properties of the total generalized variation (TGV) functional for the solution of linear inverse problems by means of Tikhonov regularization are studied. Considering the associated minimization problem for general symmetric tensor fields, the well-posedness is established in the space of symmetric tensor fields of bounded deformation, a generalization of the space of functions of bounded variation. Convergence for vanishing noise level is shown in a multiple regularization parameter framework in terms of the naturally arising notion of TGV-strict convergence. Finally, some basic properties, in particular non-equivalence for different parameters, are discussed for this notion.

About this journal

This journal aims to present original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published.

Issues of the Journal of Inverse and Ill-Posed Problems contain high quality papers which have an innovative approach and topical interest.

The following topics are covered:

Inverse problems

existence and uniqueness theorems
stability estimates
optimization and identification problems
numerical methods
machine learning
Artificial Intellegence
Big Data

Ill-posed problems

regularization theory
operator equations
integral geometry

Applications

inverse problems in geophysics, electrodynamics and acoustics
inverse problems in ecology
inverse and ill-posed problems in medicine
mathematical problems of tomography