journal: Journal of Inverse and Ill-posed Problems

Impact Factor: 1.1

Published since January 1, 1993

Journal of Inverse and Ill-posed Problems

ISSN: 1569-3945

Editor-in-chief: Sergey I. Kabanikhin

Managing Editor: Maxim A. Shishlenin

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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

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Volume 31 Issue 6

December 2023

Publicly Available December 1, 2023

Frontmatter

Page range: i-iv

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October 8, 2020

Variation method solving of the inverse problems for Schrödinger-type equation

Arif Mir Calal Pashaev, Asaf Dashdamir Iskenderov, Qabil Yavar Yaqubov, Matanet Asaf Musaeva

Page range: 799-810

Abstract

A variation method for solving the inverse problems of determining of the complex quantum potential in a nonlinear non-stationary Schrödinger-type equation with final and boundary observations is considered. The existence and uniqueness theorem of the solution of the variation formulation of the inverse problem is proved, the continuity and continuous differentiability of the quality criterion are established, the formula for its gradient is found, the necessary condition of optimality is proved and iterative regularization of the solution is indicated.

Open Access March 3, 2023

Stability estimate for scalar image velocimetry

Erik Burman, Jurriaan J. J. Gillissen, Lauri Oksanen

Page range: 811-822

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Abstract

In this paper, we analyze the stability of the system of partial differential equations modelling scalar image velocimetry. We first revisit a successful numerical technique to reconstruct velocity vectors u {{u}} from images of a passive scalar field ψ by minimizing a cost functional that penalizes the difference between the reconstructed scalar field ϕ and the measured scalar field ψ, under the constraint that ϕ is advected by the reconstructed velocity field u {{u}} , which again is governed by the Navier–Stokes equations. We investigate the stability of the reconstruction by applying this method to synthetic scalar fields in two-dimensional turbulence that are generated by numerical simulation. Then we present a mathematical analysis of the nonlinear coupled problem and prove that, in the two-dimensional case, smooth solutions of the Navier–Stokes equations are uniquely determined by the measured scalar field. We also prove a conditional stability estimate showing that the map from the measured scalar field ψ to the reconstructed velocity field u , on any interior subset, is Hölder continuous.

February 28, 2023

Inverse problems for equations of a mixed parabolic-hyperbolic type with power degeneration in finding the right-hand parts that depend on time

Kamil Basirovich Sabitov, Stanislav Nikolaevich Sidorov

Page range: 823-834

Abstract

For the equation of a mixed parabolic-hyperbolic type with a power degeneration on the type change line, the inverse problems to determine the time-dependent factors of right-hand sides are studied. Based on the formula for solving a direct problem, the solution of inverse problems is equivalently reduced to the solvability of loaded integral equations. Using the theory of integral equations, the corresponding theorems for the existence and uniqueness of the solutions of the stated inverse problems are proved and the explicit formulas for the solution have been given.

November 24, 2022

Inverse problem for integro-differential Kelvin–Voigt equations

Khonatbek Khompysh, Nursaule K. Nugymanova

Page range: 835-847

Abstract

In this paper, the existence and uniqueness of a strong solution of the inverse problem of determining a coefficient of right-hand side of the integro-differential Kelvin–Voigt equation are investigated. The unknown coefficient that we search defends on space variables. Additional information on a solution of the inverse problem is given here as an integral overdetermination condition. The original inverse problem is reduced to study an equivalent inverse problem with homogeneous initial condition. Then the equivalences of the last inverse problem to an operator equation of second kind is proved. We establish the sufficient conditions for the unique solvability of the operator equation of second kind.

March 31, 2023

A projected homotopy perturbation method for nonlinear inverse problems in Banach spaces

Yuxin Xia, Bo Han, Wei Wang

Page range: 849-872

Abstract

In this paper, we propose and analyze a projected homotopy perturbation method based on sequential Bregman projections for nonlinear inverse problems in Banach spaces. To expedite convergence, the approach uses two search directions given by homotopy perturbation iteration, and the new iteration is calculated as the projection of the current iteration onto the intersection of stripes decided by above directions. The method allows to use L 1 {L^{1}} -like penalty terms, which is significant to reconstruct sparsity solutions. Under reasonable conditions, we establish the convergence and regularization properties of the method. Finally, two parameter identification problems are presented to indicate the effectiveness of capturing the property of the sparsity solutions and the acceleration effect of the proposed method.

August 25, 2023

Tensor tomography of the residual stress field in graded-index YAG’s single crystals

Alfred Puro, Egor Marin

Page range: 873-883

Abstract

This work presents an application of tensor field tomography for non-destructive reconstructions of axially symmetric residual stresses in a graded-index YAG single crystal for the case of beam deflection. The axis of the cylinder coincides with the crystallographic axis [001] of the single crystal and it has an axially symmetric refractive index distribution. The transformation of the polarization of light is measured in a plane orthogonal to the axis of the cylinder. Stresses are determined within the framework of the Maxwell piezo-optic law (linear dependence of the permittivity tensor on stresses) and small rotation of quasi principal stress axes. This paper generalizes the method of integrated photoelasticity for the case of ray deflection.

February 28, 2023

Uniqueness and stability for inverse source problem for fractional diffusion-wave equations

Xing Cheng, Zhiyuan Li

Page range: 885-904

Abstract

This paper is devoted to the inverse problem of determining the spatially dependent source in a time fractional diffusion-wave equation, with the aid of extra measurement data at a subboundary. A uniqueness result is obtained by using the analyticity and the newly established unique continuation principle provided that the coefficients are all temporally independent. We also derive a Lipschitz stability of our inverse source problem under a suitable topology whose norm is given via the adjoint system of the fractional diffusion-wave equation.

April 26, 2023

Reconstruction of modified transmission eigenvalues using Cauchy data

Juan Liu, Yanfang Liu, Jiguang Sun

Page range: 905-919

Abstract

The modified transmission eigenvalue (MTE) problem was introduced in [S. Cogar, D. Colton, S. Meng and P. Monk, Modified transmission eigenvalues in inverse scattering theory, Inverse Problems 33 2017, 12, Article ID 125002] and used as a target signature for nondestructive testing. In this paper, we study the inverse spectral problem to reconstruct the modified transmission eigenvalues using Cauchy data. We propose a reciprocity gap functional method and show that the MTEs can be determined by solving some linear ill-posed integral equations. Numerical examples for both absorbing and non-absorbing media are presented to validate the effectiveness and robustness of the proposed method.

January 31, 2023

Alternative fan-beam backprojection and adjoint operators

Patricio Guerrero, Matheus Bernardi, Eduardo Miqueles

Page range: 921-935

Abstract

We present in this work alternative analytic formulations for the fan-beam tomographic backprojection operation and its associated adjoint transform in standard (equiangular) and linear (equidistant) detector geometries. The proposed formulations are obtained from a recent backprojection theorem in parallel tomography. Such formulations are written as a Bessel–Neumann series in the frequency domain that can be implemented as an O ⁢ ( N 2.3729 ) {O(N^{2.3729})} matrix multiplication. Proofs are provided together with numerical simulations compared with conventional fan-beam O ⁢ ( N 3 ) {O(N^{3})} backprojection representations showing more robustness when dealing with highly noisy data.

June 27, 2023

The problem of determining multiple coefficients in an ultrahyperbolic equation

Fikret Gölgeleyen

Page range: 937-948

Abstract

In this article, we discuss an inverse problem of determining unknown coefficients of the first-order derivatives in an ultrahyperbolic equation. By a finite set of measurements, we prove the uniqueness of solution of the problem in semi-geodesic coordinates under some conditions on the principal coefficients of the equation. Our main tool is a Carleman estimate.

October 4, 2023

A note on the degree of ill-posedness for mixed differentiation on the d-dimensional unit cube

Bernd Hofmann, Hans-Jürgen Fischer

Page range: 949-957

Abstract

Numerical differentiation of a function over the unit interval of the real axis, which is contaminated with noise, by inverting the simple integration operator J mapping in L 2 {L^{2}} is discussed extensively in the literature. The complete singular system of the compact operator J is explicitly given with singular values σ n ⁢ ( J ) {\sigma_{n}(J)} asymptotically proportional to 1 n {\frac{1}{n}} . This indicates a degree one of ill-posedness for the associated inverse problem of differentiation. We recall the concept of the degree of ill-posedness for linear operator equations with compact forward operators in Hilbert spaces. In contrast to the one-dimensional case, there is little specific material available about the inverse problem of mixed differentiation, where the d-dimensional analog J d {J_{d}} to J , defined over unit d -cube, is to be inverted. In this note, we show for that problem that the degree of ill-posedness stays at one for all dimensions d ∈ ℕ {d\in{\mathbb{N}}} . Some more discussion refers to the two-dimensional case in order to characterize the range of the operator J 2 {J_{2}} .

October 4, 2023

A uniqueness result for the inverse problem of identifying boundaries from weighted Radon transform

Dmitrii Sergeevich Anikonov, Sergey G. Kazantsev, Dina S. Konovalova

Page range: 959-965

Abstract

We study the problem of the integral geometry, in which the functions are integrated over hyperplanes in the n -dimensional Euclidean space, n = 2 ⁢ m + 1 {n=2m+1} . The integrand is the product of a function of n variables called the density and weight function depending on 2 ⁢ n {2n} variables. Such an integration is called here the weighted Radon transform, which coincides with the classical one if the weight function is equal to one. It is proved the uniqueness for the problem of determination of the surface on which the integrand is discontinuous.

Ahead of Print / Just Accepted

Ahead of Print / Just Accepted

Volume 31 (2023)

Volume 30 (2022)

Volume 29 (2021)

Volume 28 (2020)

Volume 27 (2019)

Volume 26 (2018)

Volume 25 (2017)

Volume 24 (2016)

Volume 23 (2015)

Volume 22 (2014)

Volume 21 (2013)

Volume 20 (2012)

Volume 19 (2011)

Volume 18 (2011)

Volume 17 (2009)

Volume 16 (2008)

Volume 15 (2007)

Volume 14 (2006)

Volume 13 (2005)

Volume 12 (2004)

Volume 11 (2003)

Volume 10 (2002)

Volume 9 (2001)

Volume 8 (2000)

Volume 7 (1999)

Volume 6 (1998)

Volume 5 (1997)

Volume 4 (1996)

Volume 3 (1995)

Volume 2 (1994)

Volume 1 (1993)

Journal Impact Factor	1.1	2022, Journal Citation Reports (Clarivate, 2023)
5-year Journal Impact Factor	1.0	2022, Journal Citation Reports (Clarivate, 2023)
Journal Citation Indicator	0.82	2022, Journal Citation Reports (Clarivate, 2023)
CiteScore	2.4	2022, Scopus (Elsevier B.V., 2023)
SCImago Journal Rank	0.428	2022, SJR (Scimago Lab, 2023; Data Source: Scopus)
Source Normalized Impact per Paper	0.769	2022, CWTS Journal Indicators (CWTS B.V., 2023; Data Source: Scopus)
Mathematical Citation Quotient	0.53

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Articles in journals:
[1] A. Carbonaro, G. Metafune and C. Spina, Parabolic Schrödinger operators, J. Math
Anal. and Appl. 343 (2008), 965-974.
Articles ahead of print:
[2] D. Farley and L. Sabalka, Presentations of graph braid groups, Forum Math. (2010), doi: 10.1515/FORM.2011.086.
Books and Monographs:
[3] I. M. Isaacs, Character theory of finite groups, AMS Chelsea Publishing, 2006.
Chapters:
[4] T. De Medts, F. Haot, R. Knop and H. Van Maldeghem, On the uniqueness of the unipotent subgroups of some Moufang sets, in: Finite Geometries, Groups, and Computation, pp. 43-66, de Gruyter, Berlin, 2006.

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Editorial

Editor-in-Chief

Sergey I. Kabanikhin, Novosibirsk
Email: ksi52@mail.ru

Managing Editor

Maxim A. Shishlenin, Novosibirsk

Advisory Board

Avner Friedman, Columbus
Rainer Kress, Göttingen
Peter Lax, New York
Zuhair Nashed, Orlando
Vladimir V. Romanov, Novosibirsk
Vladimir V. Vasin, Ekaterinburg

Editorial Board

Giovanni Alessandrini, Trieste
Habib Ammari, Zurich
Gang Bao, Hangzhou
Tatiana Bubba, Bath
Alexander L. Bukhgeim, Novosibirsk/Whichita
Jin Cheng, Shanghai
Christian Clason, Graz
Dinh Nho Ha'o, Hanoi
Alemdar Hasanoglu, Izmir
Bernd Hofmann, Chemnitz
Thorsten Hohage, Göttingen
Maarten V. de Hoop, Houston
Masaru Ikehata, Higashihiroshima
Mikhail Yu. Kokurin, Yoshkar-Ola
Daniel Lesnic, Leeds
Jijun Liu, Nanjing
Alfred K. Louis, Saarbrücken
Gen Nakamura, Sapporo
Andreas Neubauer, Linz
Roman G. Novikov, Paris/Moscow
Ivan Oseledets, Skolkovo
Valery V. Pickalov, Novosibirsk
Eric Todd Quinto, Medford
Ronny Ramlau, Linz
Karl Sabelfeld, Novosibirsk
Paul Sacks, Ames
Alexander Shananin, Moscow
Otmar Scherzer, Vienna
Andrey Shkalikov, Moscow
Plamen D Stefanov, West Lafayette
Gunther Uhlmann, Seattle
Yanfei Wang, Beijing
Konstantin Vorontsov, Moscow
Anatoly G. Yagola, Moscow
Masahiro Yamamoto, Tokyo
Vyacheslav A. Yurko, Saratov
Jun Zou, Hong Kong

(Deutsch)

Online ISSN: 1569-3945
Print ISSN: 0928-0219
Type: Journal
Language: English
Publisher: De Gruyter
First published: January 1, 1993
Publication Frequency: 6 Issues per Year