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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Sergey I. Kabanikhin
Managing Editor: Maxim A. Shishlenin

Language: English

First published: January 1, 1993

Publication Frequency: 6 issues per year

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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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Weighting operators for sparsity regularization

October 1, 2025

Ole Løseth Elvetun, Bjørn Fredrik Nielsen, Niranjana Sudheer

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Standard regularization methods typically favor solutions which are in, or close to, the orthogonal complement of the null space of the forward operator/matrix 𝖠 {{\mathsf{A}}} . This particular biasedness might not be desirable in applications and can lead to severe challenges when 𝖠 {{\mathsf{A}}} is non-injective. We have therefore, in a series of papers, investigated how to “remedy” this fact, relative to a chosen basis and in a certain mathematical sense: Based on a weighting procedure, it turns out that it is possible to modify both Tikhonov and sparsity regularization such that each member of the chosen basis can be almost perfectly recovered from their image under 𝖠 {{\mathsf{A}}} . In particular, we have studied this problem for the task of using boundary data to identify the source term in an elliptic PDE. However, this weighting procedure involves 𝖠 † ⁢ 𝖠 {{\mathsf{A}}^{\dagger}{\mathsf{A}}} , where 𝖠 † {{\mathsf{A}}^{\dagger}} denotes the pseudo-inverse of 𝖠 {{\mathsf{A}}} , and can thus be CPU-demanding and lead to undesirable error amplification. We therefore, in this paper, study alternative weighting approaches and prove that some of the recovery results established for the methodology involving 𝖠 † {{\mathsf{A}}^{\dagger}} hold for a broader class of weighting schemes. In fact, it turns out that “any” linear operator 𝖡 {{\mathsf{B}}} has an associated proper weighting defined in terms of images under 𝖡𝖠 {{\mathsf{B}}{\mathsf{A}}} . We also present a series of numerical experiments, employing different choices of 𝖡 {{\mathsf{B}}} .
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Classification of cerebrovascular pathologies in real-time using nonlinear ODE-based surrogate model

October 1, 2025

Yuriy V. Bugai, Alexander A. Cherevko, Maxim A. Shishlenin

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In this paper we consider the coefficient inverse problem for a second-order nonlinear ODE surrogate model describing hemodynamic parameters during intraoperative neurosurgical measurements. This mathematical model of cerebral hemodynamics is based on the generalized Van der Pol–Duffing equation and described the local interaction of the velocity and pressure of blood flow in cerebral vessels. For each patient, the coefficients of this equation are individual and are determined from clinical data in real-time by solving the coefficient inverse problem. We apply the gradient method for optimization of the cost functional with the analytical finding of initial guess to get the coefficients by clinical data obtained during neurosurgical operation in the vicinity of arterial pathologies. A good initial guess is based on the analytical Fourier method. Statistical analysis of clinical data has shown that the surrogate model equation is sensitive to different types of pathology, which allows intraoperative monitoring of the patient’s condition and assessment of the type of pathology in real time. Numerical results are presented and it is shown that the proposed mathematical model and numerical method predict clinical data well.
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A two-stage algorithm based on convolution and attention hybrid architecture for multi-obstacle inverse scattering problems: Integrating obstacle perception and shape recognition

October 1, 2025

Pengcheng Cheng, Yiyang Guo, Yongxu Liu, Junliang Lv, Yiru Zhao

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In this paper, we consider the use of an artificial neural network approach to solve the multi-obstacle inverse scattering problems. We propose a two-step algorithm consisting of obstacle perception and shape recognition. In the obstacle perception step, we train a deep neural network to predict the number of obstacles in the system of interest. At the shape recognition step, we train multiple neural networks to invert the shape of each obstacle. A large number of numerical experiments are provided to verify the effectiveness of the proposed method. It is worth pointing out that our algorithm is not only applicable to the full-phase data problem, but also to the phaseless data problem, and even to the finite aperture problem.
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Binary sparse signal recovery with binary least squares

September 16, 2025

Haifeng Li, Qi Chen

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The problem of obtaining a 𝐾-sparse binary signal 𝐱 from y = Φ ⁢ x + ν \mathbf{y}=\boldsymbol{\Phi}\mathbf{x}+\boldsymbol{\nu} is often encountered in communications and signal processing. Based on orthogonal least squares (OLS), a greedy algorithm guaranteeing the support recovery of a 𝐾-sparse binary signal is presented in this paper. Using the mutual coherence and restricted isometric properties of Φ \boldsymbol{\Phi} , sufficient conditions ensuring the exact recovery of the support of 𝐱 are given. Finally, compared with binary matching pursuit (BMP) and OLS, the simulation experiments imply the advantage of the proposed algorithm.
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Direct application of a Gaussian process model to generate the inverse function in a parameter estimation problem

September 5, 2025

Jonathan Carter, Peter Carter, Ian Vernon

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In a parameter estimation problem, the objective is to find the input parameters for some forward model such that the outputs of the model are close to some stated values. These could be measurements from a physical system, or targets in a design problem. A typical approach is to create an optimisation problem, where we search the space of possible parameters and seek to minimise a sum of errors squared. This will often require the forward model to be run many times, which can be computationally expensive. Gaussian process models can be used to model the forward function, which can reduce the computational effort. If a forward function exists, then it may be possible to create an inverse function. This inverse function can then be modelled by a Gaussian process model. We have formulated a method that allows an approximate inverse function to exist for many typical simulation models. We then iteratively create a sequence of Gaussian process models that capture this function. This allows us to solve the parameter estimation problem. This approach uses fewer forward model calculations than typical search algorithms.
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ECALL: Estimation-calibrated learning for joint deconvolution and kernel inference from unpaired data

August 1, 2025

Markus Haltmeier, Gyeongha Hwang

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Recovering an original image from a blurred observation when the blurring kernel is unknown is a classical inverse problem with applications in astronomy, microscopy, and medical imaging. This setting is often referred to as blind deconvolution, a highly ill-posed problem that suffers from significant non-uniqueness and typically requires strong prior assumptions or training data. While recent deep learning methods have achieved impressive performance, they generally rely on access to paired datasets of original and blurred images, which is often difficult to acquire. In this paper, we propose ECALL (Estimation-Calibrated Learning), a novel framework for joint kernel and signal recovery from unpaired data. ECALL operates without paired observations, instead using separate collections of original and blurred images. The method leverages moment-based constraints to estimate the kernel and simultaneously perform deconvolution. A key component of our approach is a loss function that incorporates cycle consistency with respect to the estimated kernel and a reconstruction operator, alongside terms that match statistical properties (e.g., moments) of the data distributions. We demonstrate the effectiveness of ECALL with numerical experiments, highlighting its potential in scenarios where paired data are unavailable.
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Inverse nodal problems for Sturm–Liouville operators with many frozen arguments

August 1, 2025

Yu Ping Wang, Shahrbanoo Akbarpoor, Baki Keskin

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In this paper, we study inverse nodal problems for Sturm–Liouville operators with many frozen arguments. We obtain the asymptotics of eigenvalues and nodal points (zeros) of the corresponding eigenfunction. Moreover, we establish a uniqueness theorem concerning the potential under condition q ∈ W 2 1 ⁢ [ 0 , π ] {q\in W_{2}^{1}[0,\pi]} . Numerical examples are presented to show the validity and effectiveness of the numerical scheme. Since the potential q ⁢ ( x ) {q(x)} for q ∈ L 2 ⁢ [ 0 , π ] {q\in L_{2}[0,\pi]} cannot be recovered by the classical method, we use an alternative method to reconstruct the potential as an application of numerical methods.
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Reconstruction of the heat transfer coefficient from boundary measurements

June 19, 2025

Sergey Grigor’evich Pyatkov, Oleg A. Soldatov

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Well-posedness questions of reconstruction of the heat transfer coefficients from boundary measurements are considered. The additional measurements are the integrals of a solution with weights over the boundary of a domain. The inverse problem theory is involved. The heat transfer coefficient is sought in the form of a finite segment of the series with unknown Fourier coefficients depending on time. They satisfy a system of integral equations of Volterra type. It is demonstrated that, under certain conditions, a solution to the problem is uniquely determined locally in time and depends continuously on the data of the problem. These conditions include smoothness and compatibility conditions on the data and additional conditions on the kernels of the integral operators used in the additional measurements. The method is constructive and allows to provide a numerical algorithm for solving the problem. The proof rely on a priori estimates and the contraction mapping principle.
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Convergence of a series associated with the convexification method for coefficient inverse problems

September 9, 2020

Michael V. Klibanov, Dinh-Liem Nguyen

Article number: 000010151520200042

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This paper is concerned with the convergence of a series associated with a certain version of the convexification method. That version has been recently developed by the research group of the first author for solving coefficient inverse problems. The convexification method aims to construct a globally convex Tikhonov-like functional with a Carleman weight function in it. In the previous works, the construction of the strictly convex weighted Tikhonov-like functional assumes a truncated Fourier series (i.e. a finite series instead of an infinite one) for a function generated by the total wave field. In this paper, we prove a convergence property for this truncated Fourier series approximation. More precisely, we show that the residual of the approximate PDE obtained by using the truncated Fourier series tends to zero in L2{L^{2}} as the truncation index in the truncated Fourier series tends to infinity. The proof relies on a convergence result in the H1{H^{1}}-norm for a sequence of L2{L^{2}}-orthogonal projections on finite-dimensional subspaces spanned by elements of a special Fourier basis. However, due to the ill-posed nature of coefficient inverse problems, we cannot prove that the solution of that approximate PDE, which results from the minimization of that Tikhonov-like functional, converges to the correct solution.
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Propagation of pollutants in the atmosphere based on 3D Navier–Stokes equations: Direct and inverse source problem

January 1, 2000

Elena S. Tyatyushkina, Andrey S. Kozelkov, Robert R. Giniyatullin, Maxim A. Shishlenin

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The article presents a description of the methodology for three-dimensional modeling of the gaseous pollutants spread in the atmospheric air. The methodology is based on solving a complete hydrodynamic system of equations – the Reynolds-averaged Navier–Stokes equations together with the equations of concentration transfer based on the Fick diffusion law, in conjunction with the equations of the turbulence model. Such a technique will allow for accurate modeling of the transfer of gaseous pollutants, without taking into account empirical patterns. The advantages of such a technique include the ability to take into account the turbulence, terrain, wind rose, underlying surface and geometry of pollution sources, accounting for the compressibility of the medium and the ability to specify an accurate dependence of the stratification of atmospheric air. The article presents the results of numerical modeling of the test case of propane emission from a single source, as well as the problem of dispersion of mixed emissions from a single chimney and determination of the maximum ground concentration of harmful substances. The results of the modeling are evaluated, and compared with experimental data, using the method of calculating the maximum one-time concentrations from emissions of a single point source using the adopted empirical methodology. It is shown that the technique allows estimating all hydrodynamic parameters of polluted air distribution with the required accuracy. The empirical approach is used for the inverse source problem of determining the location of the emission source by known pollutant concentrations. We show a good localization of the source by global optimization method (differential evolution) and that this approximate solution of inverse problem can be considered as good initial guess for iterative methods to solve the inverse source pollutant problem based on the Navier–Stokes equations.
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An inverse problem of determining the fractional order in a fractional SVIRS epidemic model based on the asymptotic solution

January 1, 2000

Zhen Wang, Gongsheng Li

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In this paper we propose a fractional SVIRS epidemic model by directly choosing the survival probability distribution as a power-law decay function in the non-Markovian process, where the model incorporates with the Riemann–Liouville fractional derivative and the coefficients depending upon the order of the derivative. The asymptotic solution of the fractional epidemic model is obtained by using the Laplace Adomian decomposition method (L-ADM), and numerical simulations are performed to reveal the impact of the order of the derivative on the fractional dynamic system. An inverse problem of determining the order of the fractional derivative by one data of the recovered people is transformed to a nonlinear equation with the aid of the asymptotic solution, and a uniqueness of the inverse problem is proved by monotonicity of the nonlinear function on the order.
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An inversion formula for the Radon transform in three-dimensional space with a weight function

January 1, 2000

Mukhiddin I. Muminov, Zarijfjon K. Ochilov

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In this paper we consider an inversion problem for the Radon transform in three-dimensional space with a weight function of a special form. Under some natural conditions we prove invertibility for the considering problem and find an explicit formula for the inverse of the problem.

Special Issue Dedicated to Professor Michael Victor Klibanov

Special Issue: The Third International Workshop on Computational Inverse Problems and Applications (2013)

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Journal Impact Factor	1.0	2024, Journal Citation Reports (Clarivate, 2025)
5-year Journal Impact Factor	1.1	2024, Journal Citation Reports (Clarivate, 2025)
Journal Citation Indicator	0.78	2024, Journal Citation Reports (Clarivate, 2025)
CiteScore	2.2	2024, Scopus (Elsevier B.V., 2025)
SCImago Journal Rank	0.513	2024, SJR (Scimago Lab, 2025; Data Source: Scopus)
Source Normalized Impact per Paper	1.058	2024, CWTS Journal Indicators (CWTS B.V., 2025; 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.
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[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.
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[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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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
Evgeny Burnaev, Moscow
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
Barbara Kaltenbacher, Klagenfurt　
Mikhail Yu. Kokurin, Yoshkar-Ola
Daniel Lesnic, Leeds
Jijun Liu, Nanjing
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
Alexander Shananin, Moscow
Otmar Scherzer, Vienna
Andrey Shkalikov, Moscow
Plamen D Stefanov, West Lafayette
Gunther Uhlmann, Seattle
Yanfei Wang, Beijing
Konstantin Vorontsov, Moscow
Masahiro Yamamoto, Tokyo
Vyacheslav A. Yurko, Saratov
Ye Zhang, Shenzhen
Jun Zou, Hong Kong

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Journal information

Additional information

eISSN:

1569-3945

ISSN:

0928-0219

Language:

English

Publisher:

De Gruyter

Additional information

First published:

January 1, 1993

Publication Frequency:

6 issues per year

eISSN: 1569-3945

ISSN: 0928-0219

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Journal of Inverse and Ill-posed Problems

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