Inverse and Ill-Posed Problems

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (31 July 2020) | Viewed by 15095

Special Issue Editor


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Guest Editor
Department of Cybernetics, Tallinn University of Technology, 19086 Tallinn, Estonia
Interests: inverse problems; methods of regularization; integral equations

Special Issue Information

Dear Colleagues,

At present, mathematical modeling is used in wider and wider areas of science and technology. In this process, inverse problems have a significant role—they enable to determine not directly measurable parameters of models. This is one reason why the study of inverse problems, which started in the beginning of 20th century, has rapidly intensified in last decades. Modern areas of application of inverse problems include mathematical biology, materials science, remote sensing, medical imaging, geophysics, oceanography, mathematical finance, etc.

Often a practical solution of an inverse problem is accompanied by specific difficulties: The problem is ill-posed, or the data of the problem are abnormally large or irregular. For ill-posed problems, proper regularization procedures are elaborated. This is complemented by a development of statistical methods for inversion.

The aim of this Special Issue is to make a collection of papers that comprises the latest developments in mathematics (theory, numerics) of inverse and ill-posed problems.

Prof. Dr. Jaan Janno
Guest Editor

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Keywords

  • Inverse problem
  • Regularization
  • Statistical inversion

Published Papers (6 papers)

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Research

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10 pages, 1466 KiB  
Article
Inverse Minimum Cut Problem with Lower and Upper Bounds
by Adrian Deaconu and Laura Ciupala
Mathematics 2020, 8(9), 1494; https://doi.org/10.3390/math8091494 - 3 Sep 2020
Cited by 3 | Viewed by 2039
Abstract
The inverse minimum cut problem is one of the classical inverse optimization researches. In this paper, the inverse minimum cut with a lower and upper bounds problem is considered. The problem is to change both, the lower and upper bounds on arcs so [...] Read more.
The inverse minimum cut problem is one of the classical inverse optimization researches. In this paper, the inverse minimum cut with a lower and upper bounds problem is considered. The problem is to change both, the lower and upper bounds on arcs so that a given feasible cut becomes a minimum cut in the modified network and the distance between the initial vector of bounds and the modified one is minimized. A strongly polynomial algorithm to solve the problem under l1 norm is developed. Full article
(This article belongs to the Special Issue Inverse and Ill-Posed Problems)
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21 pages, 666 KiB  
Article
Q-Curve and Area Rules for Choosing Heuristic Parameter in Tikhonov Regularization
by Toomas Raus and Uno Hämarik
Mathematics 2020, 8(7), 1166; https://doi.org/10.3390/math8071166 - 16 Jul 2020
Cited by 1 | Viewed by 1805
Abstract
We consider choice of the regularization parameter in Tikhonov method if the noise level of the data is unknown. One of the best rules for the heuristic parameter choice is the quasi-optimality criterion where the parameter is chosen as the global minimizer of [...] Read more.
We consider choice of the regularization parameter in Tikhonov method if the noise level of the data is unknown. One of the best rules for the heuristic parameter choice is the quasi-optimality criterion where the parameter is chosen as the global minimizer of the quasi-optimality function. In some problems this rule fails. We prove that one of the local minimizers of the quasi-optimality function is always a good regularization parameter. For the choice of the proper local minimizer we propose to construct the Q-curve which is the analogue of the L-curve, but on the x-axis we use modified discrepancy instead of discrepancy and on the y-axis the quasi-optimality function instead of the norm of the approximate solution. In the area rule we choose for the regularization parameter such local minimizer of the quasi-optimality function for which the area of the polygon, connecting on Q-curve this minimum point with certain maximum points, is maximal. We also provide a posteriori error estimates of the approximate solution, which allows to check the reliability of the parameter chosen heuristically. Numerical experiments on an extensive set of test problems confirm that the proposed rules give much better results than previous heuristic rules. Results of proposed rules are comparable with results of the discrepancy principle and the monotone error rule, if the last two rules use the exact noise level. Full article
(This article belongs to the Special Issue Inverse and Ill-Posed Problems)
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15 pages, 285 KiB  
Article
Unique Determination of the Shape of a Scattering Screen from a Passive Measurement
by Emilia Blåsten, Lassi Päivärinta and Sadia Sadique
Mathematics 2020, 8(7), 1156; https://doi.org/10.3390/math8071156 - 15 Jul 2020
Cited by 4 | Viewed by 1857
Abstract
We consider the problem of fixed frequency acoustic scattering from a sound-soft flat screen. More precisely, the obstacle is restricted to a two-dimensional plane and interacting with an arbitrary incident wave, it scatters acoustic waves to three-dimensional space. The model is particularly relevant [...] Read more.
We consider the problem of fixed frequency acoustic scattering from a sound-soft flat screen. More precisely, the obstacle is restricted to a two-dimensional plane and interacting with an arbitrary incident wave, it scatters acoustic waves to three-dimensional space. The model is particularly relevant in the study and design of reflecting sonars and antennas, cases where one cannot assume that the incident wave is a plane wave. Our main result is that given the plane where the screen is located, the far-field pattern produced by any single arbitrary incident wave determines the exact shape of the screen, as long as it is not antisymmetric with respect to the plane. This holds even for screens whose shape is an arbitrary simply connected smooth domain. This is in contrast to earlier work where the incident wave had to be a plane wave, or more recent work where only polygonal scatterers are determined. Full article
(This article belongs to the Special Issue Inverse and Ill-Posed Problems)
17 pages, 959 KiB  
Article
Multiscale Compression Algorithm for Solving Nonlinear Ill-Posed Integral Equations via Landweber Iteration
by Rong Zhang, Fanchun Li and Xingjun Luo
Mathematics 2020, 8(2), 221; https://doi.org/10.3390/math8020221 - 9 Feb 2020
Cited by 4 | Viewed by 1725
Abstract
In this paper, Landweber iteration with a relaxation factor is proposed to solve nonlinear ill-posed integral equations. A compression multiscale Galerkin method that retains the properties of the Landweber iteration is used to discretize the Landweber iteration. This method leads to the optimal [...] Read more.
In this paper, Landweber iteration with a relaxation factor is proposed to solve nonlinear ill-posed integral equations. A compression multiscale Galerkin method that retains the properties of the Landweber iteration is used to discretize the Landweber iteration. This method leads to the optimal convergence rates under certain conditions. As a consequence, we propose a multiscale compression algorithm to solve nonlinear ill-posed integral equations. Finally, the theoretical analysis is verified by numerical results. Full article
(This article belongs to the Special Issue Inverse and Ill-Posed Problems)
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16 pages, 310 KiB  
Article
An Inverse Problem for a Generalized Fractional Derivative with an Application in Reconstruction of Time- and Space-Dependent Sources in Fractional Diffusion and Wave Equations
by Nataliia Kinash and Jaan Janno
Mathematics 2019, 7(12), 1138; https://doi.org/10.3390/math7121138 - 21 Nov 2019
Cited by 35 | Viewed by 3078
Abstract
In this article, we consider two inverse problems with a generalized fractional derivative. The first problem, IP1, is to reconstruct the function u based on its value and the value of its fractional derivative in the neighborhood of the final time. We prove [...] Read more.
In this article, we consider two inverse problems with a generalized fractional derivative. The first problem, IP1, is to reconstruct the function u based on its value and the value of its fractional derivative in the neighborhood of the final time. We prove the uniqueness of the solution to this problem. Afterwards, we investigate the IP2, which is to reconstruct a source term in an equation that generalizes fractional diffusion and wave equations, given measurements in a neighborhood of final time. The source to be determined depends on time and all space variables. The uniqueness is proved based on the results for IP1. Finally, we derive the explicit solution formulas to the IP1 and IP2 for some particular cases of the generalized fractional derivative. Full article
(This article belongs to the Special Issue Inverse and Ill-Posed Problems)

Review

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27 pages, 383 KiB  
Review
Applications of Microlocal Analysis in Inverse Problems
by Mikko Salo
Mathematics 2020, 8(7), 1184; https://doi.org/10.3390/math8071184 - 18 Jul 2020
Cited by 2 | Viewed by 3323
Abstract
This note reviews certain classical applications of microlocal analysis in inverse problems. The text is based on lecture notes for a postgraduate level minicourse on applications of microlocal analysis in inverse problems, given in Helsinki and Shanghai in June 2019. Full article
(This article belongs to the Special Issue Inverse and Ill-Posed Problems)
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