Special Issue "Numerical Analysis: Inverse Problems - Theory and Applications"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: 1 December 2019

Special Issue Editor

Guest Editor
Prof. Dr. Christine Böckmann

Institut für Mathematik, Universität Potsdam, Karl-Liebknecht-Str. 24-25, D-14476 Potsdam OT Golm, Germany
Website | E-Mail
Phone: +49-(0)331-977-1743
Fax: +49-(0)331-977-1001
Interests: Numerical and applied mathematics; linear and nonlinear inverse ill-posed problems; regularization methods; numerical methods for inverse Sturm-Liouville problems; applications in atmospheric physics

Special Issue Information

Dear Colleagues,

This Special Issue, “Numerical Analysis: Inverse Problems - Theory and Applications”, will be open for the publication of high-quality mathematical papers in the area of linear and nonlinear inverse ill-posed and well-posed problems.

A plenty of problems in mathematics, economics, physics, biology, chemistry, engineering, in particular, e.g., optics, radar, acoustics, communication theory, signal processing, medical imaging, computer vision, geophysics, oceanography, astronomy, remote sensing, natural language processing, machine learning, non-destructive testing and other disciplines can be reduced to solving an inverse problem in an abstract space, e.g., in Hilbert and Banach spaces. It is called an inverse problem because it starts with the results and then calculates the causes. Solving inverse problems is a non-trivial task that involves many areas of Mathematics and Techniques. In case the problem is ill-posed small errors in the data are greatly amplified in the solution, therefore, regularization techniques using parameter choice rules with optimal convergence rates are necessary.

While inverse problems are often formulated in infinite dimensional spaces, limitations to a finite number of often noisy data, and the practical consideration of recovering only a finite number of unknown parameters, may lead to the problems being recast in discrete form. In this case the inverse problem will typically be ill-conditioned.

Papers involving all those above mentioned topics are welcome. Moreover, this special issue gives an opportunity to researchers and practitioners to communicate their ideas.

Prof. Dr. Christine Böckmann
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 850 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Inverse problems
  • Ill- and well-posed problems
  • Regularization
  • Inverse Sturm-Liouville problems

Published Papers (2 papers)

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Research

Open AccessArticle
A Modified Asymptotical Regularization of Nonlinear Ill-Posed Problems
Mathematics 2019, 7(5), 419; https://doi.org/10.3390/math7050419
Received: 19 March 2019 / Revised: 25 April 2019 / Accepted: 30 April 2019 / Published: 10 May 2019
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Abstract
In this paper, we investigate the continuous version of modified iterative Runge–Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is [...] Read more.
In this paper, we investigate the continuous version of modified iterative Runge–Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of F ( x δ ( T ) ) y δ = τ δ + for some δ + > δ , and an appropriate source condition. We yield the optimal rate of convergence. Full article
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems - Theory and Applications)
Open AccessArticle
A Regularization Method to Solve a Cauchy Problem for the Two-Dimensional Modified Helmholtz Equation
Mathematics 2019, 7(4), 360; https://doi.org/10.3390/math7040360
Received: 31 January 2019 / Revised: 6 April 2019 / Accepted: 16 April 2019 / Published: 20 April 2019
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Abstract
In this paper, the ill-posed problem of the two-dimensional modified Helmholtz equation is investigated in a strip domain. For obtaining a stable numerical approximation solution, a mollification regularization method with the de la Vallée Poussin kernel is proposed. An error estimate between the [...] Read more.
In this paper, the ill-posed problem of the two-dimensional modified Helmholtz equation is investigated in a strip domain. For obtaining a stable numerical approximation solution, a mollification regularization method with the de la Vallée Poussin kernel is proposed. An error estimate between the exact solution and approximation solution is given under suitable choices of the regularization parameter. Two numerical experiments show that our procedure is effective and stable with respect to perturbations in the data. Full article
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems - Theory and Applications)
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