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

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: 31 January 2021.

Special Issue Editor

Prof. Dr. Christine Böckmann
Website
Guest Editor
Institut für Mathematik, Universität Potsdam, Karl-Liebknecht-Str. 24-25, D-14476 Potsdam OT Golm, Germany
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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Keywords

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

Published Papers (12 papers)

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Open AccessArticle
Convergence Rate of the Modified Landweber Method for Solving Inverse Potential Problems
Mathematics 2020, 8(4), 608; https://doi.org/10.3390/math8040608 - 16 Apr 2020
Abstract
In this paper, we present the convergence rate analysis of the modified Landweber method under logarithmic source condition for nonlinear ill-posed problems. The regularization parameter is chosen according to the discrepancy principle. The reconstructions of the shape of an unknown domain for an [...] Read more.
In this paper, we present the convergence rate analysis of the modified Landweber method under logarithmic source condition for nonlinear ill-posed problems. The regularization parameter is chosen according to the discrepancy principle. The reconstructions of the shape of an unknown domain for an inverse potential problem by using the modified Landweber method are exhibited. Full article
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)
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Open AccessArticle
The Impact of the Discrepancy Principle on the Tikhonov-Regularized Solutions with Oversmoothing Penalties
Mathematics 2020, 8(3), 331; https://doi.org/10.3390/math8030331 - 03 Mar 2020
Abstract
This paper deals with the Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. One focus is on the application of the discrepancy principle for choosing the regularization parameter and its consequences. Numerical case studies are performed in order [...] Read more.
This paper deals with the Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. One focus is on the application of the discrepancy principle for choosing the regularization parameter and its consequences. Numerical case studies are performed in order to complement analytical results concerning the oversmoothing situation. For example, case studies are presented for exact solutions of Hölder type smoothness with a low Hölder exponent. Moreover, the regularization parameter choice using the discrepancy principle, for which rate results are proven in the oversmoothing case in in reference (Hofmann, B.; Mathé, P. Inverse Probl. 2018, 34, 015007) is compared to Hölder type a priori choices. On the other hand, well-known analytical results on the existence and convergence of regularized solutions are summarized and partially augmented. In particular, a sketch for a novel proof to derive Hölder convergence rates in the case of oversmoothing penalties is given, extending ideas from in reference (Hofmann, B.; Plato, R. ETNA. 2020, 93). Full article
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)
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Open AccessArticle
Generalized-Fractional Tikhonov-Type Method for the Cauchy Problem of Elliptic Equation
Mathematics 2020, 8(1), 48; https://doi.org/10.3390/math8010048 - 01 Jan 2020
Abstract
This article researches an ill-posed Cauchy problem of the elliptic-type equation. By placing the a-priori restriction on the exact solution we establish conditional stability. Then, based on the generalized Tikhonov and fractional Tikhonov methods, we construct a generalized-fractional Tikhonov-type regularized solution to recover [...] Read more.
This article researches an ill-posed Cauchy problem of the elliptic-type equation. By placing the a-priori restriction on the exact solution we establish conditional stability. Then, based on the generalized Tikhonov and fractional Tikhonov methods, we construct a generalized-fractional Tikhonov-type regularized solution to recover the stability of the considered problem, and some sharp-type estimates of convergence for the regularized method are derived under the a-priori and a-posteriori selection rules for the regularized parameter. Finally, we verify that the proposed method is efficient and acceptable by making the corresponding numerical experiments. Full article
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)
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Open AccessArticle
The Truncation Regularization Method for Identifying the Initial Value on Non-Homogeneous Time-Fractional Diffusion-Wave Equations
Mathematics 2019, 7(11), 1007; https://doi.org/10.3390/math7111007 - 23 Oct 2019
Cited by 4
Abstract
In the essay, we consider an initial value question for a mixed initial-boundary value of time-fractional diffusion-wave equations. This matter is an ill-posed problem; the solution relies discontinuously on the measured information. The truncation regularization technique is used for restoring the initial value [...] Read more.
In the essay, we consider an initial value question for a mixed initial-boundary value of time-fractional diffusion-wave equations. This matter is an ill-posed problem; the solution relies discontinuously on the measured information. The truncation regularization technique is used for restoring the initial value functions. The convergence estimations are given in a priori regularization parameter choice regulations and a posteriori regularization parameter choice regulations. Numerical examples are given to demonstrate this is effective and practicable. Full article
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)
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Open AccessArticle
A New Method for De-Noising of Well Test Pressure Data Base on Legendre Approximation
Mathematics 2019, 7(10), 989; https://doi.org/10.3390/math7100989 - 18 Oct 2019
Abstract
In this paper, noise removing of the well test data is considered. We use the Legendre expansion to approximate well test data and a truncated strategy has been employed to reduce noise. The parameter of the truncation will be chosen by a discrepancy [...] Read more.
In this paper, noise removing of the well test data is considered. We use the Legendre expansion to approximate well test data and a truncated strategy has been employed to reduce noise. The parameter of the truncation will be chosen by a discrepancy principle and a corresponding convergence result has been obtained. The theoretical analysis shows that a well numerical approximation can be obtained by the new method. Moreover, we can directly obtain the stable numerical derivatives of the pressure data in this method. Finally, we give some numerical tests to show the effectiveness of the method. Full article
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)
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Open AccessArticle
Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum
Mathematics 2019, 7(8), 667; https://doi.org/10.3390/math7080667 - 25 Jul 2019
Cited by 1Correction
Abstract
We investigate a Cauchy problem of the modified Helmholtz equation with nonhomogeneous Dirichlet and Neumann datum, this problem is ill-posed and some regularization techniques are required to stabilize numerical computation. We established the result of conditional stability under an a priori assumption for [...] Read more.
We investigate a Cauchy problem of the modified Helmholtz equation with nonhomogeneous Dirichlet and Neumann datum, this problem is ill-posed and some regularization techniques are required to stabilize numerical computation. We established the result of conditional stability under an a priori assumption for an exact solution. A generalized Tikhonov method is proposed to solve this problem, we select the regularization parameter by a priori and a posteriori rules and derive the convergence results of sharp type for this method. The corresponding numerical experiments are implemented to verify that our regularization method is practicable and satisfied. Full article
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)
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Open AccessFeature PaperArticle
Solutions of Direct and Inverse Even-Order Sturm-Liouville Problems Using Magnus Expansion
Mathematics 2019, 7(6), 544; https://doi.org/10.3390/math7060544 - 14 Jun 2019
Abstract
In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct [...] Read more.
In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively. Full article
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)
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Open AccessArticle
A Numerical Method for Filtering the Noise in the Heat Conduction Problem
Mathematics 2019, 7(6), 502; https://doi.org/10.3390/math7060502 - 02 Jun 2019
Abstract
In this paper, we give an effective numerical method for the heat conduction problem connected with the Laplace equation. Through the use of a single-layer potential approach to the solution, we get the boundary integral equation about the density function. In order to [...] Read more.
In this paper, we give an effective numerical method for the heat conduction problem connected with the Laplace equation. Through the use of a single-layer potential approach to the solution, we get the boundary integral equation about the density function. In order to deal with the weakly singular kernel of the integral equation, we give the projection method to deal with this part, i.e., using the Lagrange trigonometric polynomials basis to give an approximation of the density function. Although the problems under investigation are well-posed, herein the Tikhonov regularization method is not used to regularize the aforementioned direct problem with noisy data, but to filter out the noise in the corresponding perturbed data. Finally, the effectiveness of the proposed method is demonstrated using a few examples, including a boundary condition with a jump discontinuity and a boundary condition with a corner. Whilst a comparative study with the method of fundamental solutions (MFS) is also given. Full article
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)
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Open AccessArticle
A Numerical Approximation Method for the Inverse Problem of the Three-Dimensional Laplace Equation
Mathematics 2019, 7(6), 487; https://doi.org/10.3390/math7060487 - 28 May 2019
Cited by 1
Abstract
In this article, an inverse problem with regards to the Laplace equation with non-homogeneous Neumann boundary conditions in a three-dimensional case is investigated. To deal with this problem, a regularization method (mollification method) with the bivariate de la Vallée Poussin kernel is proposed. [...] Read more.
In this article, an inverse problem with regards to the Laplace equation with non-homogeneous Neumann boundary conditions in a three-dimensional case is investigated. To deal with this problem, a regularization method (mollification method) with the bivariate de la Vallée Poussin kernel is proposed. Stable estimates are obtained under a priori bound assumptions and an appropriate choice of the regularization parameter. The error estimates indicate that the solution of the approximation continuously depends on the noisy data. Two experiments are presented, in order to validate the proposed method in terms of accuracy, convergence, stability, and efficiency. Full article
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)
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Open AccessArticle
A Modified Asymptotical Regularization of Nonlinear Ill-Posed Problems
Mathematics 2019, 7(5), 419; https://doi.org/10.3390/math7050419 - 10 May 2019
Cited by 1
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 - 20 Apr 2019
Cited by 3
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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Open AccessCorrection
Correction: Zhang, H.; Zhang, X. Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum. Mathematics 2019, 7, 667
Mathematics 2019, 7(10), 888; https://doi.org/10.3390/math7100888 - 24 Sep 2019
Abstract
The authors wish to make the following corrections to this paper [...] Full article
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

Title: The impact of the discrepancy principle on Tikhonov regularized solutions when the penalty is oversmoothing
Authors: Bernd Hofmann and Christopher Hofmann
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