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Mathematics
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Inverse Problems and Numerical Computation in Mathematical Physics
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Inverse Problems and Numerical Computation in Mathematical Physics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E4: Mathematical Physics".

Deadline for manuscript submissions: 10 January 2026 | Viewed by 2183

Special Issue Editor

Prof. Dr. Youjun Deng

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Central South University, Changsha 410083, China
Interests: applied mathematics; inverse problems; partial differential equations; numerical computation

Special Issue Information

Dear Colleagues,

Inverse problems and numerical computation in mathematical physics is a rapidly evolving field that continues to attract significant attention from researchers in mathematics, physics, engineering and other disciplines. Advances in this area have the potential to significantly impact various industries and scientific endeavors by enabling more accurate and reliable solutions to complex inverse problems.

The suitable topics for this Special Issue include, but are not limited to, the following:

Theory and methods for solving inverse problems in mathematical physics;

Applications of inverse problems in various fields, such as nondestructive testing, seismic imaging, medical imaging and material sciences;

Uniqueness and stability of solutions for inverse problems;

Numerical methods for solving inverse problems, including iterative sequences, optimization techniques and regularization strategies;

Recent advances and novel approaches in inverse problems and numerical computation;

Inverse problems in specific mathematical physics contexts, such as layered media, impedance, gravimetry and heat conduction;

Deep learning approaches for solving inverse problems or numerical computation in mathematical physics, including neural network architectures, training strategies and applications in various domains.

Prof. Dr. Youjun Deng
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 semimonthly 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 2600 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
  • numerical computation
  • uniqueness and stability
  • iterative methods
  • optimization
  • regularization
  • deep learning

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Published Papers (3 papers)

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Research

17 pages, 472 KiB  
Article
Detection of a Spatial Source Term Within a Multi-Dimensional, Multi-Term Time-Space Fractional Diffusion Equation
by Mofareh Alhazmi, Yasser Alrashedi, Hamed Ould Sidi and Maawiya Ould Sidi
Mathematics 2025, 13(5), 705; https://doi.org/10.3390/math13050705 - 21 Feb 2025
Viewed by 339
Abstract
The main objective of this study was to identify the undetermined source term (ST) in a fractional space-time scattering equation with multiple terms, using data obtained from the most recent observations. To address this complex problem, we reformulated the equation by adopting a [...] Read more.
The main objective of this study was to identify the undetermined source term (ST) in a fractional space-time scattering equation with multiple terms, using data obtained from the most recent observations. To address this complex problem, we reformulated the equation by adopting a regularization-based optimization approach. This methodology not only makes it possible to determine the existence of a single minimum solution, but also to assess its stability. In the numerical context, we estimate and approach the function (ST) by applying the Levenberg–Marquardt regularization method, a powerful tool for solving inverse problems. In order to demonstrate the effectiveness of the proposed approach, we performed numerical simulations in one-dimensional and two-dimensional scenarios. These simulations illustrate our method’s ability to process complex data and provide accurate and stable solutions. Through this extended approach, we aimed to discover the single source term in a multi-term space-time fractional scattering equation, ensuring robust and reliable results, supported by the most recent observational data. Full article
(This article belongs to the Special Issue Inverse Problems and Numerical Computation in Mathematical Physics)
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15 pages, 4991 KiB  
Article
Enhanced Small Reflections Sparse-Spike Seismic Inversion with Iterative Hybrid Thresholding Algorithm
by Yue Feng, Ronghuo Dai and Zidan Fan
Mathematics 2025, 13(1), 37; https://doi.org/10.3390/math13010037 - 26 Dec 2024
Viewed by 650
Abstract
Seismic inversion is a process of imaging or predicting the spatial and physical properties of underground strata. The most commonly used one is sparse-spike seismic inversion with sparse regularization. There are many effective methods to solve sparse regularization, such as L0-norm, L1-norm, weighted [...] Read more.
Seismic inversion is a process of imaging or predicting the spatial and physical properties of underground strata. The most commonly used one is sparse-spike seismic inversion with sparse regularization. There are many effective methods to solve sparse regularization, such as L0-norm, L1-norm, weighted L1-norm, etc. This paper studies the sparse-spike inversion with L0-norm. It is usually solved by the iterative hard thresholding algorithm (IHTA) or its faster variants. However, hard thresholding algorithms often lead to a sharp increase or numerical oscillation of the residual, which will affect the inversion results. In order to deal with this issue, this paper attempts the idea of the relaxed optimal thresholding algorithm (ROTA). In the solution process, due to the particularity of the sparse constraints in this convex relaxation model, this model can be considered as a L1-norm problem when dealt with the location of non-zero elements. We use a modified iterative soft thresholding algorithm (MISTA) to solve it. Hence, it forms a new algorithm called the iterative hybrid thresholding algorithm (IHyTA), which combines IHTA and MISTA. The synthetic and real seismic data tests show that, compared with IHTA, the results of IHyTA are more accurate with the same SNR. IHyTA improves the noise resistance. Full article
(This article belongs to the Special Issue Inverse Problems and Numerical Computation in Mathematical Physics)
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15 pages, 3035 KiB  
Article
Application of Kirchhoff Migration from Two-Dimensional Fresnel Dataset by Converting Unavailable Data into a Constant
by Won-Kwang Park
Mathematics 2024, 12(20), 3253; https://doi.org/10.3390/math12203253 - 17 Oct 2024
Viewed by 714
Abstract
In this contribution, we consider an application of the Kirchhoff migration (KM) technique for fast and accurate identification of small dielectric objects from two-dimensional Fresnel experimental dataset. Generally, for successful application of the KM, a complete set of elements from the so-called multi-static [...] Read more.
In this contribution, we consider an application of the Kirchhoff migration (KM) technique for fast and accurate identification of small dielectric objects from two-dimensional Fresnel experimental dataset. Generally, for successful application of the KM, a complete set of elements from the so-called multi-static response (MSR) matrix must be collected; however, in the Fresnel experimental dataset, many of the elements of an MSR matrix are not measurable. Nevertheless, the existence, location, and outline shape of small objects can be retrieved using the KM by converting unavailable data into the zero constant. However, the theoretical reason behind such conversion has not been confirmed to date. In order to explain this theoretical reason, we convert unavailable measurement data into a constant and demonstrate that the imaging function of the KM can be expressed by an infinite series of the Bessel functions of integer order of the first kind, the object’s material properties, and the converted constant. Following the theoretical result, we confirm that converting unknown data into the zero constant guarantees good results and unique determination of the objects. Finally, various numerical simulation results from Fresnel experimental dataset are presented and discussed to validate the theoretical result. Full article
(This article belongs to the Special Issue Inverse Problems and Numerical Computation in Mathematical Physics)
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