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Mathematics
Special Issues
Computational and Analytical Methods for Inverse Problems
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Computational and Analytical Methods for Inverse Problems

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E: Applied Mathematics".

Deadline for manuscript submissions: 31 December 2025 | Viewed by 2786

Special Issue Editor

Dr. Mohsen Tadi

E-Mail Website
Guest Editor
Department of Civil & Mechanical Engineering, United States Military Academy, West Point, NY 10996, USA
Interests: inverse problems; dynamic systems; feedback controls; numerical mathematics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The aim of this Special Issue is to gather recent results on analytical and computational methods for inverse problems that appear in various physical systems. It is often the case that, given a physical system, a subsurface material property is unknown. Given a set of observations, the goal of an inverse problem is to recover the material property based on the given input/output data. Important physical applications include land/sea mine detection, breast cancer identification and many more. It is often the case that the system includes an unknown boundary condition such as surface ablation in re-entry vehicles or plasma location inside a Tokamak. Such problems are often referred to as Cauchy problems. It is well known that both Cauchy problems and inverse problems are severely ill-posed.

The aim of this Special Issue is to gather recent results on such problems. In particular, new results on computational and analytical methods for such problems are welcomed. Applications of the existing methods to various fields are also welcomed, such as imaging biological systems, recovering subsurface material properties in elastic/chemical/thermal systems, and recovering interior unknowns from boundary measurements in furnaces/Tokamaks. Theoretical results on the uniqueness, regularity and identifiability of unknown functions/boundaries are also welcomed. New results on the regularization of such problems are also welcomed.

Dr. Mohsen Tadi
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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 problem
  • Cauchy problem
  • regularization
  • land/sea mine detection
  • impedance tomography
  • inverse wave scattering in elastic/electromagnetic domains

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Published Papers (2 papers)

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Research

26 pages, 17158 KiB  
Article
An Inverse Problem for Estimating Spatially and Temporally Dependent Surface Heat Flux with Thermography Techniques
by Cheng-Hung Huang and Kuan-Chieh Fang
Mathematics 2024, 12(10), 1584; https://doi.org/10.3390/math12101584 - 19 May 2024
Viewed by 964
Abstract
In this study, an inverse conjugate heat transfer problem is examined to estimate temporally and spatially the dependent unknown surface heat flux using thermography techniques with a thermal camera in a three-dimensional domain. Thermography techniques encompass a broad set of methods and procedures [...] Read more.
In this study, an inverse conjugate heat transfer problem is examined to estimate temporally and spatially the dependent unknown surface heat flux using thermography techniques with a thermal camera in a three-dimensional domain. Thermography techniques encompass a broad set of methods and procedures used for capturing and analyzing thermal data, while thermal cameras are specific tools used within those techniques to capture thermal images. In the present study, the interface conditions of the plate and air domains are obtained using perfect thermal contact conditions, and therefore we define the problem studied as an inverse conjugate heat transfer problem. Achieving the simultaneous solution of the continuity, Navier–Stokes, and energy equations within the air domain, alongside the heat conduction equation in the plate domain, presents a more intricate challenge compared to conventional inverse heat conduction problems. The validity of our inverse solutions was verified through numerical simulations, considering various inlet air velocities and plate thicknesses. Notably, it was found that due to the singularity of the gradient of the cost function at the final time point, the estimated results near the final time must be discarded, and exact measurements consistently produce accurate boundary heat fluxes under thin-plate conditions, with air velocity exhibiting no significant impact on the estimates. Additionally, an analysis of measurement errors and their influence on the inverse solutions was conducted. The numerical results conclusively demonstrated that the maximum error when estimating heat flux consistently remained below 3% and higher measurement noise resulted in the accuracy of the heat flux estimation decreasing. This underscores the inherent challenges associated with inverse problems and highlights the importance of obtaining accurate measurement data in the problem domain. Full article
(This article belongs to the Special Issue Computational and Analytical Methods for Inverse Problems)
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8 pages, 583 KiB  
Article
Deep Neural Network-Oriented Indicator Method for Inverse Scattering Problems Using Partial Data
by Yule Lin, Xiaoyi Yan, Jiguang Sun and Juan Liu
Mathematics 2024, 12(4), 522; https://doi.org/10.3390/math12040522 - 7 Feb 2024
Cited by 2 | Viewed by 1267
Abstract
We consider the inverse scattering problem to reconstruct an obstacle using partial far-field data due to one incident wave. A simple indicator function, which is negative inside the obstacle and positive outside of it, is constructed and then learned using a deep neural [...] Read more.
We consider the inverse scattering problem to reconstruct an obstacle using partial far-field data due to one incident wave. A simple indicator function, which is negative inside the obstacle and positive outside of it, is constructed and then learned using a deep neural network (DNN). The method is easy to implement and effective as demonstrated by numerical examples. Rather than developing sophisticated network structures for the classical inverse operators, we reformulate the inverse problem as a suitable operator such that standard DNNs can learn it well. The idea of the DNN-oriented indicator method can be generalized to treat other partial data inverse problems. Full article
(This article belongs to the Special Issue Computational and Analytical Methods for Inverse Problems)
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