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Symmetry
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Advanced Mathematical and Simulation Methods for Inverse Problems
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Advanced Mathematical and Simulation Methods for Inverse Problems

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (15 January 2024) | Viewed by 9450

Special Issue Editor

Dr. Stephen Kirkup

E-Mail Website
Guest Editor
School of Engineering, University of Central Lancashire, Preston PR1 2HE, UK

Special Issue Information

Dear Colleagues,

Inverse problems are studied in mathematics, science, and engineering, and they involve finding an unknown property of a medium or object from a probing excitation or observation. Inverse problems fit in with the Symmetry concept of this journal as they are the opposite of the associated forward problem, in which the causes are set and the effects are determined. Clearly, if a solution to an inverse problem is passed to the forward problem, then the original data should be returned at least approximately.

A solution of an inverse problem is useful, as it can return information that is not measurable, for example, if the source is inaccessible or in the past. They have a wide range of applications, including acoustics, medical imaging, non-destructive testing, computer vision, remote sensing, radar, oceanography, and geophysics.

Inverse problems are said to be ill-posed; there is usually no clear-cut solution, as there tends to be with forward problems. Usually, there is a range of possible solutions, even when assumptions are made about the sort of solutions that are allowable. This Special Issue of Symmetry focuses on the the current state of the art in mathematical analysis and simulation in inverse problems and how they relate to their applications in science and engineering.

Dr. Stephen Kirkup
Guest Editor

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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. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

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Published Papers (4 papers)

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Research

15 pages, 3083 KiB  
Article
Dynamic Load Identification for Structures with Unknown Parameters
by Hongzhi Tang, Jinhui Jiang, M. Shadi Mohamed, Fang Zhang and Xu Wang
Symmetry 2022, 14(11), 2449; https://doi.org/10.3390/sym14112449 - 18 Nov 2022
Cited by 5 | Viewed by 1697
Abstract
The inverse problem and the direct problem are symmetrical to each other. As a mathematical method for inverse problems, dynamic load identification is applicable to the situation when the load acting on the structure is difficult to measure directly. In addition, in most [...] Read more.
The inverse problem and the direct problem are symmetrical to each other. As a mathematical method for inverse problems, dynamic load identification is applicable to the situation when the load acting on the structure is difficult to measure directly. In addition, in most practical fields, the exact value of the structural parameters cannot be obtained precisely, which makes the inverse problem beyond the capabilities of traditional dynamic load identification methods. Hence, in this work, we propose a dynamic load identification algorithm based on the extended Kalman filter (EKF) for a structure with unknown parameters. The algorithm is discussed under different conditions where the unknown parameters are either the stiffness or the mass of the structure. Such a case has not been considered in other literature yet. In order to verify the advantages of the proposed method, the recursive least square method was also used to compare the results. A 5-Dof symmetric system with unknown coefficients was selected for numerical simulation examples, and the influence of noise on the algorithm was also considered in the simulation. The results show that the proposed algorithm is effective for structures with unknown mass and stiffness coefficients. Compared with the recursive least square method, the method proposed in this paper has the higher accuracy and a wider application scope. Full article
(This article belongs to the Special Issue Advanced Mathematical and Simulation Methods for Inverse Problems)
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22 pages, 1864 KiB  
Article
Multifrequency Topological Derivative Approach to Inverse Scattering Problems in Attenuating Media
by Ana Carpio and María-Luisa Rapún
Symmetry 2021, 13(9), 1702; https://doi.org/10.3390/sym13091702 - 15 Sep 2021
Cited by 2 | Viewed by 2143
Abstract
Detecting objects hidden in a medium is an inverse problem. Given data recorded at detectors when sources emit waves that interact with the medium, we aim to find objects that would generate similar data in the presence of the same waves. In opposition, [...] Read more.
Detecting objects hidden in a medium is an inverse problem. Given data recorded at detectors when sources emit waves that interact with the medium, we aim to find objects that would generate similar data in the presence of the same waves. In opposition, the associated forward problem describes the evolution of the waves in the presence of known objects. This gives a symmetry relation: if the true objects (the desired solution of the inverse problem) were considered for solving the forward problem, then the recorded data should be returned. In this paper, we develop a topological derivative-based multifrequency iterative algorithm to reconstruct objects buried in attenuating media with limited aperture data. We demonstrate the method on half-space configurations, which can be related to problems set in the whole space by symmetry. One-step implementations of the algorithm provide initial approximations, which are improved in a few iterations. We can locate object components of sizes smaller than the main components, or buried deeper inside. However, attenuation effects hinder object detection depending on the size and depth for fixed ranges of frequencies. Full article
(This article belongs to the Special Issue Advanced Mathematical and Simulation Methods for Inverse Problems)
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26 pages, 9402 KiB  
Article
Online Dynamic Load Identification Based on Extended Kalman Filter for Structures with Varying Parameters
by Hongqiu Li, Jinhui Jiang and M Shadi Mohamed
Symmetry 2021, 13(8), 1372; https://doi.org/10.3390/sym13081372 - 28 Jul 2021
Cited by 9 | Viewed by 2062
Abstract
Dynamic load identification is an inverse problem concerned with finding the load applied on a structure when the dynamic characteristics and the response of the structure are known. In engineering applications, some of the structure parameters such as the mass or the stiffness [...] Read more.
Dynamic load identification is an inverse problem concerned with finding the load applied on a structure when the dynamic characteristics and the response of the structure are known. In engineering applications, some of the structure parameters such as the mass or the stiffness may be unknown and/or may change in time. In this paper, an online dynamic load identification algorithm based on an extended Kalman filter is proposed. The algorithm not only identifies the load by measuring the structural response but also identifies the unknown structure parameters and tracks their changes. We discuss the proposed algorithm for the cases when the unknown parameters are the stiffness or the mass coefficients. Furthermore, for a system with many degrees of freedom and to achieve online computations, we implement the model reduction theory. Thus, we reduce the number of degrees of freedom in the resulting symmetric system before applying the proposed extended Kalman filter algorithm. The algorithm is used to recover the dynamic loads in three numerical examples. It is also used to identify the dynamic load in a lab experiment for a structure with varying parameters. The simulations and the experimental results show that the proposed algorithm is effective and can simultaneously identify the parameters and any changes in them as well as the applied dynamic load. Full article
(This article belongs to the Special Issue Advanced Mathematical and Simulation Methods for Inverse Problems)
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23 pages, 2765 KiB  
Article
Application of Anti-Diagonal Averaging in Response Reconstruction
by Bradley Dean Collins, Stephan Heyns, Schalk Kok and Daniel Nico Wilke
Symmetry 2021, 13(7), 1165; https://doi.org/10.3390/sym13071165 - 28 Jun 2021
Viewed by 1498
Abstract
Response reconstruction is used to obtain accurate replication of vehicle structural responses of field recorded measurements in a laboratory environment, a crucial step in the process of Accelerated Destructive Testing (ADA). Response Reconstruction is cast as an inverse problem whereby an input signal [...] Read more.
Response reconstruction is used to obtain accurate replication of vehicle structural responses of field recorded measurements in a laboratory environment, a crucial step in the process of Accelerated Destructive Testing (ADA). Response Reconstruction is cast as an inverse problem whereby an input signal is inferred to generate the desired outputs of a system. By casting the problem as an inverse problem we veer away from the familiarity of symmetry in physical systems since multiple inputs may generate the same output. We differ in our approach from standard force reconstruction problems in that the optimisation goal is the recreated output of the system. This alleviates the need for highly accurate inputs. We focus on offline non-causal linear regression methods to obtain input signals. A new windowing method called AntiDiagonal Averaging (ADA) is proposed to improve the regression techniques’ performance. ADA introduces overlaps within the predicted time signal windows and averages them. The newly proposed method is tested on a numerical quarter car model and shown to accurately reproduce the system’s outputs, which outperform related Finite Impulse Response (FIR) methods. In the nonlinear configuration of the numerical quarter car, ADA achieved a recreated output Mean Fit Function Error (MFFE) score of 0.40% compared to the next best performing FIR method, which generated a score of 4.89%. Similar performance was shown for the linear case. Full article
(This article belongs to the Special Issue Advanced Mathematical and Simulation Methods for Inverse Problems)
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