Mathematical Methods in Waves-Based Inverse Problems at Different Scales

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 26 September 2024 | Viewed by 6783

Special Issue Editor


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Guest Editor
Radon Institute, Austrian Academy of Sciences, Science Park 2, Altenbergerstrasse 69, A 4040 Linz, Austria
Interests: wave propagation in complex media; inverse problems; mathematical imaging; material sciences

Special Issue Information

Dear Colleagues, 

This Special Issue focuses on the mathematical analysis of inverse problems related to wave propagation-based modern imaging modalities with motivations coming from real-world applications. Such modalities can apply at different scales: larges scales (as in geophysics), moderate scales (as in tomography in the broader sense) and small scales (as in microscopy). We welcome contributions to the modeling, analysis and computational aspects of these inverse problems. We expect contributions from authors coming from different backgrounds to discuss different aspects of such imaging modalities.

The following tentative of classifying these inverse problems, according to the related scales, can be instructive:

At the large scale, we expect to estimate the first-order modes of the object to the image as ‘simple’ equivalent shapes or/and averages of the material parameters. Here, the mathematical techniques are related to the asymptotic modeling and analysis of the related functionals in terms of the large-scale parameters.

At the moderate scales, which are also related to the resonance regimes, different approaches are expected. Here, we deal with the intermediate modes of the objects to image. Without being exhaustive, we can cite techniques based on regularizations, optimizations, localized perturbations, Carleman estimates, spectral theory, and the use of exponential-type solutions or singular (or Green-type) solutions.

At a small scale, at least two approaches can be discussed. In the first one, we use higher-order functionals to detect inhomogeneities with finer details. In the second one, we use small perturbations (as contrasting agents) to highlight the small contrasts of the internal images. Therefore, one can extract higher-order modes of the object to the image (modeling finer details). The mathematical methods are related to the asymptotic modeling and analysis of the related functionals in terms of the small-scale parameters.

Of course, these scales are not sharply distinguishable. The techniques developed for some modalities (at a certain scale) might be used to initiate or improve the results related to the others. In addition, combinations of these techniques, whenever they are applicable, might be fruitful.

Prof. Dr. Mourad Sini
Guest Editor

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Keywords

  • inverse problems
  • mathematical imaging
  • waves in complex media
  • integral equations
  • regularization
  • inverse scattering
  • shape reconstruction
  • parameter estimation
  • carleman estimates

Published Papers (7 papers)

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Research

15 pages, 234 KiB  
Article
Covariance and Uncertainty Principle for Dispersive Pulse Propagation
by Leon Cohen
Axioms 2024, 13(4), 242; https://doi.org/10.3390/axioms13040242 - 8 Apr 2024
Viewed by 604
Abstract
We develop the concept of covariance for waves and show that it plays a fundamental role in understanding the evolution of a propagating pulse. The concept clarifies several issues regarding the spread of a pulse and the motion of the mean. Exact results [...] Read more.
We develop the concept of covariance for waves and show that it plays a fundamental role in understanding the evolution of a propagating pulse. The concept clarifies several issues regarding the spread of a pulse and the motion of the mean. Exact results are obtained for the time dependence of the covariance between position and wavenumber and the covariance between position and group velocity. We also derive relevant uncertainty principles for waves. Full article
19 pages, 344 KiB  
Article
Uniqueness Results for Some Inverse Electromagnetic Scattering Problems with Phaseless Far-Field Data
by Xianghe Zhu, Jun Guo and Haibing Wang
Axioms 2023, 12(12), 1069; https://doi.org/10.3390/axioms12121069 - 22 Nov 2023
Viewed by 789
Abstract
Consider three electromagnetic scattering models, namely, electromagnetic scattering by an elastic body, by a chiral medium, and by a cylinder at oblique incidence. We are concerned with the corresponding inverse problems of determining the locations and shapes of the scatterers from phaseless far-field [...] Read more.
Consider three electromagnetic scattering models, namely, electromagnetic scattering by an elastic body, by a chiral medium, and by a cylinder at oblique incidence. We are concerned with the corresponding inverse problems of determining the locations and shapes of the scatterers from phaseless far-field patterns. There are certain essential differences from the usual inverse electromagnetic scattering problems, and some fundamental conclusions need to be proved. First, we show that the phaseless far-field data are invariant under the translation of the scatterers and prove the reciprocity relations of the scattering data. Then, we justify the unique determination of the scatterers by utilizing the reference ball approach and the superpositions of a fixed point source and plane waves as the incident fields. The proofs are based on the reciprocity relations, Green’s formulas, and the analyses of the wave fields in the reference ball. Full article
16 pages, 328 KiB  
Article
On Unique Determination of Polyhedral Sets
by Luca Rondi
Axioms 2023, 12(11), 1035; https://doi.org/10.3390/axioms12111035 - 5 Nov 2023
Viewed by 915
Abstract
In this paper, we develop in detail the geometric constructions that lead to many uniqueness results for the determination of polyhedral sets, typically scatterers, by a finite minimal number of measurements. We highlight how unique continuation and a suitable reflection principle are enough [...] Read more.
In this paper, we develop in detail the geometric constructions that lead to many uniqueness results for the determination of polyhedral sets, typically scatterers, by a finite minimal number of measurements. We highlight how unique continuation and a suitable reflection principle are enough to proceed with the constructions without any other assumption on the underlying partial differential equation or the boundary condition. We also aim to keep the geometric constructions and their proofs as simple as possible. To illustrate the applicability of this theory, we show how several uniqueness results present in the literature immediately follow from our arguments. Indeed, we believe that this theory may serve as a roadmap for establishing similar uniqueness results for other partial differential equations or boundary conditions. Full article
13 pages, 2623 KiB  
Article
2-D Elastodynamic Time-Reversal Analysis for Surface Defects on Thin Plate Using Topological Sensitivity
by Takahiro Saitoh
Axioms 2023, 12(10), 920; https://doi.org/10.3390/axioms12100920 - 27 Sep 2023
Viewed by 683
Abstract
In recent years, there has been increasing attention on the development of non-destructive evaluation (NDE) methods using guided waves for long-length materials such as thin plates and pipes. The guided waves are capable of long-distance propagation in thin plates and pipes, and they [...] Read more.
In recent years, there has been increasing attention on the development of non-destructive evaluation (NDE) methods using guided waves for long-length materials such as thin plates and pipes. The guided waves are capable of long-distance propagation in thin plates and pipes, and they exhibit properties such as multimodality and dispersion. These characteristics of the guided waves make inspection using guided waves challenging. In this study, we apply a 2-D elastodynamic time-reversal method to detect surface breaking cracks of a thin plate where guided waves are present. The finite element method (FEM) is used to calculate the scattered waves from surface breaking cracks and their corresponding time-reversal waves. We also employ topological sensitivity as an assessment index for detecting surface breaking cracks using the time-reversal method. As numerical examples, we demonstrate guided wave propagation, scattering, and the time-reversal wave propagation obtained by using the FEM. Finally, we present the results of surface breaking crack detection in a thin plate and discuss the validity and effectiveness of the proposed method. Full article
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15 pages, 321 KiB  
Article
Using Alternating Minimization and Convexified Carleman Weighted Objective Functional for a Time-Domain Inverse Scattering Problem
by Nguyen Trung Thành
Axioms 2023, 12(7), 642; https://doi.org/10.3390/axioms12070642 - 28 Jun 2023
Viewed by 746
Abstract
This paper considers a 1D time-domain inverse scattering problem for the Helmholtz equation in which penetrable scatterers are to be determined from boundary measurements of the scattering data. It is formulated as a coefficient identification problem for a wave equation. Using the Laplace [...] Read more.
This paper considers a 1D time-domain inverse scattering problem for the Helmholtz equation in which penetrable scatterers are to be determined from boundary measurements of the scattering data. It is formulated as a coefficient identification problem for a wave equation. Using the Laplace transform, the inverse problem is converted into an overdetermined nonlinear system of partial differential equations. To solve this system, a Carleman weighted objective functional, which is proved to be strictly convex in an arbitrary set in a Hilbert space, is constructed. An alternating minimization algorithm is used to minimize the Carleman weighted objective functional. Numerical results are presented to illustrate the performance of the proposed algorithm. Full article
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20 pages, 8061 KiB  
Article
Reconstructing Loads in Nanoplates from Dynamic Data
by Alexandre Kawano and Antonino Morassi
Axioms 2023, 12(4), 398; https://doi.org/10.3390/axioms12040398 - 20 Apr 2023
Viewed by 986
Abstract
It was recently proved that the knowledge of the transverse displacement of a nanoplate in an open subset of its mid-plane, measured for any interval of time, allows for the unique determination of the spatial components [...] Read more.
It was recently proved that the knowledge of the transverse displacement of a nanoplate in an open subset of its mid-plane, measured for any interval of time, allows for the unique determination of the spatial components {fm(x,y)}m=1M of the transverse load m=1Mgm(t)fm(x,y), where M1 and {gm(t)}m=1M is a known set of linearly independent functions of the time variable. The nanoplate mechanical model is built within the strain gradient linear elasticity theory, according to the Kirchhoff–Love kinematic assumptions. In this paper, we derive a reconstruction algorithm for the above inverse source problem, and we implement a numerical procedure based on a finite element spatial discretization to approximate the loads {fm(x,y)}m=1M. The computations are developed for a uniform rectangular nanoplate clamped at the boundary. The sensitivity of the results with respect to the main parameters that influence the identification is analyzed in detail. The adoption of a regularization scheme based on the singular value decomposition turns out to be decisive for the accuracy and stability of the reconstruction. Full article
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16 pages, 3030 KiB  
Article
Wave Patterns inside Transparent Scatterers
by Youzi He, Hongyu Liu and Xianchao Wang
Axioms 2022, 11(12), 661; https://doi.org/10.3390/axioms11120661 - 22 Nov 2022
Viewed by 1035
Abstract
It may happen that under a certain wave interrogation, a medium scatterer produces no scattering. In such a case, the scattering field is trapped inside the scatterer and forms a certain interior resonant mode. We are concerned with the behavior of the wave [...] Read more.
It may happen that under a certain wave interrogation, a medium scatterer produces no scattering. In such a case, the scattering field is trapped inside the scatterer and forms a certain interior resonant mode. We are concerned with the behavior of the wave propagation inside a transparent scatterer. It turns out that the study can be boiled down to analyzing the interior transmission eigenvalue problem. For isotropic mediums, it is shown in a series of recent works that the transmission eigenfunctions possess rich patterns. In this paper, we show that those spectral patterns also hold for anisotropic mediums. Full article
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