We consider an inverse spectral problem on a quantum graph associated with the square lattice. Assuming that the potentials on the edges are compactly supported and symmetric, we show that the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph uniquely determines the potentials. We obtain a reconstruction procedure, which is based on the reduction of the differential Schrödinger operator to a discrete one. As a corollary of the main results, it is proved that the S-matrix for all energies in any given open set in the continuous spectrum uniquely specifies the potentials on the square lattice.
我们考虑了与方阵相关的量子图上的反谱问题。假设边上的势是紧凑支撑和对称的，我们证明了图的有限部分上边界值问题的 Dirichlet 到 Neumann 映射唯一地决定了势。我们获得了一种基于将微分薛定谔算子还原为离散算子的重构程序。作为主要结果的一个推论，我们证明了连续谱中任何给定开集的所有能量的 S 矩阵唯一地指定了方格上的势。
{"title":"Inverse spectral problem for the Schrödinger operator on the square lattice","authors":"Dongjie Wu, Chuan-Fu Yang, Natalia Pavlovna Bondarenko","doi":"10.1088/1361-6420/ad3332","DOIUrl":"https://doi.org/10.1088/1361-6420/ad3332","url":null,"abstract":"We consider an inverse spectral problem on a quantum graph associated with the square lattice. Assuming that the potentials on the edges are compactly supported and symmetric, we show that the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph uniquely determines the potentials. We obtain a reconstruction procedure, which is based on the reduction of the differential Schrödinger operator to a discrete one. As a corollary of the main results, it is proved that the S-matrix for all energies in any given open set in the continuous spectrum uniquely specifies the potentials on the square lattice.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"63 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-25DOI: 10.1088/1361-6420/ad2cf8
Phuoc-Truong Huynh, Konstantin Pieper, Daniel Walter
The objective of this work is to quantify the reconstruction error in sparse inverse problems with measures and stochastic noise, motivated by optimal sensor placement. To be useful in this context, the error quantities must be explicit in the sensor configuration and robust with respect to the source, yet relatively easy to compute in practice, compared to a direct evaluation of the error by a large number of samples. In particular, we consider the identification of a measure consisting of an unknown linear combination of point sources from a finite number of measurements contaminated by Gaussian noise. The statistical framework for recovery relies on two main ingredients: first, a convex but non-smooth variational Tikhonov point estimator over the space of Radon measures and, second, a suitable mean-squared error based on its Hellinger–Kantorovich distance to the ground truth. To quantify the error, we employ a non-degenerate source condition as well as careful linearization arguments to derive a computable upper bound. This leads to asymptotically sharp error estimates in expectation that are explicit in the sensor configuration. Thus they can be used to estimate the expected reconstruction error for a given sensor configuration and guide the placement of sensors in sparse inverse problems.
{"title":"Towards optimal sensor placement for inverse problems in spaces of measures","authors":"Phuoc-Truong Huynh, Konstantin Pieper, Daniel Walter","doi":"10.1088/1361-6420/ad2cf8","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2cf8","url":null,"abstract":"The objective of this work is to quantify the reconstruction error in sparse inverse problems with measures and stochastic noise, motivated by optimal sensor placement. To be useful in this context, the error quantities must be explicit in the sensor configuration and robust with respect to the source, yet relatively easy to compute in practice, compared to a direct evaluation of the error by a large number of samples. In particular, we consider the identification of a measure consisting of an unknown linear combination of point sources from a finite number of measurements contaminated by Gaussian noise. The statistical framework for recovery relies on two main ingredients: first, a convex but non-smooth variational Tikhonov point estimator over the space of Radon measures and, second, a suitable mean-squared error based on its Hellinger–Kantorovich distance to the ground truth. To quantify the error, we employ a non-degenerate source condition as well as careful linearization arguments to derive a computable upper bound. This leads to asymptotically sharp error estimates in expectation that are explicit in the sensor configuration. Thus they can be used to estimate the expected reconstruction error for a given sensor configuration and guide the placement of sensors in sparse inverse problems.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"6 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140316622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-20DOI: 10.1088/1361-6420/ad2cfa
Kevin Bui, Zichao (Wendy) Di
Ptychography, a prevalent imaging technique in fields such as biology and optics, poses substantial challenges in its reconstruction process, characterized by nonconvexity and large-scale requirements. This paper presents a novel approach by introducing a class of variational models that incorporate the weighted difference of anisotropic–isotropic total variation. This formulation enables the handling of measurements corrupted by Gaussian or Poisson noise, effectively addressing the nonconvex challenge. To tackle the large-scale nature of the problem, we propose an efficient stochastic alternating direction method of multipliers, which guarantees convergence under mild conditions. Numerical experiments validate the superiority of our approach by demonstrating its capability to successfully reconstruct complex-valued images, especially in recovering the phase components even in the presence of highly corrupted measurements.
{"title":"A stochastic ADMM algorithm for large-scale ptychography with weighted difference of anisotropic and isotropic total variation","authors":"Kevin Bui, Zichao (Wendy) Di","doi":"10.1088/1361-6420/ad2cfa","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2cfa","url":null,"abstract":"Ptychography, a prevalent imaging technique in fields such as biology and optics, poses substantial challenges in its reconstruction process, characterized by nonconvexity and large-scale requirements. This paper presents a novel approach by introducing a class of variational models that incorporate the weighted difference of anisotropic–isotropic total variation. This formulation enables the handling of measurements corrupted by Gaussian or Poisson noise, effectively addressing the nonconvex challenge. To tackle the large-scale nature of the problem, we propose an efficient stochastic alternating direction method of multipliers, which guarantees convergence under mild conditions. Numerical experiments validate the superiority of our approach by demonstrating its capability to successfully reconstruct complex-valued images, especially in recovering the phase components even in the presence of highly corrupted measurements.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"40 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-19DOI: 10.1088/1361-6420/ad3087
Huaian Diao, Hongyu Liu, Longyue Tao
We establish sharp stability estimates of logarithmic type in determining an impedance obstacle in R2. The obstacle is the polygonal shape and the surface impedance parameter is non-zero constant. We establish the stability results using a single far-field pattern, constituting a longstanding problem in the inverse scattering theory. This is the first stability result in the literature in determining an impedance obstacle by a single far-field measurement. The stability in simultaneously determining the obstacle and the boundary impedance is established in terms of the classical Hausdorff distance. Several technical novelties and developments in the mathematical strategy developed for establishing the aforementioned stability results exist. First, the stability analysis is conducted around a corner point in a micro-local manner. Second, our stability estimates establish explicit relationships between the obstacle’s geometric configurations and the wave field’s vanishing order at the corner point. Third, we develop novel error propagation techniques to tackle singularities of the wave field at a corner with the impedance boundary condition.
{"title":"Stable determination of an impedance obstacle by a single far-field measurement","authors":"Huaian Diao, Hongyu Liu, Longyue Tao","doi":"10.1088/1361-6420/ad3087","DOIUrl":"https://doi.org/10.1088/1361-6420/ad3087","url":null,"abstract":"We establish sharp stability estimates of logarithmic type in determining an impedance obstacle in <inline-formula>\u0000<tex-math><?CDATA $mathbb{R}^2$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math>\u0000<inline-graphic xlink:href=\"ipad3087ieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>. The obstacle is the polygonal shape and the surface impedance parameter is non-zero constant. We establish the stability results using a single far-field pattern, constituting a longstanding problem in the inverse scattering theory. This is the first stability result in the literature in determining an impedance obstacle by a single far-field measurement. The stability in simultaneously determining the obstacle and the boundary impedance is established in terms of the classical Hausdorff distance. Several technical novelties and developments in the mathematical strategy developed for establishing the aforementioned stability results exist. First, the stability analysis is conducted around a corner point in a micro-local manner. Second, our stability estimates establish explicit relationships between the obstacle’s geometric configurations and the wave field’s vanishing order at the corner point. Third, we develop novel error propagation techniques to tackle singularities of the wave field at a corner with the impedance boundary condition.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"7 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-19DOI: 10.1088/1361-6420/ad2eca
P B de Castro, E C N Silva, E A Fancello
This paper presents a multiple material-phase level-set approach for acoustic full-waveform inversion in the time domain. By using a single level set (LS) function, several level values are used to define virtual boundaries between material phases with different (and known) wave propagation velocities. The aim of the proposed approach is to provide a suitable framework to identify multiple/nested inclusions or a finite number of almost homogeneous sedimentary layers with sharp interfaces between them. The use of a single LS function provides a significant reduction in the number of variables to be identified, when compared with the usual multi-material phase approaches defined by multiple functions, especially for problems with a high number of degrees of freedom. Numerical experiments show satisfactory results in identifying simultaneously different interfaces. Cases with and without inverse crime are evaluated, showing that the approach is reasonably robust in dealing with such a condition.
本文介绍了一种用于时域声学全波形反演的多材料相位水平集方法。通过使用单电平集（LS）函数，使用多个电平值来定义具有不同（和已知）波传播速度的材料相之间的虚拟边界。所提议方法的目的是提供一个合适的框架，以识别多层/嵌套夹杂物或数量有限的几乎均质的沉积层（它们之间有尖锐的界面）。与通常由多个函数定义的多物质相位方法相比，使用单一 LS 函数可显著减少需要识别的变量数量，特别是对于自由度较高的问题。数值实验表明，同时识别不同界面的结果令人满意。对有反向犯罪和无反向犯罪的情况进行了评估，结果表明该方法在处理这种情况时相当稳健。
{"title":"A single level set function approach for multiple material-phases applied to full-waveform inversion in the time domain","authors":"P B de Castro, E C N Silva, E A Fancello","doi":"10.1088/1361-6420/ad2eca","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2eca","url":null,"abstract":"This paper presents a multiple material-phase level-set approach for acoustic full-waveform inversion in the time domain. By using a single level set (LS) function, several level values are used to define virtual boundaries between material phases with different (and known) wave propagation velocities. The aim of the proposed approach is to provide a suitable framework to identify multiple/nested inclusions or a finite number of almost homogeneous sedimentary layers with sharp interfaces between them. The use of a single LS function provides a significant reduction in the number of variables to be identified, when compared with the usual multi-material phase approaches defined by multiple functions, especially for problems with a high number of degrees of freedom. Numerical experiments show satisfactory results in identifying simultaneously different interfaces. Cases with and without inverse crime are evaluated, showing that the approach is reasonably robust in dealing with such a condition.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"6 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315582","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-19DOI: 10.1088/1361-6420/ad3164
Sombuddha Bhattacharyya, Pranav Kumar
In this article, we study an inverse problem with local data for a linear polyharmonic operator with several lower order tensorial perturbations. We consider our domain to have an inaccessible portion of the boundary where neither the input can be prescribed nor the output can be measured. We prove the unique determination of all the tensorial coefficients of the operator from the knowledge of the Dirichlet and Neumann map on the accessible part of the boundary, under suitable geometric assumptions on the domain.
{"title":"Local data inverse problem for the polyharmonic operator with anisotropic perturbations","authors":"Sombuddha Bhattacharyya, Pranav Kumar","doi":"10.1088/1361-6420/ad3164","DOIUrl":"https://doi.org/10.1088/1361-6420/ad3164","url":null,"abstract":"In this article, we study an inverse problem with local data for a linear polyharmonic operator with several lower order tensorial perturbations. We consider our domain to have an inaccessible portion of the boundary where neither the input can be prescribed nor the output can be measured. We prove the unique determination of all the tensorial coefficients of the operator from the knowledge of the Dirichlet and Neumann map on the accessible part of the boundary, under suitable geometric assumptions on the domain.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"21 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-15DOI: 10.1088/1361-6420/ad308a
Piermarco Cannarsa, Anna Doubova, Masahiro Yamamoto
We consider an inverse problem of reconstructing a degeneracy point in the diffusion coefficient in a one-dimensional parabolic equation by measuring the normal derivative on one side of the domain boundary. We analyze the sensitivity of the inverse problem to the initial data. We give sufficient conditions on the initial data for uniqueness and stability for the one-point measurement and show some examples of positive and negative results. On the other hand, we present more general uniqueness results, also for the identification of an initial data by measurements distributed over time. The proofs are based on an explicit form of the solution by means of Bessel functions of the first type. Finally, the theoretical results are supported by numerical experiments.
{"title":"Reconstruction of degenerate conductivity region for parabolic equations","authors":"Piermarco Cannarsa, Anna Doubova, Masahiro Yamamoto","doi":"10.1088/1361-6420/ad308a","DOIUrl":"https://doi.org/10.1088/1361-6420/ad308a","url":null,"abstract":"We consider an inverse problem of reconstructing a degeneracy point in the diffusion coefficient in a one-dimensional parabolic equation by measuring the normal derivative on one side of the domain boundary. We analyze the sensitivity of the inverse problem to the initial data. We give sufficient conditions on the initial data for uniqueness and stability for the one-point measurement and show some examples of positive and negative results. On the other hand, we present more general uniqueness results, also for the identification of an initial data by measurements distributed over time. The proofs are based on an explicit form of the solution by means of Bessel functions of the first type. Finally, the theoretical results are supported by numerical experiments.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"53 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315442","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-15DOI: 10.1088/1361-6420/ad2d72
Zhonghua Liao, Qi Lü
Stochastic parabolic equations are widely used to model many random phenomena in natural sciences, such as the temperature distribution in a noisy medium, the dynamics of a chemical reaction in a noisy environment, or the evolution of the density of bacteria population. In many cases, the equation may involve an unknown moving boundary which could represent a change of phase, a reaction front, or an unknown population. In this paper, we focus on an inverse problem with the goal is to determine an unknown moving boundary based on data observed in a specific interior subdomain for the stochastic parabolic equation. The uniqueness of the solution of this problem is proved, and furthermore a stability estimate of log type is derived. This allows us, theoretically, to track and to monitor the behavior of the unknown boundary from observation in an arbitrary interior domain. The primary tool is a new Carleman estimate for stochastic parabolic equations. As a byproduct, we obtain a quantitative unique continuation property for stochastic parabolic equations.
{"title":"Stability estimate for an inverse stochastic parabolic problem of determining unknown time-varying boundary *","authors":"Zhonghua Liao, Qi Lü","doi":"10.1088/1361-6420/ad2d72","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2d72","url":null,"abstract":"Stochastic parabolic equations are widely used to model many random phenomena in natural sciences, such as the temperature distribution in a noisy medium, the dynamics of a chemical reaction in a noisy environment, or the evolution of the density of bacteria population. In many cases, the equation may involve an unknown moving boundary which could represent a change of phase, a reaction front, or an unknown population. In this paper, we focus on an inverse problem with the goal is to determine an unknown moving boundary based on data observed in a specific interior subdomain for the stochastic parabolic equation. The uniqueness of the solution of this problem is proved, and furthermore a stability estimate of log type is derived. This allows us, theoretically, to track and to monitor the behavior of the unknown boundary from observation in an arbitrary interior domain. The primary tool is a new Carleman estimate for stochastic parabolic equations. As a byproduct, we obtain a quantitative unique continuation property for stochastic parabolic equations.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"31 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315579","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-13DOI: 10.1088/1361-6420/ad2ec9
Yasmina Zaky, Nicolas Fortino, Benoit Miramond, Jean-Yves Dauvignac
This study addresses the classification of objects using their electromagnetic signatures with convolutional neural networks (CNNs) trained on noiseless data. The singularity expansion method (SEM) was applied to establish a compact model that accurately represents the ultra-wideband scattered field (SF) of an object, independently of its orientation and observation angle. To perform the classification, we used a CNN associated with a noise-robust SEM technique to classify different objects based on their characteristic parameters. To validate this approach, we compared the performance of the classifier with and without SEM pre-processing of the SF for different noise levels and for object sizes not present in the training set. Moreover, we propose a procedure that determines the direction of the receiving antenna and orientation of an object based on the residues associated with each complex natural resonance. This classification procedure using pre-processed SEM data is promising and easy to train, especially when generalizing to object sizes not included in the training set.
本研究利用在无噪声数据上训练的卷积神经网络（CNN），利用物体的电磁特征对其进行分类。应用奇异性扩展法（SEM）建立了一个紧凑的模型，该模型能准确地表示物体的超宽带散射场（SF），不受其方向和观测角度的影响。为了进行分类，我们使用了与噪声抑制 SEM 技术相关的 CNN，根据不同物体的特征参数对其进行分类。为了验证这种方法，我们比较了分类器在对 SF 进行 SEM 预处理和未进行 SEM 预处理的情况下，针对不同噪声水平和训练集中不存在的物体大小的性能。此外，我们还提出了一种程序，可根据与每个复杂自然共振相关的残差来确定接收天线的方向和物体的方位。这种使用预处理 SEM 数据的分类程序前景广阔且易于训练，尤其是在对训练集中未包含的物体尺寸进行泛化时。
{"title":"Shape and orientation classification of objects based on their electromagnetic signatures using convolutional neural networks","authors":"Yasmina Zaky, Nicolas Fortino, Benoit Miramond, Jean-Yves Dauvignac","doi":"10.1088/1361-6420/ad2ec9","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2ec9","url":null,"abstract":"This study addresses the classification of objects using their electromagnetic signatures with convolutional neural networks (CNNs) trained on noiseless data. The singularity expansion method (SEM) was applied to establish a compact model that accurately represents the ultra-wideband scattered field (SF) of an object, independently of its orientation and observation angle. To perform the classification, we used a CNN associated with a noise-robust SEM technique to classify different objects based on their characteristic parameters. To validate this approach, we compared the performance of the classifier with and without SEM pre-processing of the SF for different noise levels and for object sizes not present in the training set. Moreover, we propose a procedure that determines the direction of the receiving antenna and orientation of an object based on the residues associated with each complex natural resonance. This classification procedure using pre-processed SEM data is promising and easy to train, especially when generalizing to object sizes not included in the training set.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"53 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-12DOI: 10.1088/1361-6420/ad2ecb
Rohit Kumar Mishra, Chandni Thakkar
In this paper, a restricted transverse ray transform acting on vector and symmetric m-tensor fields is studied. We developed inversion algorithms using restricted transverse ray transform data to recover symmetric m-tensor fields in R3 and vector fields in Rn. We restrict the transverse ray transform to all lines going through a fixed curve γ that satisfies the Kirillov–Tuy condition. We show that the known restricted data can be used to reconstruct a specific weighted Radon transform of the unknown vector/tensor field’s components, which we then use to explicitly recover the unknown field.
本文研究了作用于矢量场和对称 m 张量场的受限横向射线变换。我们利用受限横射线变换数据开发了反演算法，以恢复 R3 中的对称 m 张量场和 Rn 中的矢量场。我们将横向射线变换限制为通过满足基里洛夫-图伊条件的固定曲线 γ 的所有线段。我们证明，已知的受限数据可用来重建未知矢量/张量场分量的特定加权拉顿变换，然后用它来明确恢复未知场。
{"title":"Inversion of a restricted transverse ray transform with sources on a curve","authors":"Rohit Kumar Mishra, Chandni Thakkar","doi":"10.1088/1361-6420/ad2ecb","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2ecb","url":null,"abstract":"In this paper, a restricted transverse ray transform acting on vector and symmetric <italic toggle=\"yes\">m</italic>-tensor fields is studied. We developed inversion algorithms using restricted transverse ray transform data to recover symmetric <italic toggle=\"yes\">m</italic>-tensor fields in <inline-formula>\u0000<tex-math><?CDATA $mathbb{R}^3$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math>\u0000<inline-graphic xlink:href=\"ipad2ecbieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> and vector fields in <inline-formula>\u0000<tex-math><?CDATA $mathbb{R}^n$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math>\u0000<inline-graphic xlink:href=\"ipad2ecbieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>. We restrict the transverse ray transform to all lines going through a fixed curve <italic toggle=\"yes\">γ</italic> that satisfies the Kirillov–Tuy condition. We show that the known restricted data can be used to reconstruct a specific weighted Radon transform of the unknown vector/tensor field’s components, which we then use to explicitly recover the unknown field.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"33 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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