Pub Date : 2023-12-12DOI: 10.1088/1361-6420/ad10c8
Deyue Zhang, Yan Chang, Yukun Guo
A numerical method is developed for recovering both the source locations and the obstacle from the scattered Cauchy data of the time-harmonic acoustic field. First of all, the incident and scattered components are decomposed from the coupled Cauchy data by the representation of the single-layer potentials and the solution to the resulting linear integral system. As a consequence of this decomposition, the original problem of joint inversion is reformulated into two decoupled subproblems: an inverse source problem and an inverse obstacle scattering problem. Then, two sampling-type schemes are proposed to recover the shape of the obstacle and the source locations, respectively. The sampling methods rely on the specific indicator functions defined on target-oriented probing domains of circular shape. The error estimates of the decoupling procedure are established and the asymptotic behaviors of the indicator functions are analyzed. Extensive numerical experiments are also conducted to verify the performance of the sampling schemes.
{"title":"Jointly determining the point sources and obstacle from Cauchy data","authors":"Deyue Zhang, Yan Chang, Yukun Guo","doi":"10.1088/1361-6420/ad10c8","DOIUrl":"https://doi.org/10.1088/1361-6420/ad10c8","url":null,"abstract":"A numerical method is developed for recovering both the source locations and the obstacle from the scattered Cauchy data of the time-harmonic acoustic field. First of all, the incident and scattered components are decomposed from the coupled Cauchy data by the representation of the single-layer potentials and the solution to the resulting linear integral system. As a consequence of this decomposition, the original problem of joint inversion is reformulated into two decoupled subproblems: an inverse source problem and an inverse obstacle scattering problem. Then, two sampling-type schemes are proposed to recover the shape of the obstacle and the source locations, respectively. The sampling methods rely on the specific indicator functions defined on target-oriented probing domains of circular shape. The error estimates of the decoupling procedure are established and the asymptotic behaviors of the indicator functions are analyzed. Extensive numerical experiments are also conducted to verify the performance of the sampling schemes.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"6 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688216","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 : 2023-12-12DOI: 10.1088/1361-6420/ad149e
Ye Zhang, Chuchu Chen
The a posteriori stopping rule plays a significant role in the design of efficient stochastic algorithms for various tasks in computational mathematics, such as inverse problems, optimization, and machine learning. Through the lens of classical regularization theory, this paper describes a novel analysis of Morozov’s discrepancy principle for the stochastic generalized Landweber iteration and its continuous analog of generalized stochastic asymptotical regularization. Unlike existing results relating to convergence in probability, we prove the strong convergence of the regularization error using tools from stochastic analysis, namely the theory of martingales. Numerical experiments are conducted to verify the convergence of the discrepancy principle and demonstrate two new capabilities of stochastic generalized Landweber iteration, which should also be valid for other stochastic/statistical approaches: improved accuracy by selecting the optimal path and the identification of multi-solutions by clustering samples of obtained approximate solutions.
{"title":"Stochastic linear regularization methods: random discrepancy principle and applications","authors":"Ye Zhang, Chuchu Chen","doi":"10.1088/1361-6420/ad149e","DOIUrl":"https://doi.org/10.1088/1361-6420/ad149e","url":null,"abstract":"\u0000 The a posteriori stopping rule plays a significant role in the design of efficient stochastic algorithms for various tasks in computational mathematics, such as inverse problems, optimization, and machine learning. Through the lens of classical regularization theory, this paper describes a novel analysis of Morozov’s discrepancy principle for the stochastic generalized Landweber iteration and its continuous analog of generalized stochastic asymptotical regularization. Unlike existing results relating to convergence in probability, we prove the strong convergence of the regularization error using tools from stochastic analysis, namely the theory of martingales. Numerical experiments are conducted to verify the convergence of the discrepancy principle and demonstrate two new capabilities of stochastic generalized Landweber iteration, which should also be valid for other stochastic/statistical approaches: improved accuracy by selecting the optimal path and the identification of multi-solutions by clustering samples of obtained approximate solutions.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"15 11","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139009579","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 : 2023-12-12DOI: 10.1088/1361-6420/ad14a1
Lorenzo Della Cioppa, Michela Tartaglione, Annalisa Pascarella, F. Pitolli
Electroencephalography (EEG) source imaging aims to reconstruct brain activity maps from the neuroelectric potential difference measured on the skull. To obtain the brain activity map, we need to solve an ill-posed and ill-conditioned inverse problem that requires regularization techniques to make the solution viable. When dealing with real-time applications, dimensionality reduction techniques can be used to reduce the computational load required to evaluate the numerical solution of the EEG inverse problem. To this end, in this paper we use the random dipole sampling method, in which a Monte Carlo technique is used to reduce the number of neural sources. This is equivalent to reducing the number of the unknowns in the inverse problem and can be seen as a first regularization step. Then, we solve the reduced EEG inverse problem with two popular inversion methods, the weighted Minimum Norm Estimate (wMNE) and the standardized LOw Resolution brain Electromagnetic TomogrAphy (sLORETA). The main result of this paper is the error estimates of the reconstructed activity map obtained with the randomized version of wMNE and sLORETA. Numerical experiments on synthetic EEG data demonstrate the effectiveness of the random dipole sampling method.
{"title":"Solution of the EEG inverse problem by random dipole sampling","authors":"Lorenzo Della Cioppa, Michela Tartaglione, Annalisa Pascarella, F. Pitolli","doi":"10.1088/1361-6420/ad14a1","DOIUrl":"https://doi.org/10.1088/1361-6420/ad14a1","url":null,"abstract":"\u0000 Electroencephalography (EEG) source imaging aims to reconstruct brain activity maps from the neuroelectric potential difference measured on the skull. To obtain the brain activity map, we need to solve an ill-posed and ill-conditioned inverse problem that requires regularization techniques to make the solution viable. When dealing with real-time applications, dimensionality reduction techniques can be used to reduce the computational load required to evaluate the numerical solution of the EEG inverse problem. To this end, in this paper we use the random dipole sampling method, in which a Monte Carlo technique is used to reduce the number of neural sources. This is equivalent to reducing the number of the unknowns in the inverse problem and can be seen as a first regularization step. Then, we solve the reduced EEG inverse problem with two popular inversion methods, the weighted Minimum Norm Estimate (wMNE) and the standardized LOw Resolution brain Electromagnetic TomogrAphy (sLORETA). The main result of this paper is the error estimates of the reconstructed activity map obtained with the randomized version of wMNE and sLORETA. Numerical experiments on synthetic EEG data demonstrate the effectiveness of the random dipole sampling method.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"80 2","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139008079","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 : 2023-12-11DOI: 10.1088/1361-6420/ad0fad
Xiaofan Lu, Linan Zhang, Hongjin He
Automated model selection is an important application in science and engineering. In this work, we develop a learning approach for identifying structured dynamical systems from undersampled and noisy spatiotemporal data. The learning is performed by a sparse least-squares fitting over a large set of candidate functions via a nonconvex ℓ1−ℓ2 sparse optimization solved by the alternating direction method of multipliers. We show that if the set of candidate functions forms a structured random sampling matrix of a bounded orthogonal system, the recovery is stable and the error is bounded. The learning approach is validated on synthetic data generated by the viscous Burgers’ equation and two reaction–diffusion equations. The computational results demonstrate the theoretical guarantees of success and the efficiency with respect to the number of candidate functions.
{"title":"Structured model selection via ℓ1−ℓ2 optimization","authors":"Xiaofan Lu, Linan Zhang, Hongjin He","doi":"10.1088/1361-6420/ad0fad","DOIUrl":"https://doi.org/10.1088/1361-6420/ad0fad","url":null,"abstract":"Automated model selection is an important application in science and engineering. In this work, we develop a learning approach for identifying structured dynamical systems from undersampled and noisy spatiotemporal data. The learning is performed by a sparse least-squares fitting over a large set of candidate functions via a nonconvex <inline-formula>\u0000<tex-math><?CDATA $ell_1-ell_2$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msub><mml:mi>ℓ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>ℓ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>\u0000<inline-graphic xlink:href=\"ipad0fadieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> sparse optimization solved by the alternating direction method of multipliers. We show that if the set of candidate functions forms a structured random sampling matrix of a bounded orthogonal system, the recovery is stable and the error is bounded. The learning approach is validated on synthetic data generated by the viscous Burgers’ equation and two reaction–diffusion equations. The computational results demonstrate the theoretical guarantees of success and the efficiency with respect to the number of candidate functions.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"24 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688101","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 : 2023-12-11DOI: 10.1088/1361-6420/ad1132
Remo Kretschmann, Daniel Wachsmuth, Frank Werner
Testing of hypotheses is a well studied topic in mathematical statistics. Recently, this issue has also been addressed in the context of inverse problems, where the quantity of interest is not directly accessible but only after the inversion of a (potentially) ill-posed operator. In this study, we propose a regularized approach to hypothesis testing in inverse problems in the sense that the underlying estimators (or test statistics) are allowed to be biased. Under mild source-condition type assumptions, we derive a family of tests with prescribed level α and subsequently analyze how to choose the test with maximal power out of this family. As one major result we prove that regularized testing is always at least as good as (classical) unregularized testing. Furthermore, using tools from convex optimization, we provide an adaptive test by maximizing the power functional, which then outperforms previous unregularized tests in numerical simulations by several orders of magnitude.
{"title":"Optimal regularized hypothesis testing in statistical inverse problems","authors":"Remo Kretschmann, Daniel Wachsmuth, Frank Werner","doi":"10.1088/1361-6420/ad1132","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1132","url":null,"abstract":"Testing of hypotheses is a well studied topic in mathematical statistics. Recently, this issue has also been addressed in the context of inverse problems, where the quantity of interest is not directly accessible but only after the inversion of a (potentially) ill-posed operator. In this study, we propose a regularized approach to hypothesis testing in inverse problems in the sense that the underlying estimators (or test statistics) are allowed to be biased. Under mild source-condition type assumptions, we derive a family of tests with prescribed level <italic toggle=\"yes\">α</italic> and subsequently analyze how to choose the test with maximal power out of this family. As one major result we prove that regularized testing is always at least as good as (classical) unregularized testing. Furthermore, using tools from convex optimization, we provide an adaptive test by maximizing the power functional, which then outperforms previous unregularized tests in numerical simulations by several orders of magnitude.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"41 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688109","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 : 2023-12-11DOI: 10.1088/1361-6420/ad1131
Jun-Liang Fu, Jijun Liu
We propose a double-parameter regularization scheme for dealing with the backward diffusion process. Considering the smoothing effect of Yosida approximation for PDE, we propose to regularize this ill-posed problem by modifying original governed system in terms of a pseudoparabolic equation together with a quasi-boundary condition simultaneously, which consequently contains two regularizing parameters. Theoretically, we establish the optimal error estimates between the regularizing solution and the exact one in terms of suitable choice strategy for the regularizing parameters, under a-priori regularity assumptions on the exact solution. The a-posteriori choice strategy for the regularizing parameters based on the discrepancy principle is also studied. To weaken the heavy computational cost for solving the discrete nonsymmetric linear regularizing system by finite difference scheme, especially in higher spatial dimensional cases, the block divide-and-conquer method together with the properties of the Schur complement is applied to decompose the linear system into two half-size linear systems, one of which can be solved by the diagonalization technique, and consequently an efficient parallel-in-time algorithm originally developed for direct problem is applicable. Our proposed method is of much lower complexity than the standard solver for the corresponding linear system. Finally, some numerical examples are presented to verify the efficiency of our proposed method.
{"title":"Double-parameter regularization for solving the backward diffusion problem with parallel-in-time algorithm","authors":"Jun-Liang Fu, Jijun Liu","doi":"10.1088/1361-6420/ad1131","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1131","url":null,"abstract":"We propose a double-parameter regularization scheme for dealing with the backward diffusion process. Considering the smoothing effect of Yosida approximation for PDE, we propose to regularize this ill-posed problem by modifying original governed system in terms of a pseudoparabolic equation together with a quasi-boundary condition simultaneously, which consequently contains two regularizing parameters. Theoretically, we establish the optimal error estimates between the regularizing solution and the exact one in terms of suitable choice strategy for the regularizing parameters, under <italic toggle=\"yes\">a-priori</italic> regularity assumptions on the exact solution. The <italic toggle=\"yes\">a-posteriori</italic> choice strategy for the regularizing parameters based on the discrepancy principle is also studied. To weaken the heavy computational cost for solving the discrete nonsymmetric linear regularizing system by finite difference scheme, especially in higher spatial dimensional cases, the block divide-and-conquer method together with the properties of the Schur complement is applied to decompose the linear system into two half-size linear systems, one of which can be solved by the diagonalization technique, and consequently an efficient parallel-in-time algorithm originally developed for direct problem is applicable. Our proposed method is of much lower complexity than the standard solver for the corresponding linear system. Finally, some numerical examples are presented to verify the efficiency of our proposed method.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"42 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688099","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 : 2023-12-11DOI: 10.1088/1361-6420/ad1133
Jorge Aguayo, Cristóbal Bertoglio, Axel Osses
We present a parameter identification problem for a scalar permeability field and the maximum velocity in an inflow, following a reference profile. We utilize a modified version of the Navier–Stokes equations, incorporating a permeability term described by the Brinkman’s Law into the momentum equation. This modification takes into account the presence of obstacles on some parts of the boundary. For the outflow, we implement a directional do-nothing condition as a means of stabilizing the backflow. This work extends our previous research published in (Aguayo et al 2021 Inverse Problems37 025010), where we considered a similar inverse problem for a linear Oseen model with do-nothing boundary conditions on the outlet and numerical simulations in 2D. Here we consider the more realistic case of Navier–Stokes equations with a backflow correction on the outflow and 3D simulations of the identification of a more realistic tricuspid cardiac valve. From a reference velocity that could have some noise or be obtained in low resolution, we define a suitable quadratic cost functional with some regularization terms. Existence of minimizers and first and second order optimality conditions are derived through the differentiability of the solutions of the Navier–Stokes equations with respect to the permeability and maximum velocity in the inflow. Finally, we present some synthetic numerical test based of recovering a 2D and 3D shape of a cardiac valve from total and local velocity measurements, inspired from 2D and 3D MRI.
我们提出了一个标量渗透场的参数识别问题，以及根据参考剖面确定流入流中最大速度的问题。我们利用纳维-斯托克斯方程的修正版，在动量方程中加入布林克曼定律描述的渗透项。这一修改考虑到了边界某些部分存在障碍物的情况。对于流出的气流，我们采用了定向无为条件，作为稳定回流的一种手段。这项工作扩展了我们之前发表在《逆问题 37 025010》（Aguayo et al 2021 Inverse Problems37）上的研究，在该论文中，我们考虑了一个线性奥森模型的类似逆问题，该模型在出口处设置了无边界条件，并进行了二维数值模拟。在此，我们考虑了更现实的纳维-斯托克斯方程，并对流出流进行了回流修正，还对更现实的三尖瓣心脏瓣膜进行了三维模拟识别。参考速度可能会有一些噪声或以低分辨率获得，我们根据参考速度定义了一个合适的二次成本函数，其中包含一些正则项。通过纳维-斯托克斯方程解与渗透率和流入口最大速度的可微分性，推导出最小值的存在以及一阶和二阶最优条件。最后，我们介绍了一些基于二维和三维核磁共振成像的总速度和局部速度测量恢复心脏瓣膜的二维和三维形状的合成数值测试。
{"title":"Distributed parameter identification for the Navier–Stokes equations for obstacle detection","authors":"Jorge Aguayo, Cristóbal Bertoglio, Axel Osses","doi":"10.1088/1361-6420/ad1133","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1133","url":null,"abstract":"We present a parameter identification problem for a scalar permeability field and the maximum velocity in an inflow, following a reference profile. We utilize a modified version of the Navier–Stokes equations, incorporating a permeability term described by the Brinkman’s Law into the momentum equation. This modification takes into account the presence of obstacles on some parts of the boundary. For the outflow, we implement a directional do-nothing condition as a means of stabilizing the backflow. This work extends our previous research published in (Aguayo <italic toggle=\"yes\">et al</italic> 2021 <italic toggle=\"yes\">Inverse Problems</italic>\u0000<bold>37</bold> 025010), where we considered a similar inverse problem for a linear Oseen model with do-nothing boundary conditions on the outlet and numerical simulations in 2D. Here we consider the more realistic case of Navier–Stokes equations with a backflow correction on the outflow and 3D simulations of the identification of a more realistic tricuspid cardiac valve. From a reference velocity that could have some noise or be obtained in low resolution, we define a suitable quadratic cost functional with some regularization terms. Existence of minimizers and first and second order optimality conditions are derived through the differentiability of the solutions of the Navier–Stokes equations with respect to the permeability and maximum velocity in the inflow. Finally, we present some synthetic numerical test based of recovering a 2D and 3D shape of a cardiac valve from total and local velocity measurements, inspired from 2D and 3D MRI.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"27 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688105","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 : 2023-12-07DOI: 10.1088/1361-6420/ad1348
Silja L Christensen, N. A. B. Riis, Marcelo Pereyra, J. S. Jørgensen
Subsea pipelines can be inspected via 2D cross-sectional X-ray computed tomography (CT). Traditional reconstruction methods produce an image of the pipe's interior that can be post-processed for detection of possible defects. In this paper we propose a novel Bayesian CT reconstruction method with built-in defect detection. We decompose the reconstruction into a sum of two images; one containing the overall pipe structure, and one containing defects, and infer the images simultaneously in a Gibbs scheme. Our method requires that prior information about the two images is very distinct, i.e. the first image should contain the large-scale and layered pipe structure, and the second image should contain small, coherent defects. We demonstrate our methodology with numerical experiments using synthetic and real CT data from scans of subsea pipes in cases with full and limited data. Experiments demonstrate the effectiveness of the proposed method in various data settings, with reconstruction quality comparable to existing techniques, while also providing defect detection with uncertainty quantification.
{"title":"A Bayesian approach for CT reconstruction with defect detection for subsea pipelines","authors":"Silja L Christensen, N. A. B. Riis, Marcelo Pereyra, J. S. Jørgensen","doi":"10.1088/1361-6420/ad1348","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1348","url":null,"abstract":"\u0000 Subsea pipelines can be inspected via 2D cross-sectional X-ray computed tomography (CT). Traditional reconstruction methods produce an image of the pipe's interior that can be post-processed for detection of possible defects. In this paper we propose a novel Bayesian CT reconstruction method with built-in defect detection. We decompose the reconstruction into a sum of two images; one containing the overall pipe structure, and one containing defects, and infer the images simultaneously in a Gibbs scheme. Our method requires that prior information about the two images is very distinct, i.e. the first image should contain the large-scale and layered pipe structure, and the second image should contain small, coherent defects. We demonstrate our methodology with numerical experiments using synthetic and real CT data from scans of subsea pipes in cases with full and limited data. Experiments demonstrate the effectiveness of the proposed method in various data settings, with reconstruction quality comparable to existing techniques, while also providing defect detection with uncertainty quantification.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"19 9","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138592553","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 : 2023-12-05DOI: 10.1088/1361-6420/ad0fac
Ivan E Svetov, Anna P Polyakova
Currently, theory of the ray transforms of vector and tensor fields is well developed, but the generalized Radon transforms of such fields have not been fully studied. We consider the normal, longitudinal and mixed Radon transforms (with integration over planes) acting on three-dimensional vector and symmetric tensor fields. We prove that these operators are continuous. In case when values of all generalized Radon transforms are known, inversion formulas are derived for componentwise reconstruction of vector and symmetric m-tensor fields, m⩾2. Novel detailed decompositions of 3D vector and symmetric 2-tensor fields as a sum of pairwise orthogonal terms are obtained. For construction of each term in the sum only one function is required. With usage of these decompositions we have described the kernels and images of the generalized Radon transforms. In addition, we have obtained inversion formulas for each of the generalized Radon transforms acting on 3D vector and symmetric 2-tensor fields and have formulated theorems similar to the projection theorem for the Radon transform. For the cases m⩾3 similar statements are formulated as hypotheses. In addition, we consider the weighted longitudinal Radon transforms of 3D vector fields. Formulas are obtained for reconstructing the potential part of a 3D vector field from the known values of the longitudinal Radon transforms and one weighted Radon transform. Finally, we discuss the problem of vector fields reconstruction in Rn, n⩾4.
{"title":"Inversion of generalized Radon transforms acting on 3D vector and symmetric tensor fields","authors":"Ivan E Svetov, Anna P Polyakova","doi":"10.1088/1361-6420/ad0fac","DOIUrl":"https://doi.org/10.1088/1361-6420/ad0fac","url":null,"abstract":"Currently, theory of the ray transforms of vector and tensor fields is well developed, but the generalized Radon transforms of such fields have not been fully studied. We consider the normal, longitudinal and mixed Radon transforms (with integration over planes) acting on three-dimensional vector and symmetric tensor fields. We prove that these operators are continuous. In case when values of all generalized Radon transforms are known, inversion formulas are derived for componentwise reconstruction of vector and symmetric <italic toggle=\"yes\">m</italic>-tensor fields, <inline-formula>\u0000<tex-math><?CDATA $municode{x2A7E}2$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mi>m</mml:mi><mml:mtext>⩾</mml:mtext><mml:mn>2</mml:mn></mml:math>\u0000<inline-graphic xlink:href=\"ipad0facieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>. Novel detailed decompositions of 3D vector and symmetric 2-tensor fields as a sum of pairwise orthogonal terms are obtained. For construction of each term in the sum only one function is required. With usage of these decompositions we have described the kernels and images of the generalized Radon transforms. In addition, we have obtained inversion formulas for each of the generalized Radon transforms acting on 3D vector and symmetric 2-tensor fields and have formulated theorems similar to the projection theorem for the Radon transform. For the cases <inline-formula>\u0000<tex-math><?CDATA $municode{x2A7E}3$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mi>m</mml:mi><mml:mtext>⩾</mml:mtext><mml:mn>3</mml:mn></mml:math>\u0000<inline-graphic xlink:href=\"ipad0facieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> similar statements are formulated as hypotheses. In addition, we consider the weighted longitudinal Radon transforms of 3D vector fields. Formulas are obtained for reconstructing the potential part of a 3D vector field from the known values of the longitudinal Radon transforms and one weighted Radon transform. Finally, we discuss the problem of vector fields reconstruction 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=\"ipad0facieqn3.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>, <inline-formula>\u0000<tex-math><?CDATA $nunicode{x2A7E}4$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mi>n</mml:mi><mml:mtext>⩾</mml:mtext><mml:mn>4</mml:mn></mml:math>\u0000<inline-graphic xlink:href=\"ipad0facieqn4.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"21 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688110","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 : 2023-12-01DOI: 10.1088/1361-6420/ad0e25
Wansheng Wang, Chengyu Jin, Yunqing Huang
The purpose of this study is to recover the diffuse interface width parameter for nonlinear Allen–Cahn equation by a continuous data assimilation algorithm proposed recently. We obtain the large-time error between the true solution of the Allen–Cahn equation and the data assimilated solution produced by implicit–explicit one-leg fully discrete finite element methods due to discrepancy between an approximate diffuse interface width and the physical interface width. The strongly A-stability of the one-leg methods plays key roles in proving the exponential decay of initial error. Based on the long-time error estimates, we develop several algorithms to recover both the true solution and the true diffuse interface width using only spatially discrete phase field function measurements. Numerical experiments confirm our theoretical results and verify the effectiveness of the proposed methods.
本研究的目的是通过最近提出的一种连续数据同化算法来恢复非线性 Allen-Cahn 方程的扩散界面宽度参数。我们得到了 Allen-Cahn 方程的真实解与隐式-显式单腿完全离散有限元方法产生的数据同化解之间由于近似扩散界面宽度与物理界面宽度之间的差异而产生的大时间误差。单腿方法的强 A 稳定性在证明初始误差指数衰减方面发挥了关键作用。基于长期误差估计，我们开发了几种算法，仅使用空间离散相场函数测量就能恢复真实解和真实扩散界面宽度。数值实验证实了我们的理论结果，并验证了所提方法的有效性。
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