Abstract In this article, we discuss an inverse problem of determining unknown coefficients of the first-order derivatives in an ultrahyperbolic equation. By a finite set of measurements, we prove the uniqueness of solution of the problem in semi-geodesic coordinates under some conditions on the principal coefficients of the equation. Our main tool is a Carleman estimate.
{"title":"The problem of determining multiple coefficients in an ultrahyperbolic equation","authors":"Fikret Gölgeleyen","doi":"10.1515/jiip-2022-0091","DOIUrl":"https://doi.org/10.1515/jiip-2022-0091","url":null,"abstract":"Abstract In this article, we discuss an inverse problem of determining unknown coefficients of the first-order derivatives in an ultrahyperbolic equation. By a finite set of measurements, we prove the uniqueness of solution of the problem in semi-geodesic coordinates under some conditions on the principal coefficients of the equation. Our main tool is a Carleman estimate.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49094549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we study the convergence analysis of the inexact Newton–Landweber iteration method (INLIM) with frozen derivative in Hilbert as well as Banach spaces. To study the convergence analysis, we incorporate the Hölder stability of the inverse mapping and Lipschitz continuity of the Fréchet derivative of the forward mapping. Moreover, we derive the convergence rates of INLIM in Hilbert as well as Banach spaces without using any extra smoothness condition. Finally, we compare our convergence rates results with that of several other frozen methods proposed in the literature to solve inverse problems.
{"title":"Convergence analysis of Inexact Newton–Landweber iteration with frozen derivative in Banach spaces","authors":"Gaurav Mittal, Ankik Kumar Giri","doi":"10.1515/jiip-2023-0002","DOIUrl":"https://doi.org/10.1515/jiip-2023-0002","url":null,"abstract":"Abstract In this paper, we study the convergence analysis of the inexact Newton–Landweber iteration method (INLIM) with frozen derivative in Hilbert as well as Banach spaces. To study the convergence analysis, we incorporate the Hölder stability of the inverse mapping and Lipschitz continuity of the Fréchet derivative of the forward mapping. Moreover, we derive the convergence rates of INLIM in Hilbert as well as Banach spaces without using any extra smoothness condition. Finally, we compare our convergence rates results with that of several other frozen methods proposed in the literature to solve inverse problems.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43993780","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We establish Lipschitz stability properties for a class of inverse problems. In that class, the associated direct problem is formulated by an integral operator Am mathcal{A}_{m} depending nonlinearly on a parameter 𝑚 and operating on a function 𝑢. In the inversion step, both 𝑢 and 𝑚 are unknown, but we are only interested in recovering 𝑚. We discuss examples of such inverse problems for the elasticity equation with applications to seismology and for the inverse scattering problem in electromagnetic theory. Assuming a few injectivity and regularity properties for Am mathcal{A}_{m} , we prove that the inverse problem with a finite number of data points is solvable and that the solution is Lipschitz stable in the data. We show a reconstruction example illustrating the use of neural networks.
摘要建立了一类逆问题的Lipschitz稳定性性质。在该类中，相关的直接问题由一个积分算子A m mathcal{A}_{m}非线性地依赖于一个参数𝑚并作用于一个函数𝑢来表述。在反演步骤中，𝑢和𝑚都是未知的，但我们只对收回𝑚感兴趣。我们讨论了应用于地震学的弹性方程反问题和电磁理论中的反散射问题的例子。假设a m 数学{a}_{m}的一些注入性和正则性，证明了有限个数数据点的反问题是可解的，且解在数据中是Lipschitz稳定的。我们展示了一个重建的例子来说明神经网络的使用。
{"title":"Stability properties for a class of inverse problems","authors":"Darko Volkov","doi":"10.1515/jiip-2022-0015","DOIUrl":"https://doi.org/10.1515/jiip-2022-0015","url":null,"abstract":"Abstract We establish Lipschitz stability properties for a class of inverse problems. In that class, the associated direct problem is formulated by an integral operator <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"script\">A</m:mi> <m:mi>m</m:mi> </m:msub> </m:math> mathcal{A}_{m} depending nonlinearly on a parameter 𝑚 and operating on a function 𝑢. In the inversion step, both 𝑢 and 𝑚 are unknown, but we are only interested in recovering 𝑚. We discuss examples of such inverse problems for the elasticity equation with applications to seismology and for the inverse scattering problem in electromagnetic theory. Assuming a few injectivity and regularity properties for <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"script\">A</m:mi> <m:mi>m</m:mi> </m:msub> </m:math> mathcal{A}_{m} , we prove that the inverse problem with a finite number of data points is solvable and that the solution is Lipschitz stable in the data. We show a reconstruction example illustrating the use of neural networks.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135860435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this note, we shall compare two important concepts of “regularization operators” and “regularization strategies” that appear in different classical monographs. The definition of a regularization operator is related to the Moore–Penrose inverse of the operator. In general, a regularization operator is a regularization strategy. We shall show that the converse is also true under some conditions. It is interesting to note that these two systems share analogous properties.
{"title":"Regularization operators versus regularization strategies","authors":"Thi-An Nguyen, C. Law","doi":"10.1515/jiip-2022-0073","DOIUrl":"https://doi.org/10.1515/jiip-2022-0073","url":null,"abstract":"Abstract In this note, we shall compare two important concepts of “regularization operators” and “regularization strategies” that appear in different classical monographs. The definition of a regularization operator is related to the Moore–Penrose inverse of the operator. In general, a regularization operator is a regularization strategy. We shall show that the converse is also true under some conditions. It is interesting to note that these two systems share analogous properties.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"31 1","pages":"625 - 629"},"PeriodicalIF":1.1,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47438635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Consider a source detection problem for a diffusion system at its stationary status, which is stated as the inverse source problem for an elliptic equation from the measurement of the solution specified only in part of the domain. For this linear ill-posed problem, we propose to reconstruct the interior source applying neural network algorithm, which projects the problem into a finite-dimensional space by approximating both the unknown source and the corresponding solution in terms of two neural networks. By minimizing a novel loss function consisting of PDE-fit and data-fit terms but without the boundary condition fit, the modified deep Galerkin method (MDGM) is applied to solve this problem numerically. Based on the stability result for the analytic extension of the solution, we strictly estimate the generalization error caused by the MDGM algorithm employing the property of conditional stability and the regularity of the solution. Numerical experiments show that we can obtain satisfactory reconstructions even in higher-dimensional cases, and validate the effectiveness of the proposed algorithm for different model configurations. Moreover, our algorithm is stable with respect to noisy inversion input data for the noise in various structures.
{"title":"On the recovery of internal source for an elliptic system by neural network approximation","authors":"Hui Zhang, Jijun Liu","doi":"10.1515/jiip-2022-0005","DOIUrl":"https://doi.org/10.1515/jiip-2022-0005","url":null,"abstract":"Abstract Consider a source detection problem for a diffusion system at its stationary status, which is stated as the inverse source problem for an elliptic equation from the measurement of the solution specified only in part of the domain. For this linear ill-posed problem, we propose to reconstruct the interior source applying neural network algorithm, which projects the problem into a finite-dimensional space by approximating both the unknown source and the corresponding solution in terms of two neural networks. By minimizing a novel loss function consisting of PDE-fit and data-fit terms but without the boundary condition fit, the modified deep Galerkin method (MDGM) is applied to solve this problem numerically. Based on the stability result for the analytic extension of the solution, we strictly estimate the generalization error caused by the MDGM algorithm employing the property of conditional stability and the regularity of the solution. Numerical experiments show that we can obtain satisfactory reconstructions even in higher-dimensional cases, and validate the effectiveness of the proposed algorithm for different model configurations. Moreover, our algorithm is stable with respect to noisy inversion input data for the noise in various structures.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49415441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Image denoising and edge extraction are two main tasks in image processing. In this paper, a game model is proposed to solve the image denoising and edge extraction, which combines an adaptive improved total variation (AdITV) model for image denoising and a global sparse gradient (GSG) model for edge extraction. The AdITV model is a forward-and-backward diffusion model. In fact, forward diffusion is applied to the homogeneous region to denoise, and backward diffusion is applied to the edge region to enhance the edge. A unified explicit discrete scheme is established in this paper to solve the AdITV model, which is compatible to forward diffusion and backward diffusion. The stability of the scheme is proved. On the other hand, GSG is a functional model based on sparse representation, which is robust to extract edges under the influence of noise. AdITV and GSG are chosen as two components of the game model. The alternate iteration method is used to solve the game problem. The convergence of the algorithm is proved and numerical experiments show the effectiveness of the model.
{"title":"The game model with multi-task for image denoising and edge extraction","authors":"Wenyan Wei, Xiangchu Feng, Bingzhe Wei","doi":"10.1515/jiip-2022-0051","DOIUrl":"https://doi.org/10.1515/jiip-2022-0051","url":null,"abstract":"Abstract Image denoising and edge extraction are two main tasks in image processing. In this paper, a game model is proposed to solve the image denoising and edge extraction, which combines an adaptive improved total variation (AdITV) model for image denoising and a global sparse gradient (GSG) model for edge extraction. The AdITV model is a forward-and-backward diffusion model. In fact, forward diffusion is applied to the homogeneous region to denoise, and backward diffusion is applied to the edge region to enhance the edge. A unified explicit discrete scheme is established in this paper to solve the AdITV model, which is compatible to forward diffusion and backward diffusion. The stability of the scheme is proved. On the other hand, GSG is a functional model based on sparse representation, which is robust to extract edges under the influence of noise. AdITV and GSG are chosen as two components of the game model. The alternate iteration method is used to solve the game problem. The convergence of the algorithm is proved and numerical experiments show the effectiveness of the model.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41733770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The theory of inverse problems is an actively studied area of modern differential equation theory. This paper studies the solvability of the inverse problem for a linearized system of Navier–Stokes equations in a cylindrical domain with a final overdetermination condition. Our approach is to reduce the inverse problem to a direct problem for a loaded equation. In contrast to the well-known works in this field, our approach is to find an equation for a loaded term whose solvability condition provides the solvability of the original inverse problem. At the same time, the classical theory of spectral decomposition of unbounded self-adjoint operators is actively used. Concrete examples demonstrate that the assertions of our theorems naturally develop and complement the known results on inverse problems. Various cases are considered when the known coefficient on the right-hand side of the equation depends only on time or both on time and a spatial variable. Theorems establishing new sufficient conditions for the unique solvability of the inverse problem under consideration are proved.
{"title":"On an inverse problem for a linearized system of Navier–Stokes equations with a final overdetermination condition","authors":"M. Jenaliyev, M. Bektemesov, M. Yergaliyev","doi":"10.1515/jiip-2022-0065","DOIUrl":"https://doi.org/10.1515/jiip-2022-0065","url":null,"abstract":"Abstract The theory of inverse problems is an actively studied area of modern differential equation theory. This paper studies the solvability of the inverse problem for a linearized system of Navier–Stokes equations in a cylindrical domain with a final overdetermination condition. Our approach is to reduce the inverse problem to a direct problem for a loaded equation. In contrast to the well-known works in this field, our approach is to find an equation for a loaded term whose solvability condition provides the solvability of the original inverse problem. At the same time, the classical theory of spectral decomposition of unbounded self-adjoint operators is actively used. Concrete examples demonstrate that the assertions of our theorems naturally develop and complement the known results on inverse problems. Various cases are considered when the known coefficient on the right-hand side of the equation depends only on time or both on time and a spatial variable. Theorems establishing new sufficient conditions for the unique solvability of the inverse problem under consideration are proved.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"31 1","pages":"611 - 624"},"PeriodicalIF":1.1,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43362515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Recently in [V. Markaki, D. Kourounis and A. Charalambopoulos, A dual self-monitored reconstruction scheme on the TV mathrm{TV} -regularized inverse conductivity problem, IMA J. Appl. Math. 86 2021, 3, 604–630], a novel reconstruction scheme has been developed for the solution of the inclusion problem in the inverse conductivity problem on the basis of a weighted self-guided regularization process generalizing the total variation approach. The present work extends this concept by implementing and investigating its applicability in the two-dimensional elasticity setting. To this end, we employ the framework of the reconstruction of linear and isotropic elastic structures described by their Lamé parameters. Numerical examples of increasingly challenging geometric complexities illustrate the enhanced accuracy and efficiency of the method.
{"title":"On the identification of Lamé parameters in linear isotropic elasticity via a weighted self-guided TV-regularization method","authors":"V. Markaki, D. Kourounis, A. Charalambopoulos","doi":"10.1515/jiip-2021-0050","DOIUrl":"https://doi.org/10.1515/jiip-2021-0050","url":null,"abstract":"Abstract Recently in [V. Markaki, D. Kourounis and A. Charalambopoulos, A dual self-monitored reconstruction scheme on the TV mathrm{TV} -regularized inverse conductivity problem, IMA J. Appl. Math. 86 2021, 3, 604–630], a novel reconstruction scheme has been developed for the solution of the inclusion problem in the inverse conductivity problem on the basis of a weighted self-guided regularization process generalizing the total variation approach. The present work extends this concept by implementing and investigating its applicability in the two-dimensional elasticity setting. To this end, we employ the framework of the reconstruction of linear and isotropic elastic structures described by their Lamé parameters. Numerical examples of increasingly challenging geometric complexities illustrate the enhanced accuracy and efficiency of the method.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44699685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we consider the numerical method for an optimal control problem governed by an obstacle problem. An approximate optimization problem is proposed by regularizing the original non-differentiable constrained problem with a simple method. The connection between the two formulations is established through some convergence results. A sufficient condition is derived to decide whether a solution of the first-order optimality system is a global minimum. The method with a second-order in time dissipative system is developed to solve the optimality system numerically. Several numerical examples are reported to show the effectiveness of the proposed method.
{"title":"A dynamical method for optimal control of the obstacle problem","authors":"Qinghua Ran, Xiaoliang Cheng, R. Gong, Ye Zhang","doi":"10.1515/jiip-2020-0135","DOIUrl":"https://doi.org/10.1515/jiip-2020-0135","url":null,"abstract":"Abstract In this paper, we consider the numerical method for an optimal control problem governed by an obstacle problem. An approximate optimization problem is proposed by regularizing the original non-differentiable constrained problem with a simple method. The connection between the two formulations is established through some convergence results. A sufficient condition is derived to decide whether a solution of the first-order optimality system is a global minimum. The method with a second-order in time dissipative system is developed to solve the optimality system numerically. Several numerical examples are reported to show the effectiveness of the proposed method.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"31 1","pages":"577 - 594"},"PeriodicalIF":1.1,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48984158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper is concerned with the inverse scattering of acoustic waves by an unbounded periodic elastic medium in the three-dimensional case. A novel uniqueness theorem is proved for the inverse problem of recovering a bi-periodic interface between acoustic and elastic waves using the near-field data measured only from the acoustic side of the interface, corresponding to a countably infinite number of quasi-periodic incident acoustic waves. The proposed method depends only on a fundamental a priori estimate established for the acoustic and elastic wave fields and a new mixed-reciprocity relation established in this paper for the solutions of the fluid-solid interaction scattering problem.
{"title":"On recovery of an unbounded bi-periodic interface for the inverse fluid-solid interaction scattering problem","authors":"Yan-li Cui, F. Qu, C. Wei","doi":"10.1515/jiip-2021-0070","DOIUrl":"https://doi.org/10.1515/jiip-2021-0070","url":null,"abstract":"Abstract This paper is concerned with the inverse scattering of acoustic waves by an unbounded periodic elastic medium in the three-dimensional case. A novel uniqueness theorem is proved for the inverse problem of recovering a bi-periodic interface between acoustic and elastic waves using the near-field data measured only from the acoustic side of the interface, corresponding to a countably infinite number of quasi-periodic incident acoustic waves. The proposed method depends only on a fundamental a priori estimate established for the acoustic and elastic wave fields and a new mixed-reciprocity relation established in this paper for the solutions of the fluid-solid interaction scattering problem.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"31 1","pages":"431 - 440"},"PeriodicalIF":1.1,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41799710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：