We investigate the inverse scattering problem of the perturbed biharmonic operator by studying the recovery process of the magnetic field begin{document}$ {mathbf{A}} $end{document} and the potential field begin{document}$ V $end{document} . We show that the high-frequency asymptotic of the scattering amplitude of the biharmonic operator uniquely determines begin{document}$ {rm{curl}} {mathbf{A}} $end{document} and begin{document}$ V-frac{1}{2}nablacdot{mathbf{A}} $end{document} . We study the near-field scattering problem and show that the high-frequency asymptotic expansion up to an error begin{document}$ mathcal{O}(lambda^{-4}) $end{document} recovers above two quantities with no additional information about begin{document}$ {mathbf{A}} $end{document} and begin{document}$ V $end{document} . We also establish stability estimates for begin{document}$ {rm{curl}} {mathbf{A}} $end{document} and begin{document}$ V-frac{1}{2}nablacdot{mathbf{A}} $end{document} .
We investigate the inverse scattering problem of the perturbed biharmonic operator by studying the recovery process of the magnetic field begin{document}$ {mathbf{A}} $end{document} and the potential field begin{document}$ V $end{document} . We show that the high-frequency asymptotic of the scattering amplitude of the biharmonic operator uniquely determines begin{document}$ {rm{curl}} {mathbf{A}} $end{document} and begin{document}$ V-frac{1}{2}nablacdot{mathbf{A}} $end{document} . We study the near-field scattering problem and show that the high-frequency asymptotic expansion up to an error begin{document}$ mathcal{O}(lambda^{-4}) $end{document} recovers above two quantities with no additional information about begin{document}$ {mathbf{A}} $end{document} and begin{document}$ V $end{document} . We also establish stability estimates for begin{document}$ {rm{curl}} {mathbf{A}} $end{document} and begin{document}$ V-frac{1}{2}nablacdot{mathbf{A}} $end{document} .
{"title":"Inverse scattering and stability for the biharmonic operator","authors":"Siamak Rabieniaharatbar","doi":"10.3934/ipi.2020064","DOIUrl":"https://doi.org/10.3934/ipi.2020064","url":null,"abstract":"We investigate the inverse scattering problem of the perturbed biharmonic operator by studying the recovery process of the magnetic field begin{document}$ {mathbf{A}} $end{document} and the potential field begin{document}$ V $end{document} . We show that the high-frequency asymptotic of the scattering amplitude of the biharmonic operator uniquely determines begin{document}$ {rm{curl}} {mathbf{A}} $end{document} and begin{document}$ V-frac{1}{2}nablacdot{mathbf{A}} $end{document} . We study the near-field scattering problem and show that the high-frequency asymptotic expansion up to an error begin{document}$ mathcal{O}(lambda^{-4}) $end{document} recovers above two quantities with no additional information about begin{document}$ {mathbf{A}} $end{document} and begin{document}$ V $end{document} . We also establish stability estimates for begin{document}$ {rm{curl}} {mathbf{A}} $end{document} and begin{document}$ V-frac{1}{2}nablacdot{mathbf{A}} $end{document} .","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81955834","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}
This paper considers the problem of identifying dislocation lines of curvilinear form in three-dimensional materials from boundary measurements, when the areas surrounded by the dislocation lines are assumed to be small-sized. The objective of this inverse problem is to reconstruct the number, the initial position and certain characteristics of these dislocations and establish, using certain test functions, a Hölder stability of the centers. This paper can be considered as a generalization of [9], where instead of reconstructing point-wise dislocations, as done in the latter paper, our aim is to recover the parameters of line dislocations by employing a direct algebraic algorithm.
{"title":"Identification and stability of small-sized dislocations using a direct algorithm","authors":"Batoul Abdelaziz, A. E. Badia, A. Hajj","doi":"10.3934/ipi.2021046","DOIUrl":"https://doi.org/10.3934/ipi.2021046","url":null,"abstract":"This paper considers the problem of identifying dislocation lines of curvilinear form in three-dimensional materials from boundary measurements, when the areas surrounded by the dislocation lines are assumed to be small-sized. The objective of this inverse problem is to reconstruct the number, the initial position and certain characteristics of these dislocations and establish, using certain test functions, a Hölder stability of the centers. This paper can be considered as a generalization of [9], where instead of reconstructing point-wise dislocations, as done in the latter paper, our aim is to recover the parameters of line dislocations by employing a direct algebraic algorithm.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89632964","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}
In this paper, we shall study the convergence rates of Tikhonov regularizations for the recovery of the growth rates in a Lotka-Volterra competition model with diffusion. The ill-posed inverse problem is transformed into a nonlinear minimization system by an appropriately selected version of Tikhonov regularization. The existence of the minimizers to the minimization system is demonstrated. We shall propose a new variational source condition, which will be rigorously verified under a Hölder type stability estimate. We will also derive the reasonable convergence rates under the new variational source condition.
{"title":"Convergence rates of Tikhonov regularization for recovering growth rates in a Lotka-Volterra competition model with diffusion","authors":"De-Han Chen, Daijun Jiang","doi":"10.3934/IPI.2021023","DOIUrl":"https://doi.org/10.3934/IPI.2021023","url":null,"abstract":"In this paper, we shall study the convergence rates of Tikhonov regularizations for the recovery of the growth rates in a Lotka-Volterra competition model with diffusion. The ill-posed inverse problem is transformed into a nonlinear minimization system by an appropriately selected version of Tikhonov regularization. The existence of the minimizers to the minimization system is demonstrated. We shall propose a new variational source condition, which will be rigorously verified under a Hölder type stability estimate. We will also derive the reasonable convergence rates under the new variational source condition.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77699656","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}
We present a non-iterative algorithm to reconstruct the isotropic acoustic wave speed from the measurement of the Neumann-to-Dirichlet map. The algorithm is designed based on the boundary control method and involves only computations that are stable. We prove the convergence of the algorithm and present its numerical implementation. The effectiveness of the algorithm is validated on both constant speed and variable speed, with full and partial boundary measurement as well as different levels of noise.
{"title":"A stable non-iterative reconstruction algorithm for the acoustic inverse boundary value problem","authors":"Tianyu Yang, Yang Yang","doi":"10.3934/IPI.2021038","DOIUrl":"https://doi.org/10.3934/IPI.2021038","url":null,"abstract":"We present a non-iterative algorithm to reconstruct the isotropic acoustic wave speed from the measurement of the Neumann-to-Dirichlet map. The algorithm is designed based on the boundary control method and involves only computations that are stable. We prove the convergence of the algorithm and present its numerical implementation. The effectiveness of the algorithm is validated on both constant speed and variable speed, with full and partial boundary measurement as well as different levels of noise.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81936120","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}
This paper is concerned with the inverse source problem of the time-harmonic elastic waves. A novel non-iterative reconstruction scheme is proposed for determining the elastic body force by using the multi-frequency Fourier expansion. The key ingredient of the approach is to choose appropriate admissible frequencies and establish an relationship between the Fourier coefficients and the coupled-wave field of compressional wave and shear wave. Both theoretical justifications and numerical examples are presented to verify the validity and robustness of the proposed method.
{"title":"Fourier method for reconstructing elastic body force from the coupled-wave field","authors":"Xianchao Wang, Jiaqi Zhu, Minghui Song, Wei Wu","doi":"10.3934/ipi.2021052","DOIUrl":"https://doi.org/10.3934/ipi.2021052","url":null,"abstract":"This paper is concerned with the inverse source problem of the time-harmonic elastic waves. A novel non-iterative reconstruction scheme is proposed for determining the elastic body force by using the multi-frequency Fourier expansion. The key ingredient of the approach is to choose appropriate admissible frequencies and establish an relationship between the Fourier coefficients and the coupled-wave field of compressional wave and shear wave. Both theoretical justifications and numerical examples are presented to verify the validity and robustness of the proposed method.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80304687","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}
In the development of imaging science and image processing request in our daily life, inpainting large regions always plays an important role. However, the existing local regularized models and some patch manifold based non-local models are often not effective in restoring the features and patterns in the large missing regions. In this paper, we will apply a strategy of inpainting from outside to inside and propose a re-weighted matching algorithm by closest patch (RWCP), contributing to further enhancing the features in the missing large regions. Additionally, we propose another re-weighted matching algorithm by distance-based weighted average (RWWA), leading to a result with higher PSNR value in some cases. Numerical simulations will demonstrate that for large region inpainting, the proposed method is more applicable than most canonical methods. Moreover, combined with image denoising methods, the proposed model is also applicable for noisy image restoration with large missing regions.
{"title":"Large region inpainting by re-weighted regularized methods","authors":"Yiting Chen, Jia Li, Qingyun Yu","doi":"10.3934/IPI.2021015","DOIUrl":"https://doi.org/10.3934/IPI.2021015","url":null,"abstract":"In the development of imaging science and image processing request in our daily life, inpainting large regions always plays an important role. However, the existing local regularized models and some patch manifold based non-local models are often not effective in restoring the features and patterns in the large missing regions. In this paper, we will apply a strategy of inpainting from outside to inside and propose a re-weighted matching algorithm by closest patch (RWCP), contributing to further enhancing the features in the missing large regions. Additionally, we propose another re-weighted matching algorithm by distance-based weighted average (RWWA), leading to a result with higher PSNR value in some cases. Numerical simulations will demonstrate that for large region inpainting, the proposed method is more applicable than most canonical methods. Moreover, combined with image denoising methods, the proposed model is also applicable for noisy image restoration with large missing regions.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78176441","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}
In this paper, the Bayesian method is proposed for the interior inverse scattering problem to reconstruct the interface of a two-layered cavity. The scattered field is measured by the point sources located on a closed curve inside the interior interface. The well-posedness of the posterior distribution in the Bayesian framework is proved. The Markov Chain Monte Carlo algorithm is employed to explore the posterior density. Some numerical experiments are presented to demonstrate the effectiveness of the proposed method.
{"title":"The interior inverse scattering problem for a two-layered cavity using the Bayesian method","authors":"Yunwen Yin, W. Yin, P. Meng, Hongyu Liu","doi":"10.3934/ipi.2021069","DOIUrl":"https://doi.org/10.3934/ipi.2021069","url":null,"abstract":"In this paper, the Bayesian method is proposed for the interior inverse scattering problem to reconstruct the interface of a two-layered cavity. The scattered field is measured by the point sources located on a closed curve inside the interior interface. The well-posedness of the posterior distribution in the Bayesian framework is proved. The Markov Chain Monte Carlo algorithm is employed to explore the posterior density. Some numerical experiments are presented to demonstrate the effectiveness of the proposed method.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73697629","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}
We consider the dynamical inverse problem for the wave equation on a metric tree graph and describe the dynamical Leaf Peeling (LP) method. The main step of the method is recalculating the response operator from the original tree to a peeled tree. The LP method allows us to recover the connectivity, potential function on a tree graph and the lengths of its edges from the response operator given on a finite time interval.
{"title":"Leaf Peeling method for the wave equation on metric tree graphs","authors":"S. Avdonin, Yuanyuan Zhao","doi":"10.3934/ipi.2020060","DOIUrl":"https://doi.org/10.3934/ipi.2020060","url":null,"abstract":"We consider the dynamical inverse problem for the wave equation on a metric tree graph and describe the dynamical Leaf Peeling (LP) method. The main step of the method is recalculating the response operator from the original tree to a peeled tree. The LP method allows us to recover the connectivity, potential function on a tree graph and the lengths of its edges from the response operator given on a finite time interval.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76180488","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}
Patch-based methods, which take the advantage of the redundancy and similarity among image patches, have attracted much attention in recent years. However, these methods are mainly limited to Gaussian noise removal. In this paper, the Poisson noise removal problem is considered. Unlike Gaussian noise which has an identical and independent distribution, Poisson noise is signal dependent, which makes the problem more challenging. By incorporating the prior that a group of similar patches should possess a low-rank structure, and applying the maximum a posterior (MAP) estimation, the Poisson noise removal problem is formulated as an optimization one. Then, an alternating minimization algorithm is developed to find the minimizer of the objective function efficiently. Convergence of the minimizing sequence will be established, and the efficiency and effectiveness of the proposed algorithm will be demonstrated by numerical experiments.
{"title":"A nonlocal low rank model for poisson noise removal","authors":"Mingchao Zhao, Y. Wen, Michael K. Ng, Hongwei Li","doi":"10.3934/ipi.2021003","DOIUrl":"https://doi.org/10.3934/ipi.2021003","url":null,"abstract":"Patch-based methods, which take the advantage of the redundancy and similarity among image patches, have attracted much attention in recent years. However, these methods are mainly limited to Gaussian noise removal. In this paper, the Poisson noise removal problem is considered. Unlike Gaussian noise which has an identical and independent distribution, Poisson noise is signal dependent, which makes the problem more challenging. By incorporating the prior that a group of similar patches should possess a low-rank structure, and applying the maximum a posterior (MAP) estimation, the Poisson noise removal problem is formulated as an optimization one. Then, an alternating minimization algorithm is developed to find the minimizer of the objective function efficiently. Convergence of the minimizing sequence will be established, and the efficiency and effectiveness of the proposed algorithm will be demonstrated by numerical experiments.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91047154","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}
We consider the recovering of the shape of a cavity from the Cauchy datum on an accessible boundary in case of semilinear boundary value problems. Existence and a characterization of the domain derivative of solutions of semilinear elliptic equations are proven. Furthermore, the result is applied to solve an inverse obstacle problem with an iterative regularization scheme. By some numerical examples its performance in case of a Kerr type nonlinearity is illustrated.
{"title":"The domain derivative for semilinear elliptic inverse obstacle problems","authors":"F. Hettlich","doi":"10.3934/ipi.2021071","DOIUrl":"https://doi.org/10.3934/ipi.2021071","url":null,"abstract":"We consider the recovering of the shape of a cavity from the Cauchy datum on an accessible boundary in case of semilinear boundary value problems. Existence and a characterization of the domain derivative of solutions of semilinear elliptic equations are proven. Furthermore, the result is applied to solve an inverse obstacle problem with an iterative regularization scheme. By some numerical examples its performance in case of a Kerr type nonlinearity is illustrated.","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83525606","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}
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