Pub Date : 2024-09-11DOI: 10.1088/1361-6420/ad7495
Florian Bossmann, Jianwei Ma and Wenze Wu
Identifying and tracking objects over multiple observations is a frequent task in many applications. Traffic monitoring requires the tracking of vehicles or pedestrians in video data and geophysical exploration relies on identifying seismic wave fronts from data of multiple sensors, only to mention two examples. In many cases, the object changes its shape or position within the given data from one observation to another. Vehicles can change their position and angle relative to the camera while seismic waves have different arrival times, frequencies, or intensities depending on the sensor position. This complicates the task at hand. In a previous work, the authors presented a new algorithm to solve this problem—object reconstruction using K-approximation (ORKA). This algorithm is hindered by two conflicting limitations: the tracked movement is limited by the sampling grid while the complexity increases exponentially with the resolution. We introduce an iterative variant of the ORKA algorithm that is able to overcome this conflict. We also give a brief introduction on the original ORKA algorithm. Knowledge of the previous work is thus not required. We give theoretical error bounds and a complexity analysis which we validate with several numerical experiments. Moreover, we discuss the influence of different parameter choices in detail. The results clearly show that the iterative approach can outperform ORKA in both accuracy and efficiency. On the example of video processing we show that the new method can be applied where the original algorithm is too time and memory intensive. Furthermore, we demonstrate on seismic exploration data that we are now able to recover much finer details on the wave front movement then before.
在许多应用中，识别和跟踪多个观测对象是一项经常性任务。交通监控需要跟踪视频数据中的车辆或行人，地球物理勘探需要从多个传感器的数据中识别地震波前沿，这只是其中的两个例子。在许多情况下，从一次观测到另一次观测，物体在给定数据中的形状或位置都会发生变化。车辆相对于摄像机的位置和角度会发生变化，而地震波的到达时间、频率或强度则会因传感器位置的不同而不同。这使得手头的工作变得更加复杂。在之前的一项研究中，作者提出了一种解决这一问题的新算法--使用 K 近似法（ORKA）进行目标重建。该算法受到两个相互冲突的限制的阻碍：跟踪运动受到采样网格的限制，而复杂度则随着分辨率的增加呈指数增长。我们介绍了 ORKA 算法的迭代变体，它能够克服这一矛盾。我们还简要介绍了原始 ORKA 算法。因此不需要了解以前的工作。我们给出了理论误差范围和复杂性分析，并通过几个数值实验进行了验证。此外，我们还详细讨论了不同参数选择的影响。结果清楚地表明，迭代法在精度和效率上都优于 ORKA。以视频处理为例，我们表明新方法可以应用于原始算法时间和内存消耗过大的地方。此外，我们还在地震勘探数据上证明，我们现在能够恢复比以前更精细的波前运动细节。
{"title":"fg-ORKA: fast and gridless reconstruction of moving and deforming objects in multidimensional data","authors":"Florian Bossmann, Jianwei Ma and Wenze Wu","doi":"10.1088/1361-6420/ad7495","DOIUrl":"https://doi.org/10.1088/1361-6420/ad7495","url":null,"abstract":"Identifying and tracking objects over multiple observations is a frequent task in many applications. Traffic monitoring requires the tracking of vehicles or pedestrians in video data and geophysical exploration relies on identifying seismic wave fronts from data of multiple sensors, only to mention two examples. In many cases, the object changes its shape or position within the given data from one observation to another. Vehicles can change their position and angle relative to the camera while seismic waves have different arrival times, frequencies, or intensities depending on the sensor position. This complicates the task at hand. In a previous work, the authors presented a new algorithm to solve this problem—object reconstruction using K-approximation (ORKA). This algorithm is hindered by two conflicting limitations: the tracked movement is limited by the sampling grid while the complexity increases exponentially with the resolution. We introduce an iterative variant of the ORKA algorithm that is able to overcome this conflict. We also give a brief introduction on the original ORKA algorithm. Knowledge of the previous work is thus not required. We give theoretical error bounds and a complexity analysis which we validate with several numerical experiments. Moreover, we discuss the influence of different parameter choices in detail. The results clearly show that the iterative approach can outperform ORKA in both accuracy and efficiency. On the example of video processing we show that the new method can be applied where the original algorithm is too time and memory intensive. Furthermore, we demonstrate on seismic exploration data that we are now able to recover much finer details on the wave front movement then before.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"58 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142210289","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 : 2024-09-05DOI: 10.1088/1361-6420/ad7497
Kui Ren, Nathan Soedjak, Kewei Wang, Hongyu Zhai
In this short note, we consider an inverse problem to a mean-field games (MFGs) system where we are interested in reconstructing the state-independent running cost function from observed value-function data. We provide an elementary proof of a uniqueness result for the inverse problem using the standard multilinearization technique. One of the main features of our work is that we insist that the population distribution be a probability measure, a requirement that is not enforced in some of the existing literature on theoretical inverse MFGs.
{"title":"Reconstructing a state-independent cost function in a mean-field game model","authors":"Kui Ren, Nathan Soedjak, Kewei Wang, Hongyu Zhai","doi":"10.1088/1361-6420/ad7497","DOIUrl":"https://doi.org/10.1088/1361-6420/ad7497","url":null,"abstract":"In this short note, we consider an inverse problem to a mean-field games (MFGs) system where we are interested in reconstructing the state-independent running cost function from observed value-function data. We provide an elementary proof of a uniqueness result for the inverse problem using the standard multilinearization technique. One of the main features of our work is that we insist that the population distribution be a probability measure, a requirement that is not enforced in some of the existing literature on theoretical inverse MFGs.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"258 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142210288","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 : 2024-09-05DOI: 10.1088/1361-6420/ad7496
Qinian Jin, Yun Zhang
We consider Newton-type methods for solving nonlinear ill-posed inverse problems in Hilbert spaces where the forward operators are not necessarily Gâteaux differentiable. Modifications are proposed with the non-existing Fréchet derivatives replaced by a family of bounded linear operators satisfying suitable properties. These bounded linear operators can be constructed by the Bouligand subderivatives which are defined as limits of Fréchet derivatives of the forward operator in differentiable points. The Bouligand subderivative mapping in general is not continuous unless the forward operator is Gâteaux differentiable which introduces challenges for convergence analysis of the corresponding Bouligand–Newton type methods. In this paper we will show that, under the discrepancy principle, these Bouligand–Newton type methods are iterative regularization methods of optimal order. Numerical results for an inverse problem arising from a non-smooth semi-linear elliptic equation are presented to test the performance of the methods.
{"title":"Bouligand–Newton type methods for non-smooth ill-posed problems","authors":"Qinian Jin, Yun Zhang","doi":"10.1088/1361-6420/ad7496","DOIUrl":"https://doi.org/10.1088/1361-6420/ad7496","url":null,"abstract":"We consider Newton-type methods for solving nonlinear ill-posed inverse problems in Hilbert spaces where the forward operators are not necessarily Gâteaux differentiable. Modifications are proposed with the non-existing Fréchet derivatives replaced by a family of bounded linear operators satisfying suitable properties. These bounded linear operators can be constructed by the Bouligand subderivatives which are defined as limits of Fréchet derivatives of the forward operator in differentiable points. The Bouligand subderivative mapping in general is not continuous unless the forward operator is Gâteaux differentiable which introduces challenges for convergence analysis of the corresponding Bouligand–Newton type methods. In this paper we will show that, under the discrepancy principle, these Bouligand–Newton type methods are iterative regularization methods of optimal order. Numerical results for an inverse problem arising from a non-smooth semi-linear elliptic equation are presented to test the performance of the methods.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"9 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226723","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}
We consider the inverse source problem for the time-dependent, constant-coefficient wave equation with Cauchy data and passive cross-correlation data.We propose to consider the cross-correlation as a wave equation itself and reconstruct the cross-correlation in the support of the source for the original Cauchy wave equation. Having access to the cross-correlation in the support of the source, we show that the cross-correlation solves a wave equation, and we reconstruct the cross-correlation from boundary data to recover the source in the original Cauchy wave equation. In addition, we show the inverse source problem is ill-posed and suffers from non-uniqueness when the mean of the source is zero and provide a uniqueness result and stability estimate in case of non-zero mean sources.
{"title":"Fourier method for inverse source problem using correlation of passive measurements*","authors":"Faouzi Triki, Kristoffer Linder-Steinlein, Mirza Karamehmedović","doi":"10.1088/1361-6420/ad6fc7","DOIUrl":"https://doi.org/10.1088/1361-6420/ad6fc7","url":null,"abstract":"We consider the inverse source problem for the time-dependent, constant-coefficient wave equation with Cauchy data and passive cross-correlation data.We propose to consider the cross-correlation as a wave equation itself and reconstruct the cross-correlation in the support of the source for the original Cauchy wave equation. Having access to the cross-correlation in the support of the source, we show that the cross-correlation solves a wave equation, and we reconstruct the cross-correlation from boundary data to recover the source in the original Cauchy wave equation. In addition, we show the inverse source problem is ill-posed and suffers from non-uniqueness when the mean of the source is zero and provide a uniqueness result and stability estimate in case of non-zero mean sources.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"21 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226717","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 : 2024-09-02DOI: 10.1088/1361-6420/ad7283
Hongyu Liu, Zhi-Qiang Miao, Guang-Hui Zheng
In this paper, we develop a general mathematical framework for enhanced hydrodynamic near-cloaking of electro-osmotic flow for more complex shapes, which is obtained by simultaneously perturbing the inner and outer boundaries of the perfect cloaking structure. We first derive the asymptotic expansions of perturbed fields and obtain a first-order coupled system. We then establish the representation formula of the solution to the first-order coupled system using the layer potential techniques. Based on the asymptotic analysis, the enhanced hydrodynamic near-cloaking conditions are derived for the control region with general cross-sectional shape. The conditions reveal the inner relationship between the shapes of the object and the control region. Especially, for the shape of a deformed annulus or confocal ellipses cylinder, the relationship of shapes is quantified more accurately by recursive formulas. Our theoretical findings are validated and supplemented by a variety of numerical results. The results in this paper also provide a mathematical foundation for more complex hydrodynamic cloaking. Additionally, the concept of cloaking has efficient applications in the field of microfluidics, including drag reduction, microfluidic manipulation, and biological tissue coculture.
{"title":"Enhanced microscale hydrodynamic near-cloaking using electro-osmosis","authors":"Hongyu Liu, Zhi-Qiang Miao, Guang-Hui Zheng","doi":"10.1088/1361-6420/ad7283","DOIUrl":"https://doi.org/10.1088/1361-6420/ad7283","url":null,"abstract":"In this paper, we develop a general mathematical framework for enhanced hydrodynamic near-cloaking of electro-osmotic flow for more complex shapes, which is obtained by simultaneously perturbing the inner and outer boundaries of the perfect cloaking structure. We first derive the asymptotic expansions of perturbed fields and obtain a first-order coupled system. We then establish the representation formula of the solution to the first-order coupled system using the layer potential techniques. Based on the asymptotic analysis, the enhanced hydrodynamic near-cloaking conditions are derived for the control region with general cross-sectional shape. The conditions reveal the inner relationship between the shapes of the object and the control region. Especially, for the shape of a deformed annulus or confocal ellipses cylinder, the relationship of shapes is quantified more accurately by recursive formulas. Our theoretical findings are validated and supplemented by a variety of numerical results. The results in this paper also provide a mathematical foundation for more complex hydrodynamic cloaking. Additionally, the concept of cloaking has efficient applications in the field of microfluidics, including drag reduction, microfluidic manipulation, and biological tissue coculture.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"22 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142210290","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 : 2024-08-30DOI: 10.1088/1361-6420/ad6fc6
Thorsten Hohage, Roman G Novikov, Vladimir N Sivkin
We consider the problem of finding a compactly supported potential in the multidimensional Schrödinger equation from its differential scattering cross section (squared modulus of the scattering amplitude) at fixed energy. In the Born approximation this problem simplifies to the phase retrieval problem of reconstructing the potential from the absolute value of its Fourier transform on a ball. To compensate for the missing phase information we use the method of a priori known background scatterers. In particular, we propose an iterative scheme for finding the potential from measurements of a single differential scattering cross section corresponding to the sum of the unknown potential and a known background potential, which is sufficiently disjoint. If this condition is relaxed, then we give similar results for finding the potential from additional monochromatic measurements of the differential scattering cross section of the unknown potential without the background potential. The performance of the proposed algorithms is demonstrated in numerical examples. In the present work we significantly advance theoretically and numerically studies of Agaltsov et al (2019 Inverse Problems35 24001) and Novikov and Sivkin (2021 Inverse Problems37 055011).
{"title":"Phase retrieval and phaseless inverse scattering with background information","authors":"Thorsten Hohage, Roman G Novikov, Vladimir N Sivkin","doi":"10.1088/1361-6420/ad6fc6","DOIUrl":"https://doi.org/10.1088/1361-6420/ad6fc6","url":null,"abstract":"We consider the problem of finding a compactly supported potential in the multidimensional Schrödinger equation from its differential scattering cross section (squared modulus of the scattering amplitude) at fixed energy. In the Born approximation this problem simplifies to the phase retrieval problem of reconstructing the potential from the absolute value of its Fourier transform on a ball. To compensate for the missing phase information we use the method of <italic toggle=\"yes\">a priori</italic> known background scatterers. In particular, we propose an iterative scheme for finding the potential from measurements of a single differential scattering cross section corresponding to the sum of the unknown potential and a known background potential, which is sufficiently disjoint. If this condition is relaxed, then we give similar results for finding the potential from additional monochromatic measurements of the differential scattering cross section of the unknown potential without the background potential. The performance of the proposed algorithms is demonstrated in numerical examples. In the present work we significantly advance theoretically and numerically studies of Agaltsov <italic toggle=\"yes\">et al</italic> (2019 <italic toggle=\"yes\">Inverse Problems</italic> <bold>35</bold> 24001) and Novikov and Sivkin (2021 <italic toggle=\"yes\">Inverse Problems</italic> <bold>37</bold> 055011).","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"10 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142210298","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 : 2024-08-29DOI: 10.1088/1361-6420/ad6fc8
Daniel Rabinovich, Dan Givoli
Kirchhoff Migration (KM), sometimes called Arrival (or Travel) Time Imaging, is a basic and popular imaging technique based on the arrival time of waves from given sources to given sensors. It is commonly used in the fields of underwater acoustics and solid earth geophysics, for both subsurface structure analysis and for identifying unknown local obstacles (scatterers) in the medium. The present paper concentrates on the latter application. For acoustics, the KM algorithm is extremely simple and efficient, although it usually produces a rather crude image, which is the reason for its use as the method of choice when high resolution is not needed, or as a fast technique to produce an initial guess for a more sophisticated imaging method. For elasticity, KM is much more involved, as the arrival-time algorithm is not obvious, mainly since there is more than one wave speed at each spatial point. In this paper, a new KM scheme is proposed for obstacle identification in an isotropic piecewise-homogeneous elastic medium. The scheme is based on measuring two quantities that are second-order operators of the displacement field, which are related to P and S waves, and applying the acoustic KM algorithm to each of them, with the appropriate wave speed. It is demonstrated numerically that the operator related to S waves results in very good identification in many cases. The fact that measurements based on the S-related operator are preferred over those based on the P-related operator is an empirical observation, and awaits full analysis, although a partial explanation is given here.
基尔霍夫迁移（Kirchhoff Migration，KM），有时也称为到达（或移动）时间成像（Arrival (or Travel) Time Imaging），是一种基于波从给定源到达给定传感器的时间的基本且流行的成像技术。它通常用于水下声学和固体地球物理学领域，既可用于地下结构分析，也可用于识别介质中未知的局部障碍物（散射体）。本文主要讨论后一种应用。在声学方面，KM 算法非常简单高效，尽管它通常生成的图像相当粗糙，这也是它在不需要高分辨率时作为首选方法，或作为为更复杂的成像方法生成初始猜测的快速技术的原因。对于弹性而言，KM 涉及的问题要多得多，因为到达时间算法并不明显，主要是因为每个空间点的波速不止一种。本文提出了一种新的 KM 方案，用于在各向同性的片状均质弹性介质中识别障碍物。该方案基于测量与 P 波和 S 波相关的位移场二阶算子的两个量，并将声学 KM 算法与适当的波速分别应用于这两个量。数值结果表明，与 S 波相关的算子在许多情况下都能实现很好的识别。基于 S 波相关算子的测量结果优于基于 P 波相关算子的测量结果，这是一个经验观察结果，有待全面分析，但本文给出了部分解释。
{"title":"A Kirchhoff Migration scheme for elastic obstacle identification","authors":"Daniel Rabinovich, Dan Givoli","doi":"10.1088/1361-6420/ad6fc8","DOIUrl":"https://doi.org/10.1088/1361-6420/ad6fc8","url":null,"abstract":"Kirchhoff Migration (KM), sometimes called Arrival (or Travel) Time Imaging, is a basic and popular imaging technique based on the arrival time of waves from given sources to given sensors. It is commonly used in the fields of underwater acoustics and solid earth geophysics, for both subsurface structure analysis and for identifying unknown local obstacles (scatterers) in the medium. The present paper concentrates on the latter application. For acoustics, the KM algorithm is extremely simple and efficient, although it usually produces a rather crude image, which is the reason for its use as the method of choice when high resolution is not needed, or as a fast technique to produce an initial guess for a more sophisticated imaging method. For elasticity, KM is much more involved, as the arrival-time algorithm is not obvious, mainly since there is more than one wave speed at each spatial point. In this paper, a new KM scheme is proposed for obstacle identification in an isotropic piecewise-homogeneous elastic medium. The scheme is based on measuring two quantities that are second-order operators of the displacement field, which are related to P and S waves, and applying the acoustic KM algorithm to each of them, with the appropriate wave speed. It is demonstrated numerically that the operator related to S waves results in very good identification in many cases. The fact that measurements based on the S-related operator are preferred over those based on the P-related operator is an empirical observation, and awaits full analysis, although a partial explanation is given here.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226718","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 : 2024-08-28DOI: 10.1088/1361-6420/ad7054
Roman Novikov, Basant Lal Sharma
We consider multi-frequency inverse source problem for the discrete Helmholtz operator on the square lattice Zd, d⩾1. We consider this problem for the cases with and without phase information. We prove uniqueness results and present examples of non-uniqueness for this problem for the case of compactly supported source function, and a Lipshitz stability estimate for the phased case is established. Relations with inverse scattering problem for the discrete Schrödinger operators in the Born approximation are also provided.
{"title":"Inverse source problem for discrete Helmholtz equation","authors":"Roman Novikov, Basant Lal Sharma","doi":"10.1088/1361-6420/ad7054","DOIUrl":"https://doi.org/10.1088/1361-6420/ad7054","url":null,"abstract":"We consider multi-frequency inverse source problem for the discrete Helmholtz operator on the square lattice <inline-formula>\u0000<tex-math><?CDATA $mathbb{Z}^d$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">Z</mml:mi></mml:mrow><mml:mi>d</mml:mi></mml:msup></mml:mrow></mml:math><inline-graphic xlink:href=\"ipad7054ieqn1.gif\"></inline-graphic></inline-formula>, <inline-formula>\u0000<tex-math><?CDATA $d unicode{x2A7E} 1$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mi>d</mml:mi><mml:mtext>⩾</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href=\"ipad7054ieqn2.gif\"></inline-graphic></inline-formula>. We consider this problem for the cases with and without phase information. We prove uniqueness results and present examples of non-uniqueness for this problem for the case of compactly supported source function, and a Lipshitz stability estimate for the phased case is established. Relations with inverse scattering problem for the discrete Schrödinger operators in the Born approximation are also provided.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"61 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142210299","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 : 2024-08-28DOI: 10.1088/1361-6420/ad7056
Ke Chen, Jasen Lai, Chunmei Wang
Partial differential equations (PDEs) play a foundational role in modeling physical phenomena. This study addresses the challenging task of determining variable coefficients within PDEs from measurement data. We introduce a novel neural network, ‘pseudo-differential IAEnet’ (pd-IAEnet), which draws inspiration from pseudo-differential operators. pd-IAEnet achieves significantly enhanced computational speed and accuracy with fewer parameters compared to conventional models. Extensive benchmark evaluations are conducted across a range of inverse problems, including electrical impedance tomography, optical tomography, and seismic imaging, consistently demonstrating pd-IAEnet’s superior accuracy. Notably, pd-IAEnet exhibits robustness in the presence of measurement noise, a critical characteristic for real-world applications. An exceptional feature is its discretization invariance, enabling effective training on data from diverse discretization schemes while maintaining accuracy on different meshes. In summary, pd-IAEnet offers a potent and efficient solution for addressing inverse PDE problems, contributing to improved computational efficiency, robustness, and adaptability to a wide array of data sources.
{"title":"Pseudo-differential integral autoencoder network for inverse PDE operators","authors":"Ke Chen, Jasen Lai, Chunmei Wang","doi":"10.1088/1361-6420/ad7056","DOIUrl":"https://doi.org/10.1088/1361-6420/ad7056","url":null,"abstract":"Partial differential equations (PDEs) play a foundational role in modeling physical phenomena. This study addresses the challenging task of determining variable coefficients within PDEs from measurement data. We introduce a novel neural network, ‘pseudo-differential IAEnet’ (pd-IAEnet), which draws inspiration from pseudo-differential operators. pd-IAEnet achieves significantly enhanced computational speed and accuracy with fewer parameters compared to conventional models. Extensive benchmark evaluations are conducted across a range of inverse problems, including electrical impedance tomography, optical tomography, and seismic imaging, consistently demonstrating pd-IAEnet’s superior accuracy. Notably, pd-IAEnet exhibits robustness in the presence of measurement noise, a critical characteristic for real-world applications. An exceptional feature is its discretization invariance, enabling effective training on data from diverse discretization schemes while maintaining accuracy on different meshes. In summary, pd-IAEnet offers a potent and efficient solution for addressing inverse PDE problems, contributing to improved computational efficiency, robustness, and adaptability to a wide array of data sources.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"61 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142210310","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 : 2024-08-27DOI: 10.1088/1361-6420/ad6a35
B Harroué, J-F Giovannelli, M Pereyra
This paper considers the quantitative comparison of several alternative models to perform deconvolution in situations where there is no ground truth data available. With applications to very large data sets in mind, we focus on linear deconvolution models based on a Wiener filter. Although comparatively simple, such models are widely prevalent in large scale setting such as high-resolution image restoration because they provide an excellent trade-off between accuracy and computational effort. However, in order to deliver accurate solutions, the models need to be properly calibrated in order to capture the covariance structure of the unknown quantity of interest and of the measurement error. This calibration often requires onerous controlled experiments and extensive expert supervision, as well as regular recalibration procedures. This paper adopts an unsupervised Bayesian statistical approach to model assessment that allows comparing alternative models by using only the observed data, without the need for ground truth data or controlled experiments. Accordingly, the models are quantitatively compared based on their posterior probabilities given the data, which are derived from the marginal likelihoods or evidences of the models. The computation of these evidences is highly non-trivial and this paper consider three different strategies to address this difficulty—a Chib approach, Laplace approximations, and a truncated harmonic expectation—all of which efficiently implemented by using a Gibbs sampling algorithm specialised for this class of models. In addition to enabling unsupervised model selection, the output of the Gibbs sampler can also be used to automatically estimate unknown model parameters such as the variance of the measurement error and the power of the unknown quantity of interest. The proposed strategies are demonstrated on a range of image deconvolution problems, where they are used to compare different modelling choices for the instrument’s point spread function and covariance matrices for the unknown image and for the measurement error.
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