Pub Date : 2024-07-09DOI: 10.1088/1361-6420/ad5d0d
Yijie Yang, Qifeng Gao and Yuping Duan
The unrolling method has been investigated for learning variational models in x-ray computed tomography. However, for incomplete data reconstruction, such as sparse-view and limited-angle problems, the unrolling method of gradient descent of the energy minimization problem cannot yield satisfactory results. In this paper, we present an effective CT reconstruction model, where the low-resolution image is introduced as a regularization for incomplete data problems. In what follows, we utilize the deep equilibrium approach to unfolding of the gradient descent algorithm, thereby constructing the backbone network architecture for solving the minimization model. We theoretically discuss the convergence of the proposed low-resolution prior equilibrium (LRPE) model and provide the necessary conditions to guarantee its convergence. Experimental results on both sparse-view and limited-angle reconstruction problems are provided, demonstrating that our end-to-end LRPE model outperforms other state-of-the-art methods in terms of noise reduction, contrast-to-noise ratio, and preservation of edge details.
在 X 射线计算机断层扫描中，已经研究了用于学习变分模型的展开方法。然而，对于不完整数据重建，如稀疏视图和有限角度问题，能量最小化问题梯度下降的展开法无法获得令人满意的结果。本文提出了一种有效的 CT 重建模型，其中引入了低分辨率图像作为不完整数据问题的正则化。接下来，我们利用深度均衡方法来展开梯度下降算法，从而构建出求解最小化模型的骨干网络架构。我们从理论上讨论了所提出的低分辨率先验均衡（LRPE）模型的收敛性，并提供了保证其收敛性的必要条件。我们提供了稀疏视图和有限角度重建问题的实验结果，证明我们的端到端 LRPE 模型在降噪、对比度-噪声比和边缘细节保留方面优于其他最先进的方法。
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Pub Date : 2024-07-09DOI: 10.1088/1361-6420/ad5d0e
Alfred K Louis
Most derivations of inversion formulae for x-ray or Radon transform are based on the projection theorem, where for fixed direction the Fourier transform of x-ray or Radon transform is calculated and compared with the Fourier transform of the searched-for function. In contrast to this we start here off from the searched-for field, calculate its Fourier transform for fixed direction, which is now a vector or tensor field, that we then expand in a suitable direction dependent basis. The expansion coefficients are recognized as the Fourier transform of longitudinal, transversal or mixed ray transforms or vectorial Radon transform respectively. The inverse Fourier transform of the searched-for field then directly leads to inversion formulae for those transforms applying problem adapted backprojections. When considering the Helmholtz decomposition of the field we immediately find inversion formulae for those transversal or longitudinal transforms. First inversion formulae for the longitudinal ray transform, similar to those given by Natterer (1986 The Mathematics of Computerized Tomography (Teubner and Wiley)) for x-ray tomography, were given by Natterer-Wübbeling in 2001, Natterer and Wübbeling (2001 Mathematical Methods in Image Reconstruction (SIAM)), but then not pursued by other authors. In this paper, we present the above described method and derive in a unified way inversion formulae for the ray transforms treated in Louis (2022 Inverse Problems38 065008) containing the results from Louis (2022 Inverse Problems38 065008) as special cases. Additionally we present new inversion formulae for the vectorial Radon transform. As a consequence the inversion formulae directly give Plancherel’s formulae for the vectorial or tensorial transforms. Together with the Natterer inequality, which is independent of the ray or Radon transforms, we present the Natterer stability of those vectorial and tensorial transforms.
大多数 X 射线或拉顿变换反演公式的推导都是基于投影定理，即在固定方向上计算 X 射线或拉顿变换的傅里叶变换，并将其与搜索函数的傅里叶变换进行比较。与此相反，我们在这里从搜索到的场开始，计算其固定方向的傅里叶变换，现在它是一个矢量或张量场，然后我们在一个合适的与方向相关的基础上对其进行扩展。展开系数分别被视为纵向、横向或混合射线变换的傅里叶变换或矢量拉顿变换。搜索场的反傅里叶变换可直接得出这些变换的反演公式，并应用与问题相适应的反推。在考虑场的亥姆霍兹分解时，我们可以立即找到这些横向或纵向变换的反演公式。Natterer-Wübbeling 于 2001 年给出了纵向射线变换的反演公式，Natterer 和 Wübbeling (2001 Mathematical Methods in Image Reconstruction (SIAM))也给出了与 Natterer（1986 The Mathematics of Computerized Tomography (Teubner and Wiley)）类似的 X 射线断层摄影反演公式，但其他作者并未继续研究。在本文中，我们介绍了上述方法，并以统一的方式推导出路易斯（2022 逆问题 38 065008）中处理的射线变换的反演公式，其中包含路易斯（2022 逆问题 38 065008）中作为特例的结果。此外，我们还提出了矢量拉顿变换的新反演公式。因此，反演公式直接给出了矢量或张量变换的 Plancherel 公式。通过与射线或拉顿变换无关的纳特勒不等式，我们提出了这些矢量和张量变换的纳特勒稳定性。
{"title":"A unified approach to inversion formulae for vector and tensor ray and radon transforms and the Natterer inequality","authors":"Alfred K Louis","doi":"10.1088/1361-6420/ad5d0e","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5d0e","url":null,"abstract":"Most derivations of inversion formulae for x-ray or Radon transform are based on the projection theorem, where for fixed direction the Fourier transform of x-ray or Radon transform is calculated and compared with the Fourier transform of the searched-for function. In contrast to this we start here off from the searched-for field, calculate its Fourier transform for fixed direction, which is now a vector or tensor field, that we then expand in a suitable direction dependent basis. The expansion coefficients are recognized as the Fourier transform of longitudinal, transversal or mixed ray transforms or vectorial Radon transform respectively. The inverse Fourier transform of the searched-for field then directly leads to inversion formulae for those transforms applying problem adapted backprojections. When considering the Helmholtz decomposition of the field we immediately find inversion formulae for those transversal or longitudinal transforms. First inversion formulae for the longitudinal ray transform, similar to those given by Natterer (1986 <italic toggle=\"yes\">The Mathematics of Computerized Tomography</italic> (Teubner and Wiley)) for x-ray tomography, were given by Natterer-Wübbeling in 2001, Natterer and Wübbeling (2001 <italic toggle=\"yes\">Mathematical Methods in Image Reconstruction</italic> (SIAM)), but then not pursued by other authors. In this paper, we present the above described method and derive in a unified way inversion formulae for the ray transforms treated in Louis (2022 <italic toggle=\"yes\">Inverse Problems</italic>\u0000<bold>38</bold> 065008) containing the results from Louis (2022 <italic toggle=\"yes\">Inverse Problems</italic>\u0000<bold>38</bold> 065008) as special cases. Additionally we present new inversion formulae for the vectorial Radon transform. As a consequence the inversion formulae directly give Plancherel’s formulae for the vectorial or tensorial transforms. Together with the Natterer inequality, which is independent of the ray or Radon transforms, we present the Natterer stability of those vectorial and tensorial transforms.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"236 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141567220","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 : 2024-07-09DOI: 10.1088/1361-6420/ad5a35
Martin Burger, Thomas Schuster, Anne Wald
We consider time-dependent inverse problems in a mathematical setting using Lebesgue-Bochner spaces. Such problems arise when one aims to recover parameters from given observations where the parameters or the data depend on time. There are various important applications being subject of current research that belong to this class of problems. Typically inverse problems are ill-posed in the sense that already small noise in the data causes tremendous errors in the solution. In this article we present two different concepts of ill-posedness: temporally (pointwise) ill-posedness and uniform ill-posedness with respect to the Lebesgue-Bochner setting. We investigate the two concepts by means of a typical setting consisting of a time-depending observation operator composed by a compact operator. Furthermore we develop regularization methods that are adapted to the respective class of ill-posedness.
{"title":"Ill-posedness of time-dependent inverse problems in Lebesgue-Bochner spaces","authors":"Martin Burger, Thomas Schuster, Anne Wald","doi":"10.1088/1361-6420/ad5a35","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5a35","url":null,"abstract":"We consider time-dependent inverse problems in a mathematical setting using Lebesgue-Bochner spaces. Such problems arise when one aims to recover parameters from given observations where the parameters or the data depend on time. There are various important applications being subject of current research that belong to this class of problems. Typically inverse problems are ill-posed in the sense that already small noise in the data causes tremendous errors in the solution. In this article we present two different concepts of ill-posedness: temporally (pointwise) ill-posedness and uniform ill-posedness with respect to the Lebesgue-Bochner setting. We investigate the two concepts by means of a typical setting consisting of a time-depending observation operator composed by a compact operator. Furthermore we develop regularization methods that are adapted to the respective class of ill-posedness.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"49 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141567213","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 : 2024-07-09DOI: 10.1088/1361-6420/ad5b83
F M Watson, D Andre, W R B Lionheart
Through-wall synthetic aperture radar (SAR) imaging is of significant interest for security purposes, in particular when using multi-static SAR systems consisting of multiple distributed radar transmitters and receivers to improve resolution and the ability to recognise objects. Yet there is a significant challenge in forming focused, useful images due to multiple scattering effects through walls, whereas standard SAR imaging has an inherent single scattering assumption. This may be exacerbated with multi-static collections, since different scattering events will be observed from each angle and the data may not coherently combine well in a naive manner. To overcome this, we propose an image formation method which resolves full-wave effects through an approximately known wall or other arbitrary obstacle, which itself has some unknown ‘nuisance’ parameters that are determined as part of the reconstruction to provide well focused images. The method is more flexible and realistic than existing methods which treat a single wall as a flat layered medium, whilst being significantly computationally cheaper than full-wave methods, strongly motivated by practical considerations for through-wall SAR.
穿墙合成孔径雷达（SAR）成像在安全领域具有重要意义，尤其是在使用由多个分布式雷达发射器和接收器组成的多静态 SAR 系统来提高分辨率和识别物体的能力时。然而，由于穿墙的多重散射效应，而标准合成孔径雷达成像具有固有的单散射假设，因此在形成聚焦、有用的图像方面存在巨大挑战。由于从每个角度都会观察到不同的散射事件，数据可能无法以天真的方式连贯组合，因此多静态采集可能会加剧这一问题。为了克服这一问题，我们提出了一种图像形成方法，通过近似已知的墙壁或其他任意障碍物来解决全波效应，这些障碍物本身有一些未知的 "干扰 "参数，这些参数将作为重建的一部分来确定，以提供聚焦良好的图像。这种方法比现有的将单面墙壁视为平面分层介质的方法更灵活、更逼真，同时计算成本也比全波方法低得多，这主要是出于对穿墙合成孔径雷达的实际考虑。
{"title":"Resolving full-wave through-wall transmission effects in multi-static synthetic aperture radar","authors":"F M Watson, D Andre, W R B Lionheart","doi":"10.1088/1361-6420/ad5b83","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5b83","url":null,"abstract":"Through-wall synthetic aperture radar (SAR) imaging is of significant interest for security purposes, in particular when using multi-static SAR systems consisting of multiple distributed radar transmitters and receivers to improve resolution and the ability to recognise objects. Yet there is a significant challenge in forming focused, useful images due to multiple scattering effects through walls, whereas standard SAR imaging has an inherent single scattering assumption. This may be exacerbated with multi-static collections, since different scattering events will be observed from each angle and the data may not coherently combine well in a naive manner. To overcome this, we propose an image formation method which resolves full-wave effects through an approximately known wall or other arbitrary obstacle, which itself has some unknown ‘nuisance’ parameters that are determined as part of the reconstruction to provide well focused images. The method is more flexible and realistic than existing methods which treat a single wall as a flat layered medium, whilst being significantly computationally cheaper than full-wave methods, strongly motivated by practical considerations for through-wall SAR.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"365 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141567214","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 : 2024-07-03DOI: 10.1088/1361-6420/ad5b81
Tian Niu, Junliang Lv and Jiahui Gao
In this paper, we establish the uniqueness of identifying a smooth grating profile with a mixed boundary condition (MBC) or transmission boundary conditions (TBCs) from phaseless data. The existing uniqueness result requires the measured data to be in a bounded domain. To break this restriction, we design an incident system consisting of the superposition of point sources to reduce the measurement data from a bounded domain to a line above the grating profile. We derive reciprocity relations for point sources, diffracted fields, and total fields, respectively. Based on Rayleigh’s expansion and reciprocity relation of the total field, a grating profile with a MBC or TBCs can be uniquely determined from the phaseless total field data. An iterative algorithm is proposed to recover the Fourier modes of grating profiles at a fixed wavenumber. To implement this algorithm, we derive the Fréchet derivative of the total field operator and its adjoint operator. Some numerical examples are presented to verify the correctness of theoretical results and to show the effectiveness of our numerical algorithm.
在本文中，我们通过无相数据确定了具有混合边界条件（MBC）或传输边界条件（TBC）的光栅轮廓的唯一性。现有的唯一性结果要求测量数据位于有界域中。为了打破这一限制，我们设计了一个由点源叠加组成的入射系统，将测量数据从有界域减少到光栅轮廓上方的一条线上。我们分别推导出了点源、衍射场和总场的互易关系。根据雷利展开和总场的互易关系，可以从无相总场数据唯一确定具有 MBC 或 TBC 的光栅轮廓。我们提出了一种迭代算法来恢复光栅轮廓在固定波长下的傅里叶模式。为了实现这一算法，我们推导出了总场算子的弗雷谢特导数及其邻接算子。为了验证理论结果的正确性，并显示我们的数值算法的有效性，我们给出了一些数值示例。
{"title":"Uniqueness and numerical method for phaseless inverse diffraction grating problem with known superposition of incident point sources","authors":"Tian Niu, Junliang Lv and Jiahui Gao","doi":"10.1088/1361-6420/ad5b81","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5b81","url":null,"abstract":"In this paper, we establish the uniqueness of identifying a smooth grating profile with a mixed boundary condition (MBC) or transmission boundary conditions (TBCs) from phaseless data. The existing uniqueness result requires the measured data to be in a bounded domain. To break this restriction, we design an incident system consisting of the superposition of point sources to reduce the measurement data from a bounded domain to a line above the grating profile. We derive reciprocity relations for point sources, diffracted fields, and total fields, respectively. Based on Rayleigh’s expansion and reciprocity relation of the total field, a grating profile with a MBC or TBCs can be uniquely determined from the phaseless total field data. An iterative algorithm is proposed to recover the Fourier modes of grating profiles at a fixed wavenumber. To implement this algorithm, we derive the Fréchet derivative of the total field operator and its adjoint operator. Some numerical examples are presented to verify the correctness of theoretical results and to show the effectiveness of our numerical algorithm.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"30 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548474","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 : 2024-07-02DOI: 10.1088/1361-6420/ad5b82
Lingzheng Kong, Youjun Deng and Liyan Zhu
In this paper, we study the recovery of multi-layer structures in inverse conductivity problem by using one measurement. First, we define the concept of Generalized Polarization Tensors (GPTs) for multi-layered medium and show some important properties of the proposed GPTs. With the help of GPTs, we present the perturbation formula for general multi-layered medium. Then we derive the perturbed electric potential for multi-layer concentric disks structure in terms of the so-called generalized polarization matrix, whose dimension is the same as the number of the layers. By delicate analysis, we derive an algebraic identity involving the geometric and material configurations of multi-layer concentric disks. This enables us to reconstruct the multi-layer structures by using only one partial-order measurement.
{"title":"Inverse conductivity problem with one measurement: uniqueness of multi-layer structures","authors":"Lingzheng Kong, Youjun Deng and Liyan Zhu","doi":"10.1088/1361-6420/ad5b82","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5b82","url":null,"abstract":"In this paper, we study the recovery of multi-layer structures in inverse conductivity problem by using one measurement. First, we define the concept of Generalized Polarization Tensors (GPTs) for multi-layered medium and show some important properties of the proposed GPTs. With the help of GPTs, we present the perturbation formula for general multi-layered medium. Then we derive the perturbed electric potential for multi-layer concentric disks structure in terms of the so-called generalized polarization matrix, whose dimension is the same as the number of the layers. By delicate analysis, we derive an algebraic identity involving the geometric and material configurations of multi-layer concentric disks. This enables us to reconstruct the multi-layer structures by using only one partial-order measurement.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"10 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511938","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 : 2024-07-01DOI: 10.1088/1361-6420/ad5a34
Nagendra Kumar Chaurasia and Shubhankar Chakraborty
Among different numerical methods for modeling turbulent flow, Reynolds-averaged Navier–Stokes (RANS) is the most commonly used and computationally reasonable. However, the accuracy of RANS is lower than that of other high-fidelity numerical methods. In this work, the uncertainties associated with the coefficients of the standard RANS turbulence model are estimated and calibrated to improve the accuracy. The calibration is performed by considering the coefficients individually as well as collectively. The first three coefficients of the standard turbulence model are calibrated among the five coefficients ( and σk ). The Bayesian inference technique using the Metropolis–Hastings algorithm is applied to quantify uncertainties and calibration. Flow over a periodic hill is selected as a test case. The separation height of the bubble at and , along with the streamwise velocity at various locations, has been chosen as the quantities of interest for comparing the results with DNS. The calibration is performed using known high-fidelity data (direct numerical simulation) from the available data set. The velocity field is re-calculated from the calibrated closure coefficients and compared with the same calculated with the standard coefficients of turbulence model (baseline). The deviation of calibrated Cµ is almost 50%–60% from baseline and for and it is 3%–12% and 6%–9% respectively. The algorithm is tested for different Reynold numbers and data points. A sensitivity analysis is also performed.
{"title":"Bayesian interface technique-based inverse estimation of closure coefficients of standard k−ϵ turbulence model by limiting the number of DNS data points for flow over a periodic hill","authors":"Nagendra Kumar Chaurasia and Shubhankar Chakraborty","doi":"10.1088/1361-6420/ad5a34","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5a34","url":null,"abstract":"Among different numerical methods for modeling turbulent flow, Reynolds-averaged Navier–Stokes (RANS) is the most commonly used and computationally reasonable. However, the accuracy of RANS is lower than that of other high-fidelity numerical methods. In this work, the uncertainties associated with the coefficients of the standard RANS turbulence model are estimated and calibrated to improve the accuracy. The calibration is performed by considering the coefficients individually as well as collectively. The first three coefficients of the standard turbulence model are calibrated among the five coefficients ( and σk ). The Bayesian inference technique using the Metropolis–Hastings algorithm is applied to quantify uncertainties and calibration. Flow over a periodic hill is selected as a test case. The separation height of the bubble at and , along with the streamwise velocity at various locations, has been chosen as the quantities of interest for comparing the results with DNS. The calibration is performed using known high-fidelity data (direct numerical simulation) from the available data set. The velocity field is re-calculated from the calibrated closure coefficients and compared with the same calculated with the standard coefficients of turbulence model (baseline). The deviation of calibrated Cµ is almost 50%–60% from baseline and for and it is 3%–12% and 6%–9% respectively. The algorithm is tested for different Reynold numbers and data points. A sensitivity analysis is also performed.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"205 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511937","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 : 2024-06-25DOI: 10.1088/1361-6420/ad5583
Nathaniel Pritchard and Vivak Patel
In large-scale applications including medical imaging, collocation differential equation solvers, and estimation with differential privacy, the underlying linear inverse problem can be reformulated as a streaming problem. In theory, the streaming problem can be effectively solved using memory-efficient, exponentially-converging streaming solvers. In special cases when the underlying linear inverse problem is finite-dimensional, streaming solvers can periodically evaluate the residual norm at a substantial computational cost. When the underlying system is infinite dimensional, streaming solver can only access noisy estimates of the residual. While such noisy estimates are computationally efficient, they are useful only when their accuracy is known. In this work, we rigorously develop a general family of computationally-practical residual estimators and their uncertainty sets for streaming solvers, and we demonstrate the accuracy of our methods on a number of large-scale linear problems. Thus, we further enable the practical use of streaming solvers for important classes of linear inverse problems.
{"title":"Solving, tracking and stopping streaming linear inverse problems","authors":"Nathaniel Pritchard and Vivak Patel","doi":"10.1088/1361-6420/ad5583","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5583","url":null,"abstract":"In large-scale applications including medical imaging, collocation differential equation solvers, and estimation with differential privacy, the underlying linear inverse problem can be reformulated as a streaming problem. In theory, the streaming problem can be effectively solved using memory-efficient, exponentially-converging streaming solvers. In special cases when the underlying linear inverse problem is finite-dimensional, streaming solvers can periodically evaluate the residual norm at a substantial computational cost. When the underlying system is infinite dimensional, streaming solver can only access noisy estimates of the residual. While such noisy estimates are computationally efficient, they are useful only when their accuracy is known. In this work, we rigorously develop a general family of computationally-practical residual estimators and their uncertainty sets for streaming solvers, and we demonstrate the accuracy of our methods on a number of large-scale linear problems. Thus, we further enable the practical use of streaming solvers for important classes of linear inverse problems.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"73 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511939","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 : 2024-06-23DOI: 10.1088/1361-6420/ad575c
Antonio Corbo Esposito, Luisa Faella, Vincenzo Mottola, Gianpaolo Piscitelli, Ravi Prakash and Antonello Tamburrino
This paper deals with the Monotonicity Principle (MP) for nonlinear materials with piecewise growth exponent. The results obtained are relevant because they enable the use of a fast imaging method based on MP, applied to a wide class of problems with two or more materials, at least one of which is nonlinear. The treatment is very general and makes it possible to model a wide range of practical configurations such as superconducting (SC), perfect electrical conducting (PEC) or perfect electrical insulating (PEI) materials. A key role is played by the average Dirichlet-to-Neumann operator, introduced in Corbo Esposito et al (2021 Inverse Problems37 045012), where the MP for a single type of nonlinearity was treated. Realistic numerical examples confirm the theoretical findings.
{"title":"Piecewise nonlinear materials and Monotonicity Principle","authors":"Antonio Corbo Esposito, Luisa Faella, Vincenzo Mottola, Gianpaolo Piscitelli, Ravi Prakash and Antonello Tamburrino","doi":"10.1088/1361-6420/ad575c","DOIUrl":"https://doi.org/10.1088/1361-6420/ad575c","url":null,"abstract":"This paper deals with the Monotonicity Principle (MP) for nonlinear materials with piecewise growth exponent. The results obtained are relevant because they enable the use of a fast imaging method based on MP, applied to a wide class of problems with two or more materials, at least one of which is nonlinear. The treatment is very general and makes it possible to model a wide range of practical configurations such as superconducting (SC), perfect electrical conducting (PEC) or perfect electrical insulating (PEI) materials. A key role is played by the average Dirichlet-to-Neumann operator, introduced in Corbo Esposito et al (2021 Inverse Problems37 045012), where the MP for a single type of nonlinearity was treated. Realistic numerical examples confirm the theoretical findings.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"20 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511940","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 : 2024-06-17DOI: 10.1088/1361-6420/ad52bb
C M Wensrich, S Holman, M Courdurier, W R B Lionheart, A P Polyakova and I E Svetov
We examine the problem of Bragg-edge elastic strain tomography from energy resolved neutron transmission imaging. A new approach is developed for two-dimensional plane-stress and plane-strain systems whereby elastic strain can be reconstructed from its Longitudinal ray transform (LRT) as two parts of a Helmholtz decomposition based on the concept of an Airy stress potential. The solenoidal component of this decomposition is reconstructed using an inversion formula based on a tensor filtered back projection (FBP) algorithm whereas the potential part can be recovered using either Hooke’s law or a finite element model of the elastic system. The technique is demonstrated for two-dimensional plane-stress systems in both simulation, and on real experimental data. We also demonstrate that application of the standard scalar FBP algorithm to the LRT in these systems recovers the trace of the solenoidal component of strain and we provide physical meaning for this quantity in the case of 2D plane-stress and plane-strain systems.
{"title":"Direct inversion of the Longitudinal ray transform for 2D residual elastic strain fields","authors":"C M Wensrich, S Holman, M Courdurier, W R B Lionheart, A P Polyakova and I E Svetov","doi":"10.1088/1361-6420/ad52bb","DOIUrl":"https://doi.org/10.1088/1361-6420/ad52bb","url":null,"abstract":"We examine the problem of Bragg-edge elastic strain tomography from energy resolved neutron transmission imaging. A new approach is developed for two-dimensional plane-stress and plane-strain systems whereby elastic strain can be reconstructed from its Longitudinal ray transform (LRT) as two parts of a Helmholtz decomposition based on the concept of an Airy stress potential. The solenoidal component of this decomposition is reconstructed using an inversion formula based on a tensor filtered back projection (FBP) algorithm whereas the potential part can be recovered using either Hooke’s law or a finite element model of the elastic system. The technique is demonstrated for two-dimensional plane-stress systems in both simulation, and on real experimental data. We also demonstrate that application of the standard scalar FBP algorithm to the LRT in these systems recovers the trace of the solenoidal component of strain and we provide physical meaning for this quantity in the case of 2D plane-stress and plane-strain systems.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"49 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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