Inverse Problems最新文献

英文中文

Uniqueness and numerical method for phaseless inverse diffraction grating problem with known superposition of incident point sources 已知入射点源叠加的无相位反衍射光栅问题的唯一性和数值方法

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Inverse Problems

Pub Date : 2024-07-03 DOI: 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。

引用次数: 0

Inverse conductivity problem with one measurement: uniqueness of multi-layer structures 一次测量的反电导率问题：多层结构的唯一性

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Inverse Problems

Pub Date : 2024-07-02 DOI: 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.

在本文中，我们研究了在反电导问题中通过一次测量恢复多层结构的问题。首先，我们定义了多层介质的广义极化张量（GPT）概念，并展示了所提出的广义极化张量的一些重要特性。在 GPT 的帮助下，我们提出了一般多层介质的扰动公式。然后，我们根据所谓的广义极化矩阵（其维度与层数相同）推导出多层同心盘结构的扰动电动势。通过精细分析，我们得出了涉及多层同心圆盘的几何和材料配置的代数特性。这使得我们只需使用一次部分阶测量就能重建多层结构。

引用次数: 0

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 基于贝叶斯界面技术的标准 k-ϵ 湍流模型闭合系数的反估计，方法是限制周期性山丘上流动的 DNS 数据点数量

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Inverse Problems

Pub Date : 2024-07-01 DOI: 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.

在不同的湍流建模数值方法中，雷诺平均纳维-斯托克斯（RANS）是最常用、计算最合理的方法。然而，RANS 的精度低于其他高保真数值方法。在这项工作中，对与标准 RANS 湍流模型系数相关的不确定性进行了估计和校准，以提高精度。校准通过对系数的单独和整体考虑来进行。在五个系数（和 σk ）中，对标准湍流模型的前三个系数进行校准。使用 Metropolis-Hastings 算法的贝叶斯推理技术用于量化不确定性和校准。选择周期性山丘上的水流作为测试案例。气泡在和处的分离高度以及不同位置的流向速度被选为与 DNS 结果进行比较的相关量。校准使用现有数据集中的已知高保真数据（直接数值模拟）。根据校准后的闭合系数重新计算速度场，并与使用湍流模型标准系数（基线）计算出的速度场进行比较。校准后的 Cµ 与基线的偏差几乎为 50%-60%，而 Cµ 与基线的偏差分别为 3%-12%和 6%-9%。该算法针对不同的雷诺数和数据点进行了测试。同时还进行了敏感性分析。

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引用次数: 0

Solving, tracking and stopping streaming linear inverse problems 解决、跟踪和停止流线性逆问题

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Inverse Problems

Pub Date : 2024-06-25 DOI: 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.

在大规模应用中，包括医学成像、配位微分方程求解器和带有微分隐私的估算，基本的线性逆问题可以重新表述为流问题。从理论上讲，使用内存效率高、指数级收敛的流求解器可以有效地求解流问题。在底层线性逆问题是有限维的特殊情况下，流求解器可以定期评估残差规范，但计算成本很高。当底层系统为无限维时，流式求解器只能获取残差的噪声估计值。虽然这种噪声估计值的计算效率很高，但只有当它们的精度已知时才有用。在这项工作中，我们为流式求解器严格开发了一系列计算上实用的残差估计值及其不确定性集，并在一些大规模线性问题上证明了我们方法的准确性。因此，我们进一步推动了流求解器在重要线性逆问题类别中的实际应用。

引用次数: 0

Piecewise nonlinear materials and Monotonicity Principle 片断非线性材料和单调性原理

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Inverse Problems

Pub Date : 2024-06-23 DOI: 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.

本文论述了具有片断增长指数的非线性材料的单调性原理（MP）。所获得的结果具有重要意义，因为这些结果使基于 MP 的快速成像方法得以使用，该方法适用于具有两种或两种以上材料（其中至少一种是非线性材料）的各种问题。这种处理方法非常通用，可以模拟各种实际配置，如超导（SC）、完全导电（PEC）或完全电绝缘（PEI）材料。Corbo Esposito 等人（2021 年逆问题 37 045012）引入的平均 Dirichlet 到 Neumann 算子发挥了关键作用，该算子处理了单一类型非线性的 MP。现实的数值例子证实了理论发现。

引用次数: 0

Direct inversion of the Longitudinal ray transform for 2D residual elastic strain fields 直接反演二维残余弹性应变场的纵向射线变换

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Inverse Problems

Pub Date : 2024-06-17 DOI: 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.

我们研究了通过能量分辨中子透射成像进行布拉格边弹性应变层析成像的问题。我们为二维平面应力和平面应变系统开发了一种新方法，根据这种方法，弹性应变可以从其纵向射线变换（LRT）中重构为基于艾里应力势概念的亥姆霍兹分解的两个部分。该分解的螺线部分是通过基于张量滤波反投影算法（FBP）的反演公式重建的，而势能部分则可以通过胡克定律或弹性系统的有限元模型恢复。我们在模拟和实际实验数据中对二维平面应力系统演示了这一技术。我们还证明，将标准标量 FBP 算法应用于这些系统中的 LRT，可以恢复应变的螺线分量的迹线，并为二维平面应力和平面应变系统中的这个量提供了物理意义。

引用次数: 0

A novel Newton method for inverse elastic scattering problems 用于反弹性散射问题的新型牛顿法

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Inverse Problems

Pub Date : 2024-06-11 DOI: 10.1088/1361-6420/ad4dda

Yan Chang, Yukun Guo, Hongyu Liu and Deyue Zhang

This work is concerned with an inverse elastic scattering problem of identifying the unknown rigid obstacle embedded in an open space filled with a homogeneous and isotropic elastic medium. A Newton-type iteration method relying on the boundary condition is designed to identify the boundary curve of the obstacle. Based on the Helmholtz decomposition and the Fourier–Bessel expansion, we explicitly derive the approximate scattered field and its derivative on each iterative curve. Rigorous mathematical justifications for the proposed method are provided. Numerical examples are presented to verify the effectiveness of the proposed method.

本研究关注的是一个反弹性散射问题，即识别嵌入充满各向同性均质弹性介质的开放空间中的未知刚性障碍物。设计了一种基于边界条件的牛顿迭代法来识别障碍物的边界曲线。基于亥姆霍兹分解和傅立叶-贝塞尔展开，我们明确推导出了近似散射场及其在每条迭代曲线上的导数。我们为所提出的方法提供了严格的数学论证。我们还提供了数值示例来验证所提方法的有效性。

引用次数: 0

Reconstruction of inhomogeneous media by an iteration algorithm with a learned projector 利用迭代算法和学习投影器重建非均质介质

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Inverse Problems

Pub Date : 2024-06-10 DOI: 10.1088/1361-6420/ad4f0b

Kai Li, Bo Zhang, Haiwen Zhang

This paper is concerned with the inverse problem of reconstructing an inhomogeneous medium from the acoustic far-field data at a fixed frequency in two dimensions. This inverse problem is severely ill-posed (and also strongly nonlinear), and certain regularization strategy is thus needed. However, it is difficult to select an appropriate regularization strategy which should enforce some a priori information of the unknown scatterer. To address this issue, we plan to use a deep learning approach to learn some a priori information of the unknown scatterer from certain ground truth data, which is then combined with a traditional iteration method to solve the inverse problem. Specifically, we propose a deep learning-based iterative reconstruction algorithm for the inverse problem, based on a repeated application of a deep neural network and the iteratively regularized Gauss–Newton method (IRGNM). Our deep neural network (called the learned projector in this paper) mainly focuses on learning the a priori information of the shape of the unknown contrast with a normalization technique in the training processes and is trained to act like a projector which is helpful for projecting the solution into some feasible region. Extensive numerical experiments show that our reconstruction algorithm provides good reconstruction results even for the high contrast case and has a satisfactory generalization ability.

本文研究的是在二维范围内从固定频率下的声学远场数据重建非均质介质的逆问题。这个逆问题具有严重的求解困难（同时也是强非线性问题），因此需要一定的正则化策略。然而，要选择一种合适的正则化策略是很困难的，因为这种策略应该强制执行一些未知散射体的先验信息。为了解决这个问题，我们计划使用深度学习方法，从某些地面实况数据中学习未知散射体的一些先验信息，然后结合传统的迭代方法来解决逆问题。具体来说，我们提出了一种基于深度学习的反问题迭代重建算法，该算法基于深度神经网络和迭代正则化高斯-牛顿法（IRGNM）的重复应用。我们的深度神经网络（本文中称为 "学习投影器"）主要侧重于在训练过程中利用归一化技术学习未知对比度形状的先验信息，并被训练成一个投影器，有助于将解投影到某个可行区域。大量的数值实验表明，即使在高对比度的情况下，我们的重建算法也能提供良好的重建结果，并且具有令人满意的泛化能力。

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引用次数: 0

Feasibility of acousto-electric tomography 声电断层扫描的可行性

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Inverse Problems

Pub Date : 2024-06-03 DOI: 10.1088/1361-6420/ad4669

Bjørn Jensen, Adrian Kirkeby and Kim Knudsen

In acousto-electric tomography (AET) the goal is to reconstruct the electric conductivity in a domain from electrostatic boundary measurements of corresponding currents and voltages, while the domain is perturbed by a time-dependent acoustic wave, thus taking advantage of the acousto-electric effect. We approach the AET reconstruction in two steps: First, the interior power density is obtained from boundary measurements by solving a linear inverse and ill-posed problem; second, the interior conductivity is reconstructed from the power density by solving a non-linear and well-posed problem. Mathematically these inverse problems are fairly well understood, and reconstruction methods work well on synthetic data. This is in contrast to experimental findings. An effect can indeed be observed and data can be collected. However, the acousto-electric coupling is very weak, and consequently, the change in the measured voltage due to the acoustic perturbation might be too small compared to the background noise for viable reconstructions. In this paper, we take one step towards understanding the feasibility of AET. We provide an in-silico model of the coupled physics scenario based on standard models for the individual phenomena. Moreover, we formulate and implement numerically a full reconstruction method for the inverse problem via the two steps. We perform computational experiments with realistically chosen parameters from the context of medical imaging. The focus is on understanding the role of the acousto-electric coupling parameter and the signal-to-noise ratio (SNR). The critical signal strength is analyzed and the omnipresent Johnson–Nyquist noise is estimated. We obtain both positive and negative findings; we can reconstruct features even under severe noise conditions, but we also find that the SNR one is likely to face in practice is too low to obtain useful reconstructions.

声电层析成像（AET）的目标是通过对相应电流和电压的静电边界测量重建域中的电导率，同时域受到随时间变化的声波扰动，从而利用声电效应。我们分两步进行 AET 重建：首先，通过求解一个线性反问题和求解困难的问题，从边界测量中获得内部功率密度；其次，通过求解一个非线性和求解困难的问题，从功率密度中重建内部传导性。从数学角度来看，这些逆问题都相当容易理解，而且重建方法在合成数据上也很有效。这与实验结果截然不同。确实可以观察到效果，也可以收集数据。然而，声电耦合非常微弱，因此，与背景噪声相比，声学扰动引起的测量电压变化可能太小，无法进行可行的重建。在本文中，我们朝着了解 AET 的可行性迈出了一步。我们基于单个现象的标准模型，提供了一个耦合物理场景的内部模型。此外，我们通过两个步骤为逆问题制定并数值化了一个完整的重建方法。我们使用医学成像中实际选择的参数进行了计算实验。重点是了解声电耦合参数和信噪比（SNR）的作用。我们分析了临界信号强度，并估算了无处不在的约翰逊-奈奎斯特噪声。我们得出了正反两方面的结论：即使在严重的噪声条件下，我们也能重建特征，但我们也发现，在实践中可能面临的信噪比太低，无法获得有用的重建。

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The Effect of Electroconvulsive Therapy on Frontal QRS-T Angle in Psychiatric Patients. 电休克疗法对精神病患者额叶 QRS-T 角的影响

IF 1.1 2区数学 Q1 MATHEMATICS, APPLIED

Inverse Problems

Pub Date : 2024-05-29 eCollection Date: 2024-01-01 DOI: 10.29399/npa.28443

Ülker Atılan Fedai, Halil Fedai

Introduction: Electroconvulsive therapy (ECT) is one of the biological therapies that is well tolerated and has a low risk of complications. Acute cardiovascular complications related to ECT such as ventricular arrhythmia, myocardial infarction and cardiac arrest have been recorded. Increased frontal QRS-T (fQRS-T) angle was associated with ventricular arrhythmia, sudden cardiac death and total mortality. In this study, we aimed to evaluate the effect of ECT on the myocardium using electrocardiography (ECG) parameters such as fQRS-T angle, QRS duration, QT and QTc interval.

Methods: A total of 108 patients diagnosed with bipolar disorder (n=36), depressive disorder (n=70) and schizophrenia (n=2) who underwent ECT were included in this study. 12-lead surface ECG of all patients were taken before the ECT, 15 min. after ECT and 24 hour after ECT.

Results: QRS duration, QT interval and corrected QT (QTc) interval were not changed significantly during the follow-up period. However, we found that, fQRS-T angle was significantly increased 15 minutes after ECT compared to baseline angle (p<0.001). We also detected that this increase in fQRS-T angle 15 minutes after ECT was significantly reduced 24 hours after ECT (p=0.031). Meanwhile, there was no significant difference between baseline and 24th hour fQRS-T angle (p=0.154).

Conclusions: In our study, a significant increase in fQRS-T angle was observed 15 min after ECT. However, the fQRS-T angle was found to return to normal after 24 hours. Our findings may indicate that ECT does not have a permanent side effect on the risk of cardiovascular events according to the fQRS-T angle.

简介电休克疗法（ECT）是一种生物疗法，具有良好的耐受性和较低的并发症风险。与电休克疗法相关的急性心血管并发症，如室性心律失常、心肌梗塞和心脏骤停，均有记录。额叶 QRS-T 角（fQRS-T）增大与室性心律失常、心脏性猝死和总死亡率有关。在这项研究中，我们旨在通过心电图（ECG）参数，如 fQRS-T 角、QRS 持续时间、QT 和 QTc 间期，评估 ECT 对心肌的影响：本研究共纳入 108 名接受 ECT 治疗的双相情感障碍（36 人）、抑郁障碍（70 人）和精神分裂症（2 人）患者。所有患者在电痉挛疗法前、电痉挛疗法后 15 分钟和电痉挛疗法后 24 小时内均接受了 12 导联表面心电图检查：结果：QRS时程、QT间期和校正QT（QTc）间期在随访期间无明显变化。然而，我们发现，与基线角度相比，电痉挛治疗后 15 分钟，fQRS-T 角度明显增加（p结论：在我们的研究中，观察到 ECT 15 分钟后 fQRS-T 角明显增加。但在 24 小时后，fQRS-T 角度恢复正常。我们的研究结果可能表明，根据 fQRS-T 角度，ECT 不会对心血管事件风险产生永久性副作用。

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