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Adaptive anisotropic Bayesian meshing for inverse problems 逆问题的自适应各向异性贝叶斯网格划分
IF 2.1 2区 数学 Q1 Mathematics Pub Date : 2024-02-23 DOI: 10.1088/1361-6420/ad2696
A Bocchinfuso, D Calvetti, E Somersalo
We consider inverse problems estimating distributed parameters from indirect noisy observations through discretization of continuum models described by partial differential or integral equations. It is well understood that errors arising from the discretization can be detrimental for ill-posed inverse problems, as discretization error behaves as correlated noise. While this problem can be avoided with a discretization fine enough to decrease the modeling error level below that of the exogenous noise that is addressed, e.g. by regularization, the computational resources needed to deal with the additional degrees of freedom may increase so much as to require high performance computing environments. Following an earlier idea, we advocate the notion of the discretization as one of the unknowns of the inverse problem, which is updated iteratively together with the solution. In this approach, the discretization, defined in terms of an underlying metric, is refined selectively only where the representation power of the current mesh is insufficient. In this paper we allow the metrics and meshes to be anisotropic, and we show that this leads to significant reduction of memory allocation and computing time.
我们考虑的逆问题是,通过对偏微分方程或积分方程描述的连续模型进行离散化,从间接噪声观测中估计分布参数。众所周知,离散化产生的误差会对问题不明确的逆问题产生不利影响,因为离散化误差表现为相关噪声。虽然可以通过精细离散化(例如正则化)来避免这一问题,从而将建模误差水平降至低于外生噪声的水平,但处理额外自由度所需的计算资源可能会大幅增加,以至于需要高性能计算环境。按照早先的想法,我们主张将离散化作为逆问题的未知数之一,与解一起迭代更新。在这种方法中,只有在当前网格的表示能力不足时,才会有选择性地细化离散化,而离散化是根据基本度量定义的。在本文中,我们允许度量和网格是各向异性的,并证明这将显著减少内存分配和计算时间。
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引用次数: 0
A Bayesian approach for consistent reconstruction of inclusions 用贝叶斯方法重建内含物的一致性
IF 2.1 2区 数学 Q1 Mathematics Pub Date : 2024-02-23 DOI: 10.1088/1361-6420/ad2531
B M Afkham, K Knudsen, A K Rasmussen, T Tarvainen
This paper considers a Bayesian approach for inclusion detection in nonlinear inverse problems using two known and popular push-forward prior distributions: the star-shaped and level set prior distributions. We analyze the convergence of the corresponding posterior distributions in a small measurement noise limit. The methodology is general; it works for priors arising from any Hölder continuous transformation of Gaussian random fields and is applicable to a range of inverse problems. The level set and star-shaped prior distributions are examples of push-forward priors under Hölder continuous transformations that take advantage of the structure of inclusion detection problems. We show that the corresponding posterior mean converges to the ground truth in a proper probabilistic sense. Numerical tests on a two-dimensional quantitative photoacoustic tomography problem showcase the approach. The results highlight the convergence properties of the posterior distributions and the ability of the methodology to detect inclusions with sufficiently regular boundaries.
本文研究了一种贝叶斯方法,该方法利用两种已知且流行的前推先验分布:星形先验分布和水平集先验分布,对非线性逆问题中的包含性进行检测。我们分析了相应后验分布在小测量噪声极限下的收敛性。该方法是通用的;它适用于高斯随机场的任何赫尔德连续变换所产生的先验,并适用于一系列逆问题。水平集和星形先验分布是霍尔德连续变换下的前推先验的例子,它们利用了包含检测问题的结构。我们证明,相应的后验均值在适当的概率意义上收敛于地面实况。一个二维定量光声层析成像问题的数值测试展示了这种方法。结果凸显了后验分布的收敛特性,以及该方法检测具有足够规则边界的夹杂物的能力。
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引用次数: 0
Magnetic characterisation of steel strips using transient field measurements: global sensitivity analysis and regression from a machine-learning perspective 利用瞬态场测量对钢带进行磁性表征:从机器学习角度进行全局敏感性分析和回归
IF 2.1 2区 数学 Q1 Mathematics Pub Date : 2024-02-16 DOI: 10.1088/1361-6420/ad2a04
A. Skarlatos, R. Miorelli, C. Reboud, Frenk van den Berg
In this contribution, the magnetic characterisation of steel strips is studied using synthetic data of field-gradient transients, which have been produced via the finite integration technique (FIT). The material law is described and parametrized using the Jiles-Atherton (JA) model. The sensitivity of relevant magnetic indicators with respect to the material parameters is then analyzed using two global methods: Sobol indices and $delta$-sensitivity indices. In order to accelerate the evaluation of these quantities, a fast metamodel is built using machine learning techniques from a simulated dataset. The solution of the inverse problem based on a tailored learning framework is tested for the different proposed identifiers, and their suitability for the magnetic characterisation of the material in question is finally discussed.
本文使用有限积分技术 (FIT) 生成的场梯度瞬态合成数据研究了钢带的磁特性。使用 Jiles-Atherton (JA) 模型对材料定律进行了描述和参数化。然后使用两种全局方法分析了相关磁性指标对材料参数的敏感性:Sobol 指数和 $delta$ 敏感性指数。为了加速这些量的评估,利用机器学习技术从模拟数据集中建立了一个快速元模型。基于定制学习框架的逆问题解决方案针对不同的拟议标识符进行了测试,最后讨论了它们是否适合相关材料的磁性特征描述。
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引用次数: 0
A posterior contraction for Bayesian inverse problems in Banach spaces 巴拿赫空间中贝叶斯逆问题的后验收缩
IF 2.1 2区 数学 Q1 Mathematics Pub Date : 2024-02-16 DOI: 10.1088/1361-6420/ad2a03
De-Han Chen, Jingzhi Li, Ye Zhang
This paper features a study of statistical inference for linear inverse problems with Gaussian noise and priors in structured Banach spaces. Employing the tools of sectorial operators and Gaussian measures on Banach spaces, we overcome the theoretical difficulty of lacking the bias-variance decomposition in Banach spaces, characterize the posterior distribution of solution though its Radon-Nikodym derivative, and derive the optimal convergence rates of the corresponding square posterior contraction and the mean integrated square error. Our theoretical findings are applied to two scenarios, specifically a Volterra integral equation and an inverse source problem governed by an elliptic partial differential equation. Our investigation demonstrates the superiority of our approach over classical results. Notably, our method achieves same order of convergence rates for solutions with reduced smoothness even in a Hilbert setting.
本文主要研究结构化巴拿赫空间中具有高斯噪声和先验的线性逆问题的统计推断。利用巴拿赫空间上的扇形算子和高斯度量工具,我们克服了巴拿赫空间中缺乏偏差-方差分解的理论困难,通过其 Radon-Nikodym 导数表征了解的后验分布,并推导出相应平方后验收缩和平均积分平方误差的最优收敛率。我们的理论发现被应用于两种情况,特别是 Volterra 积分方程和由椭圆偏微分方程控制的反源问题。我们的研究表明,我们的方法优于经典结果。值得注意的是,即使在希尔伯特环境下,我们的方法也能为平滑度降低的解实现相同数量级的收敛率。
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引用次数: 0
Imaging of nonlinear materials via the Monotonicity Principle 通过单调性原理成像非线性材料
IF 2.1 2区 数学 Q1 Mathematics Pub Date : 2024-02-13 DOI: 10.1088/1361-6420/ad22e9
Vincenzo Mottola, Antonio Corbo Esposito, Gianpaolo Piscitelli, Antonello Tamburrino
Inverse problems, which are related to Maxwell’s equations, in the presence of nonlinear materials is a quite new topic in the literature. The lack of contributions in this area can be ascribed to the significant challenges that such problems pose. Retrieving the spatial behavior of some unknown physical property, from boundary measurements, is a nonlinear and highly ill-posed problem even in the presence of linear materials. Furthermore, this complexity grows exponentially in the presence of nonlinear materials. In the tomography of linear materials, the Monotonicity Principle (MP) is the foundation of a class of non-iterative algorithms able to guarantee excellent performances and compatibility with real-time applications. Recently, the MP has been extended to nonlinear materials under very general assumptions. Starting from the theoretical background for this extension, we develop a first real-time inversion method for the inverse obstacle problem in the presence of nonlinear materials. The proposed method is intendend for all problems governed by the quasilinear Laplace equation, i.e. static problems involving nonlinear materials. In this paper, we provide some preliminary results which give the foundation of our method and some extended numerical examples.
在非线性材料存在的情况下,与麦克斯韦方程相关的逆问题在文献中是一个相当新的课题。这一领域的研究成果不多,可能是因为这类问题带来了巨大的挑战。即使在线性材料存在的情况下,从边界测量中检索某些未知物理特性的空间行为,也是一个非线性和高难度问题。此外,在存在非线性材料的情况下,这种复杂性还会呈指数级增长。在线性材料层析成像中,单调性原理(MP)是一类非迭代算法的基础,能够保证卓越的性能和与实时应用的兼容性。最近,MP 在非常一般的假设条件下扩展到了非线性材料。从这一扩展的理论背景出发,我们开发了第一种非线性材料存在时的反障碍问题实时反演方法。所提出的方法适用于所有受准线性拉普拉斯方程控制的问题,即涉及非线性材料的静态问题。本文提供了一些初步结果,为我们的方法奠定了基础,并提供了一些扩展的数值示例。
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引用次数: 0
Multilevel dimension-independent likelihood-informed MCMC for large-scale inverse problems 用于大规模逆问题的多层次维度独立似然信息 MCMC
IF 2.1 2区 数学 Q1 Mathematics Pub Date : 2024-02-05 DOI: 10.1088/1361-6420/ad1e2c
Tiangang Cui, Gianluca Detommaso, Robert Scheichl
We present a non-trivial integration of dimension-independent likelihood-informed (DILI) MCMC (Cui et al 2016) and the multilevel MCMC (Dodwell et al 2015) to explore the hierarchy of posterior distributions. This integration offers several advantages: First, DILI-MCMC employs an intrinsic likelihood-informed subspace (LIS) (Cui et al 2014)—which involves a number of forward and adjoint model simulations—to design accelerated operator-weighted proposals. By exploiting the multilevel structure of the discretised parameters and discretised forward models, we design a Rayleigh–Ritz procedure to significantly reduce the computational effort in building the LIS and operating with DILI proposals. Second, the resulting DILI-MCMC can drastically improve the sampling efficiency of MCMC at each level, and hence reduce the integration error of the multilevel algorithm for fixed CPU time. Numerical results confirm the improved computational efficiency of the multilevel DILI approach.
我们提出了一种与维度无关的似然信息(DILI)MCMC(Cui 等人,2016 年)和多级 MCMC(Dodwell 等人,2015 年)的非难整合,以探索后验分布的层次结构。这种整合具有几个优势:首先,DILI-MCMC 采用内在似然信息子空间(LIS)(Cui 等人,2014 年)--其中涉及大量前向和邻接模型模拟--来设计加速算子加权建议。通过利用离散参数和离散前向模型的多层次结构,我们设计了一种 Rayleigh-Ritz 程序,以显著减少构建 LIS 和使用 DILI 建议的计算量。其次,由此产生的 DILI-MCMC 可以大幅提高各层次 MCMC 的采样效率,从而在 CPU 时间固定的情况下降低多层次算法的积分误差。数值结果证实了多级 DILI 方法提高了计算效率。
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引用次数: 0
Fourier series-based approximation of time-varying parameters in ordinary differential equations 基于傅里叶级数的常微分方程时变参数近似法
IF 2.1 2区 数学 Q1 Mathematics Pub Date : 2024-02-01 DOI: 10.1088/1361-6420/ad1fe5
Anna Fitzpatrick, Molly Folino, Andrea Arnold
Many real-world systems modeled using differential equations involve unknown or uncertain parameters. Standard approaches to address parameter estimation inverse problems in this setting typically focus on estimating constants; yet some unobservable system parameters may vary with time without known evolution models. In this work, we propose a novel approximation method inspired by the Fourier series to estimate time-varying parameters (TVPs) in deterministic dynamical systems modeled with ordinary differential equations. Using ensemble Kalman filtering in conjunction with Fourier series-based approximation models, we detail two possible implementation schemes for sequentially updating the time-varying parameter estimates given noisy observations of the system states. We demonstrate the capabilities of the proposed approach in estimating periodic parameters, both when the period is known and unknown, as well as non-periodic TVPs of different forms with several computed examples using a forced harmonic oscillator. Results emphasize the importance of the frequencies and number of approximation model terms on the time-varying parameter estimates and corresponding dynamical system predictions.
现实世界中许多使用微分方程建模的系统都涉及未知或不确定参数。在这种情况下,解决参数估计逆问题的标准方法通常侧重于估计常数;然而,在没有已知演化模型的情况下,一些不可观测的系统参数可能会随时间变化。在这项工作中,我们受傅立叶级数的启发,提出了一种新的近似方法,用于估计以常微分方程建模的确定性动态系统中的时变参数(TVPs)。我们将集合卡尔曼滤波与基于傅立叶级数的近似模型结合使用,详细介绍了两种可能的实施方案,用于在对系统状态进行噪声观测的情况下顺序更新时变参数估计。我们通过几个使用受迫谐波振荡器的计算实例,展示了所提方法在估计周期参数(包括已知和未知周期)以及不同形式的非周期性 TVP 方面的能力。结果强调了近似模型项的频率和数量对时变参数估计和相应动力系统预测的重要性。
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引用次数: 0
V-line 2-tensor tomography in the plane 平面 V 线 2 张量断层扫描
IF 2.1 2区 数学 Q1 Mathematics Pub Date : 2024-01-31 DOI: 10.1088/1361-6420/ad1f83
Gaik Ambartsoumian, Rohit Kumar Mishra, Indrani Zamindar
In this article, we introduce and study various V-line transforms (VLTs) defined on symmetric 2-tensor fields in R2. The operators of interest include the longitudinal, transverse, and mixed VLTs, their integral moments, and the star transform. With the exception of the star transform, all these operators are natural generalizations to the broken-ray trajectories of the corresponding well-studied concepts defined for straight-line paths of integration. We characterize the kernels of the VLTs and derive exact formulas for reconstruction of tensor fields from various combinations of these transforms. The star transform on tensor fields is an extension of the corresponding concepts that have been previously studied on vector fields and scalar fields (functions). We describe all injective configurations of the star transform on symmetric 2-tensor fields and derive an exact, closed-form inversion formula for that operator.
本文介绍并研究了定义在 R2 中对称 2 张量场上的各种 V 线变换(VLT)。我们感兴趣的算子包括纵向、横向和混合 VLT、它们的积分矩以及星变换。除星形变换外,所有这些算子都是对破碎射线轨迹的自然概括,而破碎射线轨迹是为直线积分路径定义的相应概念。我们描述了 VLT 的内核特征,并推导出从这些变换的各种组合中重建张量场的精确公式。张量场的星形变换是之前研究过的向量场和标量场(函数)相应概念的扩展。我们描述了对称 2 张量场上星形变换的所有注入配置,并推导出该算子的精确闭式反演公式。
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引用次数: 0
Regularization of the inverse Laplace transform by mollification 反拉普拉斯变换的规范化摩尔化
IF 2.1 2区 数学 Q1 Mathematics Pub Date : 2024-01-16 DOI: 10.1088/1361-6420/ad1609
Pierre Maréchal, Faouzi Triki, Walter C Simo Tao Lee
In this paper we study the inverse Laplace transform. We first derive a new global logarithmic stability estimate that shows that the inversion is severely ill-posed. Then we propose a regularization method to compute the inverse Laplace transform using the concept of mollification. Taking into account the exponential instability we derive a criterion for selection of the regularization parameter. We show that by taking the optimal value of this parameter we improve significantly the convergence of the method. Finally, making use of the holomorphic extension of the Laplace transform, we suggest a new PDEs based numerical method for the computation of the solution. The effectiveness of the proposed regularization method is demonstrated through several numerical examples.
本文研究反拉普拉斯变换。我们首先推导出一个新的全局对数稳定性估计值,它表明反演是一个严重的问题。然后,我们提出了一种正则化方法,利用 "钝化 "概念计算反拉普拉斯变换。考虑到指数不稳定性,我们得出了正则化参数的选择标准。我们证明,通过取该参数的最优值,可以显著改善该方法的收敛性。最后,利用拉普拉斯变换的全态扩展,我们提出了一种新的基于 PDEs 的数值方法来计算解。我们通过几个数值示例证明了所提出的正则化方法的有效性。
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引用次数: 0
Assessing the potential of using a virtual Veselago lens in quantitative microwave imaging 评估在微波定量成像中使用虚拟 Veselago 透镜的潜力
IF 2.1 2区 数学 Q1 Mathematics Pub Date : 2024-01-12 DOI: 10.1088/1361-6420/ad1e2d
Marzieh Eini Keleshteri, Vladimir Okhmatovski, Ian Jeffrey, M. Bevacqua, J. Lovetri
This study explores the potential of implementing the focusing properties of a virtual ideal Veselago lens within a standard free-space microwave imaging scenario. To achieve this, the virtual lens is introduced as an inhomogeneous numerical background for the inverse source problem. This numerical Vesealgo lens is incorporated into the incident and scattered field decomposition, resulting in a new data equation that involves the Veselago Lens Green's function. In addition to the contrast sources within the object-of-interest, the lens introduces virtual contrast sources along the lens boundaries that depend on the total tangential magnetic field. It is shown that a surface integral contribution that takes into account these surface contrast sources must be added to the collected free-space data before one can invert using the well-conditioned Veselago lens inversion operator. A preliminary investigation of the accuracy to which this surface integral contribution must be computed is performed using additive Gaussian noise. Results show that an error of less than one percent is required to achieve imaging performance similar to utilizing an actual Veselago lens. All results are performed within a 2D simulation environment.
本研究探讨了在标准自由空间微波成像场景中实现虚拟理想维塞拉哥透镜聚焦特性的潜力。为此,虚拟透镜被引入作为反源问题的非均质数值背景。这种数值维塞拉哥透镜被纳入入射场和散射场分解,从而产生一个涉及维塞拉哥透镜格林函数的新数据方程。除了感兴趣物体内部的对比源之外,透镜还沿着透镜边界引入了取决于总切向磁场的虚拟对比源。研究表明,在使用条件良好的 Veselago 透镜反演算子进行反演之前,必须将考虑到这些表面对比源的表面积分贡献添加到收集到的自由空间数据中。利用加性高斯噪声对计算表面积分贡献的精度进行了初步研究。结果表明,要达到与使用实际 Veselago 透镜类似的成像性能,误差必须小于百分之一。所有结果均在二维模拟环境中进行。
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引用次数: 0
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Inverse Problems
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