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Towards optimal sensor placement for inverse problems in spaces of measures 在度量空间中实现逆问题的最佳传感器布局
IF 2.1 2区 数学
Inverse Problems Pub Date : 2024-03-25 DOI: 10.1088/1361-6420/ad2cf8
Phuoc-Truong Huynh, Konstantin Pieper, Daniel Walter
{"title":"Towards optimal sensor placement for inverse problems in spaces of measures","authors":"Phuoc-Truong Huynh, Konstantin Pieper, Daniel Walter","doi":"10.1088/1361-6420/ad2cf8","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2cf8","url":null,"abstract":"The objective of this work is to quantify the reconstruction error in sparse inverse problems with measures and stochastic noise, motivated by optimal sensor placement. To be useful in this context, the error quantities must be explicit in the sensor configuration and robust with respect to the source, yet relatively easy to compute in practice, compared to a direct evaluation of the error by a large number of samples. In particular, we consider the identification of a measure consisting of an unknown linear combination of point sources from a finite number of measurements contaminated by Gaussian noise. The statistical framework for recovery relies on two main ingredients: first, a convex but non-smooth variational Tikhonov point estimator over the space of Radon measures and, second, a suitable mean-squared error based on its Hellinger–Kantorovich distance to the ground truth. To quantify the error, we employ a non-degenerate source condition as well as careful linearization arguments to derive a computable upper bound. This leads to asymptotically sharp error estimates in expectation that are explicit in the sensor configuration. Thus they can be used to estimate the expected reconstruction error for a given sensor configuration and guide the placement of sensors in sparse inverse problems.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140316622","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}
引用次数: 0
Microtexture region segmentation of eddy current testing data using a structural prior 利用结构先验对涡流测试数据进行微纹理区域分割
IF 2.1 2区 数学
Inverse Problems Pub Date : 2024-03-21 DOI: 10.1088/1361-6420/ad366e
Laura Homa, Tyler Lesthaeghe, Matt Cherry, J. Wertz
{"title":"Microtexture region segmentation of eddy current testing data using a structural prior","authors":"Laura Homa, Tyler Lesthaeghe, Matt Cherry, J. Wertz","doi":"10.1088/1361-6420/ad366e","DOIUrl":"https://doi.org/10.1088/1361-6420/ad366e","url":null,"abstract":"\u0000 Microtexture regions (MTR) are collections of grains with similar crystallographic orientation. Because their presence in titanium alloys can significantly impact aerospace component life, a nondestructive method to detect and characterize MTR is needed. In this work, we propose to use data from two nondestructive evaluation methods, eddy current testing (ECT) and scanning acoustic microscopy (SAM), in order to recover the boundary and dominant crystallographic orientation of each MTR in a specimen. Eddy current testing is an electromagnetic method that is sensitive to changes in crystallographic orientation associated with MTR; however, its low resolution prevents it from resolving MTR boundaries well. In contrast, scanning acoustic microscopy is a high frequency ultrasound method that is able to resolve MTR boundaries but is not sensitive to orientation. This paper proposes an algorithm to characterize MTR that makes use of a method known as covariance generalized matching component analysis. This method is used to build a surrogate linear forward model that relates MTR boundaries and orientation to ECT data. The model is inverted using the SAM data as a structural prior. We demonstrate this technique using simulated ECT and experimental SAM data from a large grain titanium specimen.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140223291","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}
引用次数: 0
Matrix recovery from nonconvex regularized least absolute deviations 从非凸正则化最小绝对偏差中恢复矩阵
IF 2.1 2区 数学
Inverse Problems Pub Date : 2024-03-20 DOI: 10.1088/1361-6420/ad35e1
Jiao Xu, Peng Li, Bing Zheng
{"title":"Matrix recovery from nonconvex regularized least absolute deviations","authors":"Jiao Xu, Peng Li, Bing Zheng","doi":"10.1088/1361-6420/ad35e1","DOIUrl":"https://doi.org/10.1088/1361-6420/ad35e1","url":null,"abstract":"\u0000 In this paper, we consider the low-rank matrix recovery problem. We propose the nonconvex regularized least absolute deviations model via $ell_1-alphaell_2 (0<alpha<1)$ minimization. We establish the theoretical analysis of the proposed model and obtain a stable error estimation. Our result is a nontrivial extension of some previous work. Different from most of the state-of-the-art methods, our method does not need any knowledge of standard deviation or any moment assumption of the noise. Numerical experiments show that our method is effective for many types of noise distributions.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140227493","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}
引用次数: 0
An inverse problem for a transmission wave equation with a flat interface in ℝn 在ℝn中具有飞行界面的透射波方程的逆问题
IF 2.1 2区 数学
Inverse Problems Pub Date : 2024-03-20 DOI: 10.1088/1361-6420/ad35e2
Alberto Mercado
{"title":"An inverse problem for a transmission wave equation with a flat interface in ℝn","authors":"Alberto Mercado","doi":"10.1088/1361-6420/ad35e2","DOIUrl":"https://doi.org/10.1088/1361-6420/ad35e2","url":null,"abstract":"\u0000 In this paper we study a wave equation with discontinuous principal coefficient within a bounded domain of arbitrary dimension. It is obtained the stability of the inverse problem of recovering a space-dependent coefficient by observing a trace of the corresponding solution on part of the boundary. We provide a precise estimate of the minimum required time, as a function of the velocity change and domain size. The main tools are new global Carleman estimates for the transmission system with a particular weight function adapted to the interface geometry, which allows to obtain an optimal estimate of the minimum time. Keywords: Carleman inequalities, Bukhgeim–Klibanov method, transmission system.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140228022","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}
引用次数: 0
A stochastic ADMM algorithm for large-scale ptychography with weighted difference of anisotropic and isotropic total variation 利用各向异性和各向同性总变化的加权差值进行大规模分层成像的随机 ADMM 算法
IF 2.1 2区 数学
Inverse Problems Pub Date : 2024-03-20 DOI: 10.1088/1361-6420/ad2cfa
Kevin Bui, Zichao (Wendy) Di
{"title":"A stochastic ADMM algorithm for large-scale ptychography with weighted difference of anisotropic and isotropic total variation","authors":"Kevin Bui, Zichao (Wendy) Di","doi":"10.1088/1361-6420/ad2cfa","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2cfa","url":null,"abstract":"Ptychography, a prevalent imaging technique in fields such as biology and optics, poses substantial challenges in its reconstruction process, characterized by nonconvexity and large-scale requirements. This paper presents a novel approach by introducing a class of variational models that incorporate the weighted difference of anisotropic–isotropic total variation. This formulation enables the handling of measurements corrupted by Gaussian or Poisson noise, effectively addressing the nonconvex challenge. To tackle the large-scale nature of the problem, we propose an efficient stochastic alternating direction method of multipliers, which guarantees convergence under mild conditions. Numerical experiments validate the superiority of our approach by demonstrating its capability to successfully reconstruct complex-valued images, especially in recovering the phase components even in the presence of highly corrupted measurements.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315608","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}
引用次数: 0
Stable determination of an impedance obstacle by a single far-field measurement 通过单次远场测量稳定确定阻抗障碍物
IF 2.1 2区 数学
Inverse Problems Pub Date : 2024-03-19 DOI: 10.1088/1361-6420/ad3087
Huaian Diao, Hongyu Liu, Longyue Tao
{"title":"Stable determination of an impedance obstacle by a single far-field measurement","authors":"Huaian Diao, Hongyu Liu, Longyue Tao","doi":"10.1088/1361-6420/ad3087","DOIUrl":"https://doi.org/10.1088/1361-6420/ad3087","url":null,"abstract":"We establish sharp stability estimates of logarithmic type in determining an impedance obstacle in <inline-formula>\u0000<tex-math><?CDATA $mathbb{R}^2$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math>\u0000<inline-graphic xlink:href=\"ipad3087ieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>. The obstacle is the polygonal shape and the surface impedance parameter is non-zero constant. We establish the stability results using a single far-field pattern, constituting a longstanding problem in the inverse scattering theory. This is the first stability result in the literature in determining an impedance obstacle by a single far-field measurement. The stability in simultaneously determining the obstacle and the boundary impedance is established in terms of the classical Hausdorff distance. Several technical novelties and developments in the mathematical strategy developed for establishing the aforementioned stability results exist. First, the stability analysis is conducted around a corner point in a micro-local manner. Second, our stability estimates establish explicit relationships between the obstacle’s geometric configurations and the wave field’s vanishing order at the corner point. Third, we develop novel error propagation techniques to tackle singularities of the wave field at a corner with the impedance boundary condition.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315443","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}
引用次数: 0
A single level set function approach for multiple material-phases applied to full-waveform inversion in the time domain 应用于时域全波形反演的多材料相单一水平集函数方法
IF 2.1 2区 数学
Inverse Problems Pub Date : 2024-03-19 DOI: 10.1088/1361-6420/ad2eca
P B de Castro, E C N Silva, E A Fancello
{"title":"A single level set function approach for multiple material-phases applied to full-waveform inversion in the time domain","authors":"P B de Castro, E C N Silva, E A Fancello","doi":"10.1088/1361-6420/ad2eca","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2eca","url":null,"abstract":"This paper presents a multiple material-phase level-set approach for acoustic full-waveform inversion in the time domain. By using a single level set (LS) function, several level values are used to define virtual boundaries between material phases with different (and known) wave propagation velocities. The aim of the proposed approach is to provide a suitable framework to identify multiple/nested inclusions or a finite number of almost homogeneous sedimentary layers with sharp interfaces between them. The use of a single LS function provides a significant reduction in the number of variables to be identified, when compared with the usual multi-material phase approaches defined by multiple functions, especially for problems with a high number of degrees of freedom. Numerical experiments show satisfactory results in identifying simultaneously different interfaces. Cases with and without inverse crime are evaluated, showing that the approach is reasonably robust in dealing with such a condition.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315582","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}
引用次数: 0
Local data inverse problem for the polyharmonic operator with anisotropic perturbations 具有各向异性扰动的多谐算子的局部数据逆问题
IF 2.1 2区 数学
Inverse Problems Pub Date : 2024-03-19 DOI: 10.1088/1361-6420/ad3164
Sombuddha Bhattacharyya, Pranav Kumar
{"title":"Local data inverse problem for the polyharmonic operator with anisotropic perturbations","authors":"Sombuddha Bhattacharyya, Pranav Kumar","doi":"10.1088/1361-6420/ad3164","DOIUrl":"https://doi.org/10.1088/1361-6420/ad3164","url":null,"abstract":"In this article, we study an inverse problem with local data for a linear polyharmonic operator with several lower order tensorial perturbations. We consider our domain to have an inaccessible portion of the boundary where neither the input can be prescribed nor the output can be measured. We prove the unique determination of all the tensorial coefficients of the operator from the knowledge of the Dirichlet and Neumann map on the accessible part of the boundary, under suitable geometric assumptions on the domain.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315590","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}
引用次数: 0
Reconstruction of degenerate conductivity region for parabolic equations 抛物方程退化传导区域的重构
IF 2.1 2区 数学
Inverse Problems Pub Date : 2024-03-15 DOI: 10.1088/1361-6420/ad308a
Piermarco Cannarsa, Anna Doubova, Masahiro Yamamoto
{"title":"Reconstruction of degenerate conductivity region for parabolic equations","authors":"Piermarco Cannarsa, Anna Doubova, Masahiro Yamamoto","doi":"10.1088/1361-6420/ad308a","DOIUrl":"https://doi.org/10.1088/1361-6420/ad308a","url":null,"abstract":"We consider an inverse problem of reconstructing a degeneracy point in the diffusion coefficient in a one-dimensional parabolic equation by measuring the normal derivative on one side of the domain boundary. We analyze the sensitivity of the inverse problem to the initial data. We give sufficient conditions on the initial data for uniqueness and stability for the one-point measurement and show some examples of positive and negative results. On the other hand, we present more general uniqueness results, also for the identification of an initial data by measurements distributed over time. The proofs are based on an explicit form of the solution by means of Bessel functions of the first type. Finally, the theoretical results are supported by numerical experiments.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315442","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}
引用次数: 0
Stability estimate for an inverse stochastic parabolic problem of determining unknown time-varying boundary * 确定未知时变边界的反随机抛物线问题的稳定性估计 *
IF 2.1 2区 数学
Inverse Problems Pub Date : 2024-03-15 DOI: 10.1088/1361-6420/ad2d72
Zhonghua Liao, Qi Lü
{"title":"Stability estimate for an inverse stochastic parabolic problem of determining unknown time-varying boundary *","authors":"Zhonghua Liao, Qi Lü","doi":"10.1088/1361-6420/ad2d72","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2d72","url":null,"abstract":"Stochastic parabolic equations are widely used to model many random phenomena in natural sciences, such as the temperature distribution in a noisy medium, the dynamics of a chemical reaction in a noisy environment, or the evolution of the density of bacteria population. In many cases, the equation may involve an unknown moving boundary which could represent a change of phase, a reaction front, or an unknown population. In this paper, we focus on an inverse problem with the goal is to determine an unknown moving boundary based on data observed in a specific interior subdomain for the stochastic parabolic equation. The uniqueness of the solution of this problem is proved, and furthermore a stability estimate of log type is derived. This allows us, theoretically, to track and to monitor the behavior of the unknown boundary from observation in an arbitrary interior domain. The primary tool is a new Carleman estimate for stochastic parabolic equations. As a byproduct, we obtain a quantitative unique continuation property for stochastic parabolic equations.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315579","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}
引用次数: 0
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