Pub Date : 2024-03-25DOI: 10.1088/1361-6420/ad2cf8
Phuoc-Truong Huynh, Konstantin Pieper, Daniel Walter
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.
{"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}
Pub Date : 2024-03-21DOI: 10.1088/1361-6420/ad366e
Laura Homa, Tyler Lesthaeghe, Matt Cherry, J. Wertz
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.
{"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}
Pub Date : 2024-03-20DOI: 10.1088/1361-6420/ad35e1
Jiao Xu, Peng Li, Bing Zheng
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
本文考虑了低阶矩阵恢复问题。我们提出了通过 $ell_1-alphaell_2 (0
{"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}
Pub Date : 2024-03-20DOI: 10.1088/1361-6420/ad35e2
Alberto Mercado
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.
{"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}
Pub Date : 2024-03-20DOI: 10.1088/1361-6420/ad2cfa
Kevin Bui, Zichao (Wendy) Di
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.
{"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}