Inverse Problems最新文献

{"title":"3D tomographic phase retrieval and unwrapping","authors":"Albert Fannjiang","doi":"10.1088/1361-6420/ad11a9","DOIUrl":"https://doi.org/10.1088/1361-6420/ad11a9","url":null,"abstract":"This paper develops uniqueness theory for 3D phase retrieval with finite, discrete measurement data for strong phase objects and weak phase objects, including: (i) <italic toggle=\"yes\">Unique determination of (phase) projections from diffraction patterns</italic>—General measurement schemes with coded and uncoded apertures are proposed and shown to ensure unique reduction of diffraction patterns to the phase projection for a strong phase object (respectively, the projection for a weak phase object) in each direction separately without the knowledge of relative orientations and locations. (ii) <italic toggle=\"yes\">Uniqueness for 3D phase unwrapping</italic>—General conditions for unique determination of a 3D strong phase object from its phase projection data are established, including, but not limited to, random tilt schemes densely sampled from a spherical triangle of vertexes in three orthogonal directions and other deterministic tilt schemes. (iii) <italic toggle=\"yes\">Uniqueness for projection tomography</italic>—Unique determination of an object of <italic toggle=\"yes\">n</italic>\u0000³ voxels from generic <italic toggle=\"yes\">n</italic> projections or <italic toggle=\"yes\">n</italic> + 1 coded diffraction patterns is proved. This approach of reducing 3D phase retrieval to the problem of (phase) projection tomography has the practical implication of enabling classification and alignment, when relative orientations are unknown, to be carried out in terms of (phase) projections, instead of diffraction patterns. The applications with the measurement schemes such as single-axis tilt, conical tilt, dual-axis tilt, random conical tilt and general random tilt are discussed.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"204 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138687947","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}

{"title":"Jointly determining the point sources and obstacle from Cauchy data","authors":"Deyue Zhang, Yan Chang, Yukun Guo","doi":"10.1088/1361-6420/ad10c8","DOIUrl":"https://doi.org/10.1088/1361-6420/ad10c8","url":null,"abstract":"A numerical method is developed for recovering both the source locations and the obstacle from the scattered Cauchy data of the time-harmonic acoustic field. First of all, the incident and scattered components are decomposed from the coupled Cauchy data by the representation of the single-layer potentials and the solution to the resulting linear integral system. As a consequence of this decomposition, the original problem of joint inversion is reformulated into two decoupled subproblems: an inverse source problem and an inverse obstacle scattering problem. Then, two sampling-type schemes are proposed to recover the shape of the obstacle and the source locations, respectively. The sampling methods rely on the specific indicator functions defined on target-oriented probing domains of circular shape. The error estimates of the decoupling procedure are established and the asymptotic behaviors of the indicator functions are analyzed. Extensive numerical experiments are also conducted to verify the performance of the sampling schemes.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"6 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688216","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}

{"title":"Stochastic linear regularization methods: random discrepancy principle and applications","authors":"Ye Zhang, Chuchu Chen","doi":"10.1088/1361-6420/ad149e","DOIUrl":"https://doi.org/10.1088/1361-6420/ad149e","url":null,"abstract":"\u0000 The a posteriori stopping rule plays a significant role in the design of efficient stochastic algorithms for various tasks in computational mathematics, such as inverse problems, optimization, and machine learning. Through the lens of classical regularization theory, this paper describes a novel analysis of Morozov’s discrepancy principle for the stochastic generalized Landweber iteration and its continuous analog of generalized stochastic asymptotical regularization. Unlike existing results relating to convergence in probability, we prove the strong convergence of the regularization error using tools from stochastic analysis, namely the theory of martingales. Numerical experiments are conducted to verify the convergence of the discrepancy principle and demonstrate two new capabilities of stochastic generalized Landweber iteration, which should also be valid for other stochastic/statistical approaches: improved accuracy by selecting the optimal path and the identification of multi-solutions by clustering samples of obtained approximate solutions.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"15 11","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139009579","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}

{"title":"Solution of the EEG inverse problem by random dipole sampling","authors":"Lorenzo Della Cioppa, Michela Tartaglione, Annalisa Pascarella, F. Pitolli","doi":"10.1088/1361-6420/ad14a1","DOIUrl":"https://doi.org/10.1088/1361-6420/ad14a1","url":null,"abstract":"\u0000 Electroencephalography (EEG) source imaging aims to reconstruct brain activity maps from the neuroelectric potential difference measured on the skull. To obtain the brain activity map, we need to solve an ill-posed and ill-conditioned inverse problem that requires regularization techniques to make the solution viable. When dealing with real-time applications, dimensionality reduction techniques can be used to reduce the computational load required to evaluate the numerical solution of the EEG inverse problem. To this end, in this paper we use the random dipole sampling method, in which a Monte Carlo technique is used to reduce the number of neural sources. This is equivalent to reducing the number of the unknowns in the inverse problem and can be seen as a first regularization step. Then, we solve the reduced EEG inverse problem with two popular inversion methods, the weighted Minimum Norm Estimate (wMNE) and the standardized LOw Resolution brain Electromagnetic TomogrAphy (sLORETA). The main result of this paper is the error estimates of the reconstructed activity map obtained with the randomized version of wMNE and sLORETA. Numerical experiments on synthetic EEG data demonstrate the effectiveness of the random dipole sampling method.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"80 2","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139008079","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}

{"title":"Structured model selection via ℓ1−ℓ2 optimization","authors":"Xiaofan Lu, Linan Zhang, Hongjin He","doi":"10.1088/1361-6420/ad0fad","DOIUrl":"https://doi.org/10.1088/1361-6420/ad0fad","url":null,"abstract":"Automated model selection is an important application in science and engineering. In this work, we develop a learning approach for identifying structured dynamical systems from undersampled and noisy spatiotemporal data. The learning is performed by a sparse least-squares fitting over a large set of candidate functions via a nonconvex <inline-formula>\u0000<tex-math><?CDATA $ell_1-ell_2$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msub><mml:mi>ℓ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>ℓ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>\u0000<inline-graphic xlink:href=\"ipad0fadieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> sparse optimization solved by the alternating direction method of multipliers. We show that if the set of candidate functions forms a structured random sampling matrix of a bounded orthogonal system, the recovery is stable and the error is bounded. The learning approach is validated on synthetic data generated by the viscous Burgers’ equation and two reaction–diffusion equations. The computational results demonstrate the theoretical guarantees of success and the efficiency with respect to the number of candidate functions.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"24 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688101","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}

{"title":"Optimal regularized hypothesis testing in statistical inverse problems","authors":"Remo Kretschmann, Daniel Wachsmuth, Frank Werner","doi":"10.1088/1361-6420/ad1132","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1132","url":null,"abstract":"Testing of hypotheses is a well studied topic in mathematical statistics. Recently, this issue has also been addressed in the context of inverse problems, where the quantity of interest is not directly accessible but only after the inversion of a (potentially) ill-posed operator. In this study, we propose a regularized approach to hypothesis testing in inverse problems in the sense that the underlying estimators (or test statistics) are allowed to be biased. Under mild source-condition type assumptions, we derive a family of tests with prescribed level <italic toggle=\"yes\">α</italic> and subsequently analyze how to choose the test with maximal power out of this family. As one major result we prove that regularized testing is always at least as good as (classical) unregularized testing. Furthermore, using tools from convex optimization, we provide an adaptive test by maximizing the power functional, which then outperforms previous unregularized tests in numerical simulations by several orders of magnitude.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"41 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688109","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}

{"title":"Double-parameter regularization for solving the backward diffusion problem with parallel-in-time algorithm","authors":"Jun-Liang Fu, Jijun Liu","doi":"10.1088/1361-6420/ad1131","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1131","url":null,"abstract":"We propose a double-parameter regularization scheme for dealing with the backward diffusion process. Considering the smoothing effect of Yosida approximation for PDE, we propose to regularize this ill-posed problem by modifying original governed system in terms of a pseudoparabolic equation together with a quasi-boundary condition simultaneously, which consequently contains two regularizing parameters. Theoretically, we establish the optimal error estimates between the regularizing solution and the exact one in terms of suitable choice strategy for the regularizing parameters, under <italic toggle=\"yes\">a-priori</italic> regularity assumptions on the exact solution. The <italic toggle=\"yes\">a-posteriori</italic> choice strategy for the regularizing parameters based on the discrepancy principle is also studied. To weaken the heavy computational cost for solving the discrete nonsymmetric linear regularizing system by finite difference scheme, especially in higher spatial dimensional cases, the block divide-and-conquer method together with the properties of the Schur complement is applied to decompose the linear system into two half-size linear systems, one of which can be solved by the diagonalization technique, and consequently an efficient parallel-in-time algorithm originally developed for direct problem is applicable. Our proposed method is of much lower complexity than the standard solver for the corresponding linear system. Finally, some numerical examples are presented to verify the efficiency of our proposed method.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"42 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688099","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}

{"title":"Distributed parameter identification for the Navier–Stokes equations for obstacle detection","authors":"Jorge Aguayo, Cristóbal Bertoglio, Axel Osses","doi":"10.1088/1361-6420/ad1133","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1133","url":null,"abstract":"We present a parameter identification problem for a scalar permeability field and the maximum velocity in an inflow, following a reference profile. We utilize a modified version of the Navier–Stokes equations, incorporating a permeability term described by the Brinkman’s Law into the momentum equation. This modification takes into account the presence of obstacles on some parts of the boundary. For the outflow, we implement a directional do-nothing condition as a means of stabilizing the backflow. This work extends our previous research published in (Aguayo <italic toggle=\"yes\">et al</italic> 2021 <italic toggle=\"yes\">Inverse Problems</italic>\u0000<bold>37</bold> 025010), where we considered a similar inverse problem for a linear Oseen model with do-nothing boundary conditions on the outlet and numerical simulations in 2D. Here we consider the more realistic case of Navier–Stokes equations with a backflow correction on the outflow and 3D simulations of the identification of a more realistic tricuspid cardiac valve. From a reference velocity that could have some noise or be obtained in low resolution, we define a suitable quadratic cost functional with some regularization terms. Existence of minimizers and first and second order optimality conditions are derived through the differentiability of the solutions of the Navier–Stokes equations with respect to the permeability and maximum velocity in the inflow. Finally, we present some synthetic numerical test based of recovering a 2D and 3D shape of a cardiac valve from total and local velocity measurements, inspired from 2D and 3D MRI.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"27 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688105","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}

{"title":"A Bayesian approach for CT reconstruction with defect detection for subsea pipelines","authors":"Silja L Christensen, N. A. B. Riis, Marcelo Pereyra, J. S. Jørgensen","doi":"10.1088/1361-6420/ad1348","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1348","url":null,"abstract":"\u0000 Subsea pipelines can be inspected via 2D cross-sectional X-ray computed tomography (CT). Traditional reconstruction methods produce an image of the pipe's interior that can be post-processed for detection of possible defects. In this paper we propose a novel Bayesian CT reconstruction method with built-in defect detection. We decompose the reconstruction into a sum of two images; one containing the overall pipe structure, and one containing defects, and infer the images simultaneously in a Gibbs scheme. Our method requires that prior information about the two images is very distinct, i.e. the first image should contain the large-scale and layered pipe structure, and the second image should contain small, coherent defects. We demonstrate our methodology with numerical experiments using synthetic and real CT data from scans of subsea pipes in cases with full and limited data. Experiments demonstrate the effectiveness of the proposed method in various data settings, with reconstruction quality comparable to existing techniques, while also providing defect detection with uncertainty quantification.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"19 9","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138592553","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}

{"title":"Inversion of generalized Radon transforms acting on 3D vector and symmetric tensor fields","authors":"Ivan E Svetov, Anna P Polyakova","doi":"10.1088/1361-6420/ad0fac","DOIUrl":"https://doi.org/10.1088/1361-6420/ad0fac","url":null,"abstract":"Currently, theory of the ray transforms of vector and tensor fields is well developed, but the generalized Radon transforms of such fields have not been fully studied. We consider the normal, longitudinal and mixed Radon transforms (with integration over planes) acting on three-dimensional vector and symmetric tensor fields. We prove that these operators are continuous. In case when values of all generalized Radon transforms are known, inversion formulas are derived for componentwise reconstruction of vector and symmetric <italic toggle=\"yes\">m</italic>-tensor fields, <inline-formula>\u0000<tex-math><?CDATA $municode{x2A7E}2$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mi>m</mml:mi><mml:mtext>⩾</mml:mtext><mml:mn>2</mml:mn></mml:math>\u0000<inline-graphic xlink:href=\"ipad0facieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>. Novel detailed decompositions of 3D vector and symmetric 2-tensor fields as a sum of pairwise orthogonal terms are obtained. For construction of each term in the sum only one function is required. With usage of these decompositions we have described the kernels and images of the generalized Radon transforms. In addition, we have obtained inversion formulas for each of the generalized Radon transforms acting on 3D vector and symmetric 2-tensor fields and have formulated theorems similar to the projection theorem for the Radon transform. For the cases <inline-formula>\u0000<tex-math><?CDATA $municode{x2A7E}3$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mi>m</mml:mi><mml:mtext>⩾</mml:mtext><mml:mn>3</mml:mn></mml:math>\u0000<inline-graphic xlink:href=\"ipad0facieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> similar statements are formulated as hypotheses. In addition, we consider the weighted longitudinal Radon transforms of 3D vector fields. Formulas are obtained for reconstructing the potential part of a 3D vector field from the known values of the longitudinal Radon transforms and one weighted Radon transform. Finally, we discuss the problem of vector fields reconstruction in <inline-formula>\u0000<tex-math><?CDATA $mathbb{R}^n$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math>\u0000<inline-graphic xlink:href=\"ipad0facieqn3.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>, <inline-formula>\u0000<tex-math><?CDATA $nunicode{x2A7E}4$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mi>n</mml:mi><mml:mtext>⩾</mml:mtext><mml:mn>4</mml:mn></mml:math>\u0000<inline-graphic xlink:href=\"ipad0facieqn4.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"21 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688110","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}