Inverse Problems最新文献

{"title":"An accelerated inexact Newton regularization scheme with a learned feature-selection rule for non-linear inverse problems","authors":"Haie Long, Ye Zhang and Guangyu Gao","doi":"10.1088/1361-6420/ad5e19","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5e19","url":null,"abstract":"With computational inverse problems, it is desirable to develop an efficient inversion algorithm to find a solution from measurement data through a mathematical model connecting the unknown solution and measurable quantity based on the first principles. However, most of mathematical models represent only a few aspects of the physical quantity of interest, and some of them are even incomplete in the sense that one measurement corresponds to many solutions satisfying the forward model. In this paper, in light of the recently developed iNETT method in (2023 Inverse Problems39 055002), we propose a novel iterative regularization method for efficiently solving non-linear ill-posed inverse problems with potentially non-injective forward mappings and (locally) non-stable inversion mappings. Our approach integrates the inexact Newton iteration, the non-stationary iterated Tikhonov regularization, the two-point gradient acceleration method, and the structure-free feature-selection rule. The main difficulty in the regularization technique is how to design an appropriate regularization penalty, capturing the key feature of the unknown solution. To overcome this difficulty, we replace the traditional regularization penalty with a deep neural network, which is structure-free and can identify the correct solution in a huge null space. A comprehensive convergence analysis of the proposed algorithm is performed under standard assumptions of regularization theory. Numerical experiments with comparisons with other state-of-the-art methods for two model problems are presented to show the efficiency of the proposed approach.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"21 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141588260","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":"Low-resolution prior equilibrium network for CT reconstruction","authors":"Yijie Yang, Qifeng Gao and Yuping Duan","doi":"10.1088/1361-6420/ad5d0d","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5d0d","url":null,"abstract":"The unrolling method has been investigated for learning variational models in x-ray computed tomography. However, for incomplete data reconstruction, such as sparse-view and limited-angle problems, the unrolling method of gradient descent of the energy minimization problem cannot yield satisfactory results. In this paper, we present an effective CT reconstruction model, where the low-resolution image is introduced as a regularization for incomplete data problems. In what follows, we utilize the deep equilibrium approach to unfolding of the gradient descent algorithm, thereby constructing the backbone network architecture for solving the minimization model. We theoretically discuss the convergence of the proposed low-resolution prior equilibrium (LRPE) model and provide the necessary conditions to guarantee its convergence. Experimental results on both sparse-view and limited-angle reconstruction problems are provided, demonstrating that our end-to-end LRPE model outperforms other state-of-the-art methods in terms of noise reduction, contrast-to-noise ratio, and preservation of edge details.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"141 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141567215","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 unified approach to inversion formulae for vector and tensor ray and radon transforms and the Natterer inequality","authors":"Alfred K Louis","doi":"10.1088/1361-6420/ad5d0e","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5d0e","url":null,"abstract":"Most derivations of inversion formulae for x-ray or Radon transform are based on the projection theorem, where for fixed direction the Fourier transform of x-ray or Radon transform is calculated and compared with the Fourier transform of the searched-for function. In contrast to this we start here off from the searched-for field, calculate its Fourier transform for fixed direction, which is now a vector or tensor field, that we then expand in a suitable direction dependent basis. The expansion coefficients are recognized as the Fourier transform of longitudinal, transversal or mixed ray transforms or vectorial Radon transform respectively. The inverse Fourier transform of the searched-for field then directly leads to inversion formulae for those transforms applying problem adapted backprojections. When considering the Helmholtz decomposition of the field we immediately find inversion formulae for those transversal or longitudinal transforms. First inversion formulae for the longitudinal ray transform, similar to those given by Natterer (1986 <italic toggle=\"yes\">The Mathematics of Computerized Tomography</italic> (Teubner and Wiley)) for x-ray tomography, were given by Natterer-Wübbeling in 2001, Natterer and Wübbeling (2001 <italic toggle=\"yes\">Mathematical Methods in Image Reconstruction</italic> (SIAM)), but then not pursued by other authors. In this paper, we present the above described method and derive in a unified way inversion formulae for the ray transforms treated in Louis (2022 <italic toggle=\"yes\">Inverse Problems</italic>\u0000<bold>38</bold> 065008) containing the results from Louis (2022 <italic toggle=\"yes\">Inverse Problems</italic>\u0000<bold>38</bold> 065008) as special cases. Additionally we present new inversion formulae for the vectorial Radon transform. As a consequence the inversion formulae directly give Plancherel’s formulae for the vectorial or tensorial transforms. Together with the Natterer inequality, which is independent of the ray or Radon transforms, we present the Natterer stability of those vectorial and tensorial transforms.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"236 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141567220","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":"Ill-posedness of time-dependent inverse problems in Lebesgue-Bochner spaces","authors":"Martin Burger, Thomas Schuster, Anne Wald","doi":"10.1088/1361-6420/ad5a35","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5a35","url":null,"abstract":"We consider time-dependent inverse problems in a mathematical setting using Lebesgue-Bochner spaces. Such problems arise when one aims to recover parameters from given observations where the parameters or the data depend on time. There are various important applications being subject of current research that belong to this class of problems. Typically inverse problems are ill-posed in the sense that already small noise in the data causes tremendous errors in the solution. In this article we present two different concepts of ill-posedness: temporally (pointwise) ill-posedness and uniform ill-posedness with respect to the Lebesgue-Bochner setting. We investigate the two concepts by means of a typical setting consisting of a time-depending observation operator composed by a compact operator. Furthermore we develop regularization methods that are adapted to the respective class of ill-posedness.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"49 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141567213","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":"Resolving full-wave through-wall transmission effects in multi-static synthetic aperture radar","authors":"F M Watson, D Andre, W R B Lionheart","doi":"10.1088/1361-6420/ad5b83","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5b83","url":null,"abstract":"Through-wall synthetic aperture radar (SAR) imaging is of significant interest for security purposes, in particular when using multi-static SAR systems consisting of multiple distributed radar transmitters and receivers to improve resolution and the ability to recognise objects. Yet there is a significant challenge in forming focused, useful images due to multiple scattering effects through walls, whereas standard SAR imaging has an inherent single scattering assumption. This may be exacerbated with multi-static collections, since different scattering events will be observed from each angle and the data may not coherently combine well in a naive manner. To overcome this, we propose an image formation method which resolves full-wave effects through an approximately known wall or other arbitrary obstacle, which itself has some unknown ‘nuisance’ parameters that are determined as part of the reconstruction to provide well focused images. The method is more flexible and realistic than existing methods which treat a single wall as a flat layered medium, whilst being significantly computationally cheaper than full-wave methods, strongly motivated by practical considerations for through-wall SAR.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"365 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141567214","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":"Uniqueness and numerical method for phaseless inverse diffraction grating problem with known superposition of incident point sources","authors":"Tian Niu, Junliang Lv and Jiahui Gao","doi":"10.1088/1361-6420/ad5b81","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5b81","url":null,"abstract":"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.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"30 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548474","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":"Inverse conductivity problem with one measurement: uniqueness of multi-layer structures","authors":"Lingzheng Kong, Youjun Deng and Liyan Zhu","doi":"10.1088/1361-6420/ad5b82","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5b82","url":null,"abstract":"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.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"10 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511938","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":"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","authors":"Nagendra Kumar Chaurasia and Shubhankar Chakraborty","doi":"10.1088/1361-6420/ad5a34","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5a34","url":null,"abstract":"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.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"205 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511937","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":"Solving, tracking and stopping streaming linear inverse problems","authors":"Nathaniel Pritchard and Vivak Patel","doi":"10.1088/1361-6420/ad5583","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5583","url":null,"abstract":"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.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"73 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511939","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":"Piecewise nonlinear materials and Monotonicity Principle","authors":"Antonio Corbo Esposito, Luisa Faella, Vincenzo Mottola, Gianpaolo Piscitelli, Ravi Prakash and Antonello Tamburrino","doi":"10.1088/1361-6420/ad575c","DOIUrl":"https://doi.org/10.1088/1361-6420/ad575c","url":null,"abstract":"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.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"20 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511940","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}