Pub Date : 2024-07-02DOI: 10.1088/1361-6420/ad5b82
Lingzheng Kong, Youjun Deng and Liyan Zhu
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.
{"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":null,"pages":null},"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}
Pub Date : 2024-07-01DOI: 10.1088/1361-6420/ad5a34
Nagendra Kumar Chaurasia and Shubhankar Chakraborty
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.
{"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":null,"pages":null},"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}
Pub Date : 2024-06-25DOI: 10.1088/1361-6420/ad5583
Nathaniel Pritchard and Vivak Patel
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.
{"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":null,"pages":null},"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}
Pub Date : 2024-06-23DOI: 10.1088/1361-6420/ad575c
Antonio Corbo Esposito, Luisa Faella, Vincenzo Mottola, Gianpaolo Piscitelli, Ravi Prakash and Antonello Tamburrino
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.
{"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":null,"pages":null},"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}
Pub Date : 2024-06-17DOI: 10.1088/1361-6420/ad52bb
C M Wensrich, S Holman, M Courdurier, W R B Lionheart, A P Polyakova and I E Svetov
We examine the problem of Bragg-edge elastic strain tomography from energy resolved neutron transmission imaging. A new approach is developed for two-dimensional plane-stress and plane-strain systems whereby elastic strain can be reconstructed from its Longitudinal ray transform (LRT) as two parts of a Helmholtz decomposition based on the concept of an Airy stress potential. The solenoidal component of this decomposition is reconstructed using an inversion formula based on a tensor filtered back projection (FBP) algorithm whereas the potential part can be recovered using either Hooke’s law or a finite element model of the elastic system. The technique is demonstrated for two-dimensional plane-stress systems in both simulation, and on real experimental data. We also demonstrate that application of the standard scalar FBP algorithm to the LRT in these systems recovers the trace of the solenoidal component of strain and we provide physical meaning for this quantity in the case of 2D plane-stress and plane-strain systems.
{"title":"Direct inversion of the Longitudinal ray transform for 2D residual elastic strain fields","authors":"C M Wensrich, S Holman, M Courdurier, W R B Lionheart, A P Polyakova and I E Svetov","doi":"10.1088/1361-6420/ad52bb","DOIUrl":"https://doi.org/10.1088/1361-6420/ad52bb","url":null,"abstract":"We examine the problem of Bragg-edge elastic strain tomography from energy resolved neutron transmission imaging. A new approach is developed for two-dimensional plane-stress and plane-strain systems whereby elastic strain can be reconstructed from its Longitudinal ray transform (LRT) as two parts of a Helmholtz decomposition based on the concept of an Airy stress potential. The solenoidal component of this decomposition is reconstructed using an inversion formula based on a tensor filtered back projection (FBP) algorithm whereas the potential part can be recovered using either Hooke’s law or a finite element model of the elastic system. The technique is demonstrated for two-dimensional plane-stress systems in both simulation, and on real experimental data. We also demonstrate that application of the standard scalar FBP algorithm to the LRT in these systems recovers the trace of the solenoidal component of strain and we provide physical meaning for this quantity in the case of 2D plane-stress and plane-strain systems.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511941","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 : 2024-06-11DOI: 10.1088/1361-6420/ad4dda
Yan Chang, Yukun Guo, Hongyu Liu and Deyue Zhang
This work is concerned with an inverse elastic scattering problem of identifying the unknown rigid obstacle embedded in an open space filled with a homogeneous and isotropic elastic medium. A Newton-type iteration method relying on the boundary condition is designed to identify the boundary curve of the obstacle. Based on the Helmholtz decomposition and the Fourier–Bessel expansion, we explicitly derive the approximate scattered field and its derivative on each iterative curve. Rigorous mathematical justifications for the proposed method are provided. Numerical examples are presented to verify the effectiveness of the proposed method.
{"title":"A novel Newton method for inverse elastic scattering problems","authors":"Yan Chang, Yukun Guo, Hongyu Liu and Deyue Zhang","doi":"10.1088/1361-6420/ad4dda","DOIUrl":"https://doi.org/10.1088/1361-6420/ad4dda","url":null,"abstract":"This work is concerned with an inverse elastic scattering problem of identifying the unknown rigid obstacle embedded in an open space filled with a homogeneous and isotropic elastic medium. A Newton-type iteration method relying on the boundary condition is designed to identify the boundary curve of the obstacle. Based on the Helmholtz decomposition and the Fourier–Bessel expansion, we explicitly derive the approximate scattered field and its derivative on each iterative curve. Rigorous mathematical justifications for the proposed method are provided. Numerical examples are presented to verify the effectiveness of the proposed method.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511942","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 : 2024-06-10DOI: 10.1088/1361-6420/ad4f0b
Kai Li, Bo Zhang, Haiwen Zhang
This paper is concerned with the inverse problem of reconstructing an inhomogeneous medium from the acoustic far-field data at a fixed frequency in two dimensions. This inverse problem is severely ill-posed (and also strongly nonlinear), and certain regularization strategy is thus needed. However, it is difficult to select an appropriate regularization strategy which should enforce some a priori information of the unknown scatterer. To address this issue, we plan to use a deep learning approach to learn some a priori information of the unknown scatterer from certain ground truth data, which is then combined with a traditional iteration method to solve the inverse problem. Specifically, we propose a deep learning-based iterative reconstruction algorithm for the inverse problem, based on a repeated application of a deep neural network and the iteratively regularized Gauss–Newton method (IRGNM). Our deep neural network (called the learned projector in this paper) mainly focuses on learning the a priori information of the shape of the unknown contrast with a normalization technique in the training processes and is trained to act like a projector which is helpful for projecting the solution into some feasible region. Extensive numerical experiments show that our reconstruction algorithm provides good reconstruction results even for the high contrast case and has a satisfactory generalization ability.
{"title":"Reconstruction of inhomogeneous media by an iteration algorithm with a learned projector","authors":"Kai Li, Bo Zhang, Haiwen Zhang","doi":"10.1088/1361-6420/ad4f0b","DOIUrl":"https://doi.org/10.1088/1361-6420/ad4f0b","url":null,"abstract":"This paper is concerned with the inverse problem of reconstructing an inhomogeneous medium from the acoustic far-field data at a fixed frequency in two dimensions. This inverse problem is severely ill-posed (and also strongly nonlinear), and certain regularization strategy is thus needed. However, it is difficult to select an appropriate regularization strategy which should enforce some <italic toggle=\"yes\">a priori</italic> information of the unknown scatterer. To address this issue, we plan to use a deep learning approach to learn some <italic toggle=\"yes\">a priori</italic> information of the unknown scatterer from certain ground truth data, which is then combined with a traditional iteration method to solve the inverse problem. Specifically, we propose a deep learning-based iterative reconstruction algorithm for the inverse problem, based on a repeated application of a deep neural network and the iteratively regularized Gauss–Newton method (IRGNM). Our deep neural network (called the learned projector in this paper) mainly focuses on learning the <italic toggle=\"yes\">a priori</italic> information of the <italic toggle=\"yes\">shape</italic> of the unknown contrast with a normalization technique in the training processes and is trained to act like a projector which is helpful for projecting the solution into some feasible region. Extensive numerical experiments show that our reconstruction algorithm provides good reconstruction results even for the high contrast case and has a satisfactory generalization ability.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511943","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 : 2024-06-03DOI: 10.1088/1361-6420/ad4669
Bjørn Jensen, Adrian Kirkeby and Kim Knudsen
In acousto-electric tomography (AET) the goal is to reconstruct the electric conductivity in a domain from electrostatic boundary measurements of corresponding currents and voltages, while the domain is perturbed by a time-dependent acoustic wave, thus taking advantage of the acousto-electric effect. We approach the AET reconstruction in two steps: First, the interior power density is obtained from boundary measurements by solving a linear inverse and ill-posed problem; second, the interior conductivity is reconstructed from the power density by solving a non-linear and well-posed problem. Mathematically these inverse problems are fairly well understood, and reconstruction methods work well on synthetic data. This is in contrast to experimental findings. An effect can indeed be observed and data can be collected. However, the acousto-electric coupling is very weak, and consequently, the change in the measured voltage due to the acoustic perturbation might be too small compared to the background noise for viable reconstructions. In this paper, we take one step towards understanding the feasibility of AET. We provide an in-silico model of the coupled physics scenario based on standard models for the individual phenomena. Moreover, we formulate and implement numerically a full reconstruction method for the inverse problem via the two steps. We perform computational experiments with realistically chosen parameters from the context of medical imaging. The focus is on understanding the role of the acousto-electric coupling parameter and the signal-to-noise ratio (SNR). The critical signal strength is analyzed and the omnipresent Johnson–Nyquist noise is estimated. We obtain both positive and negative findings; we can reconstruct features even under severe noise conditions, but we also find that the SNR one is likely to face in practice is too low to obtain useful reconstructions.
声电层析成像（AET）的目标是通过对相应电流和电压的静电边界测量重建域中的电导率，同时域受到随时间变化的声波扰动，从而利用声电效应。我们分两步进行 AET 重建：首先，通过求解一个线性反问题和求解困难的问题，从边界测量中获得内部功率密度；其次，通过求解一个非线性和求解困难的问题，从功率密度中重建内部传导性。从数学角度来看，这些逆问题都相当容易理解，而且重建方法在合成数据上也很有效。这与实验结果截然不同。确实可以观察到效果，也可以收集数据。然而，声电耦合非常微弱，因此，与背景噪声相比，声学扰动引起的测量电压变化可能太小，无法进行可行的重建。在本文中，我们朝着了解 AET 的可行性迈出了一步。我们基于单个现象的标准模型，提供了一个耦合物理场景的内部模型。此外，我们通过两个步骤为逆问题制定并数值化了一个完整的重建方法。我们使用医学成像中实际选择的参数进行了计算实验。重点是了解声电耦合参数和信噪比（SNR）的作用。我们分析了临界信号强度，并估算了无处不在的约翰逊-奈奎斯特噪声。我们得出了正反两方面的结论：即使在严重的噪声条件下，我们也能重建特征，但我们也发现，在实践中可能面临的信噪比太低，无法获得有用的重建。
{"title":"Feasibility of acousto-electric tomography","authors":"Bjørn Jensen, Adrian Kirkeby and Kim Knudsen","doi":"10.1088/1361-6420/ad4669","DOIUrl":"https://doi.org/10.1088/1361-6420/ad4669","url":null,"abstract":"In acousto-electric tomography (AET) the goal is to reconstruct the electric conductivity in a domain from electrostatic boundary measurements of corresponding currents and voltages, while the domain is perturbed by a time-dependent acoustic wave, thus taking advantage of the acousto-electric effect. We approach the AET reconstruction in two steps: First, the interior power density is obtained from boundary measurements by solving a linear inverse and ill-posed problem; second, the interior conductivity is reconstructed from the power density by solving a non-linear and well-posed problem. Mathematically these inverse problems are fairly well understood, and reconstruction methods work well on synthetic data. This is in contrast to experimental findings. An effect can indeed be observed and data can be collected. However, the acousto-electric coupling is very weak, and consequently, the change in the measured voltage due to the acoustic perturbation might be too small compared to the background noise for viable reconstructions. In this paper, we take one step towards understanding the feasibility of AET. We provide an in-silico model of the coupled physics scenario based on standard models for the individual phenomena. Moreover, we formulate and implement numerically a full reconstruction method for the inverse problem via the two steps. We perform computational experiments with realistically chosen parameters from the context of medical imaging. The focus is on understanding the role of the acousto-electric coupling parameter and the signal-to-noise ratio (SNR). The critical signal strength is analyzed and the omnipresent Johnson–Nyquist noise is estimated. We obtain both positive and negative findings; we can reconstruct features even under severe noise conditions, but we also find that the SNR one is likely to face in practice is too low to obtain useful reconstructions.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252489","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 : 2024-06-03DOI: 10.1088/1361-6420/ad5373
B. Afkham, Julianne Chung, Matthias Chung
In this work, we describe a new approach that uses variational encoder-decoder (VED) networks for efficient uncertainty quantification for goal-oriented inverse problems. Contrary to standard inverse problems, these approaches are goal-oriented in that the goal is to estimate some quantities of interest (QoI) that are functions of the solution of an inverse problem, rather than the solution itself. Moreover, we are interested in computing uncertainty metrics associated with the QoI, thus utilizing a Bayesian approach for inverse problems that incorporates the prediction operator and techniques for exploring the posterior. This may be particularly challenging, especially for nonlinear, possibly unknown, operators and nonstandard prior assumptions. We harness recent advances in machine learning, i.e., VED networks, to describe a data-driven approach to large-scale inverse problems. This enables a real-time uncertainty quantification for the QoI. One of the advantages of our approach is that we avoid the need to solve challenging inversion problems by training a network to approximate the mapping from observations to QoI. Another main benefit is that we enable uncertainty quantification for the QoI by leveraging probability distributions in the latent and target spaces. This allows us to efficiently generate QoI samples and circumvent complicated or even unknown forward models and prediction operators. Numerical results from medical tomography reconstruction and nonlinear hydraulic tomography demonstrate the potential and broad applicability of the approach.
{"title":"Uncertainty quantification for goal-oriented inverse problems via variational encoder-decoder networks","authors":"B. Afkham, Julianne Chung, Matthias Chung","doi":"10.1088/1361-6420/ad5373","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5373","url":null,"abstract":"\u0000 In this work, we describe a new approach that uses variational encoder-decoder (VED) networks for efficient uncertainty quantification for goal-oriented inverse problems. Contrary to standard inverse problems, these approaches are goal-oriented in that the goal is to estimate some quantities of interest (QoI) that are functions of the solution of an inverse problem, rather than the solution itself. Moreover, we are interested in computing uncertainty metrics associated with the QoI, thus utilizing a Bayesian approach for inverse problems that incorporates the prediction operator and techniques for exploring the posterior. This may be particularly challenging, especially for nonlinear, possibly unknown, operators and nonstandard prior assumptions. We harness recent advances in machine learning, i.e., VED networks, to describe a data-driven approach to large-scale inverse problems. This enables a real-time uncertainty quantification for the QoI. One of the advantages of our approach is that we avoid the need to solve challenging inversion problems by training a network to approximate the mapping from observations to QoI. Another main benefit is that we enable uncertainty quantification for the QoI by leveraging probability distributions in the latent and target spaces. This allows us to efficiently generate QoI samples and circumvent complicated or even unknown forward models and prediction operators. Numerical results from medical tomography reconstruction and nonlinear hydraulic tomography demonstrate the potential and broad applicability of the approach.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141271994","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 : 2024-05-27DOI: 10.1088/1361-6420/ad49cc
Hiroshi Fujiwara, David Omogbhe, Kamran Sadiq and Alexandru Tamasan
We present a reconstruction method that stably recovers the real valued, symmetric tensors compactly supported in the Euclidean plane, from knowledge of their attenuated momenta ray transform. The problem is recast as an inverse boundary value problem for a system of transport equations, which we solve by an extension of Bukhgeim’s A-analytic theory. The method of proof is constructive. To illustrate the reconstruction method, we present results obtained in the numerical implementation for the non-attenuated case of one-tensors.
{"title":"Inversion of the attenuated momenta ray transform of planar symmetric tensors","authors":"Hiroshi Fujiwara, David Omogbhe, Kamran Sadiq and Alexandru Tamasan","doi":"10.1088/1361-6420/ad49cc","DOIUrl":"https://doi.org/10.1088/1361-6420/ad49cc","url":null,"abstract":"We present a reconstruction method that stably recovers the real valued, symmetric tensors compactly supported in the Euclidean plane, from knowledge of their attenuated momenta ray transform. The problem is recast as an inverse boundary value problem for a system of transport equations, which we solve by an extension of Bukhgeim’s A-analytic theory. The method of proof is constructive. To illustrate the reconstruction method, we present results obtained in the numerical implementation for the non-attenuated case of one-tensors.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141172698","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：