Pub Date : 2024-08-30DOI: 10.1088/1361-6420/ad6fc6
Thorsten Hohage, Roman G Novikov, Vladimir N Sivkin
We consider the problem of finding a compactly supported potential in the multidimensional Schrödinger equation from its differential scattering cross section (squared modulus of the scattering amplitude) at fixed energy. In the Born approximation this problem simplifies to the phase retrieval problem of reconstructing the potential from the absolute value of its Fourier transform on a ball. To compensate for the missing phase information we use the method of a priori known background scatterers. In particular, we propose an iterative scheme for finding the potential from measurements of a single differential scattering cross section corresponding to the sum of the unknown potential and a known background potential, which is sufficiently disjoint. If this condition is relaxed, then we give similar results for finding the potential from additional monochromatic measurements of the differential scattering cross section of the unknown potential without the background potential. The performance of the proposed algorithms is demonstrated in numerical examples. In the present work we significantly advance theoretically and numerically studies of Agaltsov et al (2019 Inverse Problems35 24001) and Novikov and Sivkin (2021 Inverse Problems37 055011).
{"title":"Phase retrieval and phaseless inverse scattering with background information","authors":"Thorsten Hohage, Roman G Novikov, Vladimir N Sivkin","doi":"10.1088/1361-6420/ad6fc6","DOIUrl":"https://doi.org/10.1088/1361-6420/ad6fc6","url":null,"abstract":"We consider the problem of finding a compactly supported potential in the multidimensional Schrödinger equation from its differential scattering cross section (squared modulus of the scattering amplitude) at fixed energy. In the Born approximation this problem simplifies to the phase retrieval problem of reconstructing the potential from the absolute value of its Fourier transform on a ball. To compensate for the missing phase information we use the method of <italic toggle=\"yes\">a priori</italic> known background scatterers. In particular, we propose an iterative scheme for finding the potential from measurements of a single differential scattering cross section corresponding to the sum of the unknown potential and a known background potential, which is sufficiently disjoint. If this condition is relaxed, then we give similar results for finding the potential from additional monochromatic measurements of the differential scattering cross section of the unknown potential without the background potential. The performance of the proposed algorithms is demonstrated in numerical examples. In the present work we significantly advance theoretically and numerically studies of Agaltsov <italic toggle=\"yes\">et al</italic> (2019 <italic toggle=\"yes\">Inverse Problems</italic> <bold>35</bold> 24001) and Novikov and Sivkin (2021 <italic toggle=\"yes\">Inverse Problems</italic> <bold>37</bold> 055011).","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142210298","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 : 2024-08-29DOI: 10.1088/1361-6420/ad6fc8
Daniel Rabinovich, Dan Givoli
Kirchhoff Migration (KM), sometimes called Arrival (or Travel) Time Imaging, is a basic and popular imaging technique based on the arrival time of waves from given sources to given sensors. It is commonly used in the fields of underwater acoustics and solid earth geophysics, for both subsurface structure analysis and for identifying unknown local obstacles (scatterers) in the medium. The present paper concentrates on the latter application. For acoustics, the KM algorithm is extremely simple and efficient, although it usually produces a rather crude image, which is the reason for its use as the method of choice when high resolution is not needed, or as a fast technique to produce an initial guess for a more sophisticated imaging method. For elasticity, KM is much more involved, as the arrival-time algorithm is not obvious, mainly since there is more than one wave speed at each spatial point. In this paper, a new KM scheme is proposed for obstacle identification in an isotropic piecewise-homogeneous elastic medium. The scheme is based on measuring two quantities that are second-order operators of the displacement field, which are related to P and S waves, and applying the acoustic KM algorithm to each of them, with the appropriate wave speed. It is demonstrated numerically that the operator related to S waves results in very good identification in many cases. The fact that measurements based on the S-related operator are preferred over those based on the P-related operator is an empirical observation, and awaits full analysis, although a partial explanation is given here.
基尔霍夫迁移（Kirchhoff Migration，KM），有时也称为到达（或移动）时间成像（Arrival (or Travel) Time Imaging），是一种基于波从给定源到达给定传感器的时间的基本且流行的成像技术。它通常用于水下声学和固体地球物理学领域，既可用于地下结构分析，也可用于识别介质中未知的局部障碍物（散射体）。本文主要讨论后一种应用。在声学方面，KM 算法非常简单高效，尽管它通常生成的图像相当粗糙，这也是它在不需要高分辨率时作为首选方法，或作为为更复杂的成像方法生成初始猜测的快速技术的原因。对于弹性而言，KM 涉及的问题要多得多，因为到达时间算法并不明显，主要是因为每个空间点的波速不止一种。本文提出了一种新的 KM 方案，用于在各向同性的片状均质弹性介质中识别障碍物。该方案基于测量与 P 波和 S 波相关的位移场二阶算子的两个量，并将声学 KM 算法与适当的波速分别应用于这两个量。数值结果表明，与 S 波相关的算子在许多情况下都能实现很好的识别。基于 S 波相关算子的测量结果优于基于 P 波相关算子的测量结果，这是一个经验观察结果，有待全面分析，但本文给出了部分解释。
{"title":"A Kirchhoff Migration scheme for elastic obstacle identification","authors":"Daniel Rabinovich, Dan Givoli","doi":"10.1088/1361-6420/ad6fc8","DOIUrl":"https://doi.org/10.1088/1361-6420/ad6fc8","url":null,"abstract":"Kirchhoff Migration (KM), sometimes called Arrival (or Travel) Time Imaging, is a basic and popular imaging technique based on the arrival time of waves from given sources to given sensors. It is commonly used in the fields of underwater acoustics and solid earth geophysics, for both subsurface structure analysis and for identifying unknown local obstacles (scatterers) in the medium. The present paper concentrates on the latter application. For acoustics, the KM algorithm is extremely simple and efficient, although it usually produces a rather crude image, which is the reason for its use as the method of choice when high resolution is not needed, or as a fast technique to produce an initial guess for a more sophisticated imaging method. For elasticity, KM is much more involved, as the arrival-time algorithm is not obvious, mainly since there is more than one wave speed at each spatial point. In this paper, a new KM scheme is proposed for obstacle identification in an isotropic piecewise-homogeneous elastic medium. The scheme is based on measuring two quantities that are second-order operators of the displacement field, which are related to P and S waves, and applying the acoustic KM algorithm to each of them, with the appropriate wave speed. It is demonstrated numerically that the operator related to S waves results in very good identification in many cases. The fact that measurements based on the S-related operator are preferred over those based on the P-related operator is an empirical observation, and awaits full analysis, although a partial explanation is given here.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226718","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 : 2024-08-28DOI: 10.1088/1361-6420/ad7054
Roman Novikov, Basant Lal Sharma
We consider multi-frequency inverse source problem for the discrete Helmholtz operator on the square lattice Zd, d⩾1. We consider this problem for the cases with and without phase information. We prove uniqueness results and present examples of non-uniqueness for this problem for the case of compactly supported source function, and a Lipshitz stability estimate for the phased case is established. Relations with inverse scattering problem for the discrete Schrödinger operators in the Born approximation are also provided.
{"title":"Inverse source problem for discrete Helmholtz equation","authors":"Roman Novikov, Basant Lal Sharma","doi":"10.1088/1361-6420/ad7054","DOIUrl":"https://doi.org/10.1088/1361-6420/ad7054","url":null,"abstract":"We consider multi-frequency inverse source problem for the discrete Helmholtz operator on the square lattice <inline-formula>\u0000<tex-math><?CDATA $mathbb{Z}^d$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">Z</mml:mi></mml:mrow><mml:mi>d</mml:mi></mml:msup></mml:mrow></mml:math><inline-graphic xlink:href=\"ipad7054ieqn1.gif\"></inline-graphic></inline-formula>, <inline-formula>\u0000<tex-math><?CDATA $d unicode{x2A7E} 1$?></tex-math><mml:math overflow=\"scroll\"><mml:mrow><mml:mi>d</mml:mi><mml:mtext>⩾</mml:mtext><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href=\"ipad7054ieqn2.gif\"></inline-graphic></inline-formula>. We consider this problem for the cases with and without phase information. We prove uniqueness results and present examples of non-uniqueness for this problem for the case of compactly supported source function, and a Lipshitz stability estimate for the phased case is established. Relations with inverse scattering problem for the discrete Schrödinger operators in the Born approximation are also provided.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142210299","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 : 2024-08-28DOI: 10.1088/1361-6420/ad7056
Ke Chen, Jasen Lai, Chunmei Wang
Partial differential equations (PDEs) play a foundational role in modeling physical phenomena. This study addresses the challenging task of determining variable coefficients within PDEs from measurement data. We introduce a novel neural network, ‘pseudo-differential IAEnet’ (pd-IAEnet), which draws inspiration from pseudo-differential operators. pd-IAEnet achieves significantly enhanced computational speed and accuracy with fewer parameters compared to conventional models. Extensive benchmark evaluations are conducted across a range of inverse problems, including electrical impedance tomography, optical tomography, and seismic imaging, consistently demonstrating pd-IAEnet’s superior accuracy. Notably, pd-IAEnet exhibits robustness in the presence of measurement noise, a critical characteristic for real-world applications. An exceptional feature is its discretization invariance, enabling effective training on data from diverse discretization schemes while maintaining accuracy on different meshes. In summary, pd-IAEnet offers a potent and efficient solution for addressing inverse PDE problems, contributing to improved computational efficiency, robustness, and adaptability to a wide array of data sources.
{"title":"Pseudo-differential integral autoencoder network for inverse PDE operators","authors":"Ke Chen, Jasen Lai, Chunmei Wang","doi":"10.1088/1361-6420/ad7056","DOIUrl":"https://doi.org/10.1088/1361-6420/ad7056","url":null,"abstract":"Partial differential equations (PDEs) play a foundational role in modeling physical phenomena. This study addresses the challenging task of determining variable coefficients within PDEs from measurement data. We introduce a novel neural network, ‘pseudo-differential IAEnet’ (pd-IAEnet), which draws inspiration from pseudo-differential operators. pd-IAEnet achieves significantly enhanced computational speed and accuracy with fewer parameters compared to conventional models. Extensive benchmark evaluations are conducted across a range of inverse problems, including electrical impedance tomography, optical tomography, and seismic imaging, consistently demonstrating pd-IAEnet’s superior accuracy. Notably, pd-IAEnet exhibits robustness in the presence of measurement noise, a critical characteristic for real-world applications. An exceptional feature is its discretization invariance, enabling effective training on data from diverse discretization schemes while maintaining accuracy on different meshes. In summary, pd-IAEnet offers a potent and efficient solution for addressing inverse PDE problems, contributing to improved computational efficiency, robustness, and adaptability to a wide array of data sources.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142210310","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 : 2024-08-27DOI: 10.1088/1361-6420/ad6a35
B Harroué, J-F Giovannelli, M Pereyra
This paper considers the quantitative comparison of several alternative models to perform deconvolution in situations where there is no ground truth data available. With applications to very large data sets in mind, we focus on linear deconvolution models based on a Wiener filter. Although comparatively simple, such models are widely prevalent in large scale setting such as high-resolution image restoration because they provide an excellent trade-off between accuracy and computational effort. However, in order to deliver accurate solutions, the models need to be properly calibrated in order to capture the covariance structure of the unknown quantity of interest and of the measurement error. This calibration often requires onerous controlled experiments and extensive expert supervision, as well as regular recalibration procedures. This paper adopts an unsupervised Bayesian statistical approach to model assessment that allows comparing alternative models by using only the observed data, without the need for ground truth data or controlled experiments. Accordingly, the models are quantitatively compared based on their posterior probabilities given the data, which are derived from the marginal likelihoods or evidences of the models. The computation of these evidences is highly non-trivial and this paper consider three different strategies to address this difficulty—a Chib approach, Laplace approximations, and a truncated harmonic expectation—all of which efficiently implemented by using a Gibbs sampling algorithm specialised for this class of models. In addition to enabling unsupervised model selection, the output of the Gibbs sampler can also be used to automatically estimate unknown model parameters such as the variance of the measurement error and the power of the unknown quantity of interest. The proposed strategies are demonstrated on a range of image deconvolution problems, where they are used to compare different modelling choices for the instrument’s point spread function and covariance matrices for the unknown image and for the measurement error.
{"title":"An optimal Bayesian strategy for comparing Wiener–Hunt deconvolution models in the absence of ground truth","authors":"B Harroué, J-F Giovannelli, M Pereyra","doi":"10.1088/1361-6420/ad6a35","DOIUrl":"https://doi.org/10.1088/1361-6420/ad6a35","url":null,"abstract":"This paper considers the quantitative comparison of several alternative models to perform deconvolution in situations where there is no ground truth data available. With applications to very large data sets in mind, we focus on linear deconvolution models based on a Wiener filter. Although comparatively simple, such models are widely prevalent in large scale setting such as high-resolution image restoration because they provide an excellent trade-off between accuracy and computational effort. However, in order to deliver accurate solutions, the models need to be properly calibrated in order to capture the covariance structure of the unknown quantity of interest and of the measurement error. This calibration often requires onerous controlled experiments and extensive expert supervision, as well as regular recalibration procedures. This paper adopts an unsupervised Bayesian statistical approach to model assessment that allows comparing alternative models by using only the observed data, without the need for ground truth data or controlled experiments. Accordingly, the models are quantitatively compared based on their posterior probabilities given the data, which are derived from the marginal likelihoods or <italic toggle=\"yes\">evidences</italic> of the models. The computation of these evidences is highly non-trivial and this paper consider three different strategies to address this difficulty—a Chib approach, Laplace approximations, and a truncated harmonic expectation—all of which efficiently implemented by using a Gibbs sampling algorithm specialised for this class of models. In addition to enabling unsupervised model selection, the output of the Gibbs sampler can also be used to automatically estimate unknown model parameters such as the variance of the measurement error and the power of the unknown quantity of interest. The proposed strategies are demonstrated on a range of image deconvolution problems, where they are used to compare different modelling choices for the instrument’s point spread function and covariance matrices for the unknown image and for the measurement error.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226719","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 : 2024-08-27DOI: 10.1088/1361-6420/ad7055
Tianjiao Wang, Xiang Xu, Yue Zhao
This paper investigates inverse source problems for the Helmholtz and Navier equations in multi-layered media, considering both two and three-dimensional cases respectively. The study reveals a consistent increase in stability for each scenario, characterized by two main terms: a Hölder-type term associated with data discrepancy, and a logarithmic-type term that diminishes as more frequencies are considered. In the two-dimensional case, measurements on interfaces and far-field data are essential. By employing the fundamental solution in free-space as the test function and utilizing the asymptotic behavior of the solution and continuation principle, stability results are obtained. In the three-dimensional case, measurements on interfaces and artificial boundaries are taken, and the stability result can be derived by applying the arguments for inverse source problems in homogeneous media.
{"title":"Increasing stability of the acoustic and elastic inverse source problems in multi-layered media","authors":"Tianjiao Wang, Xiang Xu, Yue Zhao","doi":"10.1088/1361-6420/ad7055","DOIUrl":"https://doi.org/10.1088/1361-6420/ad7055","url":null,"abstract":"This paper investigates inverse source problems for the Helmholtz and Navier equations in multi-layered media, considering both two and three-dimensional cases respectively. The study reveals a consistent increase in stability for each scenario, characterized by two main terms: a Hölder-type term associated with data discrepancy, and a logarithmic-type term that diminishes as more frequencies are considered. In the two-dimensional case, measurements on interfaces and far-field data are essential. By employing the fundamental solution in free-space as the test function and utilizing the asymptotic behavior of the solution and continuation principle, stability results are obtained. In the three-dimensional case, measurements on interfaces and artificial boundaries are taken, and the stability result can be derived by applying the arguments for inverse source problems in homogeneous media.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142210300","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 : 2024-08-27DOI: 10.1088/1361-6420/ad7053
Isaac Harris, Victor Hughes, Heejin Lee
In this paper, we consider the inverse scattering problem associated with an anisotropic medium with a conductive boundary. We will assume that the corresponding far–field pattern is known/measured and we consider two inverse problems. First, we show that the far–field data uniquely determines the boundary coefficient. Next, since it is known that anisotropic coefficients are not uniquely determined by this data we will develop a qualitative method to recover the scatterer. To this end, we study the so–called monotonicity method applied to this inverse shape problem. This method has recently been applied to some inverse scattering problems but this is the first time it has been applied to an anisotropic scatterer. This method allows one to recover the scatterer by considering the eigenvalues of an operator associated with the far–field operator. We present some simple numerical reconstructions to illustrate our theory in two dimensions. For our reconstructions, we need to compute the adjoint of the Herglotz wave function as an operator mapping into H1 of a small ball.
{"title":"Analysis of the monotonicity method for an anisotropic scatterer with a conductive boundary","authors":"Isaac Harris, Victor Hughes, Heejin Lee","doi":"10.1088/1361-6420/ad7053","DOIUrl":"https://doi.org/10.1088/1361-6420/ad7053","url":null,"abstract":"In this paper, we consider the inverse scattering problem associated with an anisotropic medium with a conductive boundary. We will assume that the corresponding far–field pattern is known/measured and we consider two inverse problems. First, we show that the far–field data uniquely determines the boundary coefficient. Next, since it is known that anisotropic coefficients are not uniquely determined by this data we will develop a qualitative method to recover the scatterer. To this end, we study the so–called monotonicity method applied to this inverse shape problem. This method has recently been applied to some inverse scattering problems but this is the first time it has been applied to an anisotropic scatterer. This method allows one to recover the scatterer by considering the eigenvalues of an operator associated with the far–field operator. We present some simple numerical reconstructions to illustrate our theory in two dimensions. For our reconstructions, we need to compute the adjoint of the Herglotz wave function as an operator mapping into <italic toggle=\"yes\">H</italic><sup>1</sup> of a small ball.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226720","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}
In this paper, we place ourselves in the context of the Bayesian framework for image segmentation in the presence of varying blur. The proposed approach is based on Triplet Markov Random Fields (TMRF). This method takes into account, during segmentation, peculiarities of an image such as noise, blur, and texture. We present an unsupervised TMRF method, which jointly deals with the problem of segmentation, and that of depth estimation in order to process fluorescence microscopy images. In addition to the estimation of the depth maps using the Metropolis-Hasting and the Stochastic Parameter Estimation (SPE) algorithms, we also estimate the model parameters using the SPE algorithm. We compare our TMRF method to other MRF models on simulated images, and to an unsupervised method from the state of art on real fluorescence microscopy images. Our method offers improved results, especially when blur is important.
{"title":"Bayesian image segmentation under varying blur with triplet Markov random field","authors":"Sonia Ouali, Jean-Baptiste Courbot, Romain Pierron, Olivier Haeberlé","doi":"10.1088/1361-6420/ad6a34","DOIUrl":"https://doi.org/10.1088/1361-6420/ad6a34","url":null,"abstract":"In this paper, we place ourselves in the context of the Bayesian framework for image segmentation in the presence of varying blur. The proposed approach is based on Triplet Markov Random Fields (TMRF). This method takes into account, during segmentation, peculiarities of an image such as noise, blur, and texture. We present an unsupervised TMRF method, which jointly deals with the problem of segmentation, and that of depth estimation in order to process fluorescence microscopy images. In addition to the estimation of the depth maps using the Metropolis-Hasting and the Stochastic Parameter Estimation (SPE) algorithms, we also estimate the model parameters using the SPE algorithm. We compare our TMRF method to other MRF models on simulated images, and to an unsupervised method from the state of art on real fluorescence microscopy images. Our method offers improved results, especially when blur is important.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142210301","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 : 2024-08-11DOI: 10.1088/1361-6420/ad6a33
Matei Hanu and Simon Weissmann
The ensemble Kalman inversion (EKI), a recently introduced optimisation method for solving inverse problems, is widely employed for the efficient and derivative-free estimation of unknown parameters. Specifically in cases involving ill-posed inverse problems and high-dimensional parameter spaces, the scheme has shown promising success. However, in its general form, the EKI does not take constraints into account, which are essential and often stem from physical limitations or specific requirements. Based on a log-barrier approach, we suggest adapting the continuous-time formulation of EKI to incorporate convex inequality constraints. We underpin this adaptation with a theoretical analysis that provides lower and upper bounds on the ensemble collapse, as well as convergence to the constraint optimum for general nonlinear forward models. Finally, we showcase our results through two examples involving partial differential equations.
{"title":"On the ensemble Kalman inversion under inequality constraints","authors":"Matei Hanu and Simon Weissmann","doi":"10.1088/1361-6420/ad6a33","DOIUrl":"https://doi.org/10.1088/1361-6420/ad6a33","url":null,"abstract":"The ensemble Kalman inversion (EKI), a recently introduced optimisation method for solving inverse problems, is widely employed for the efficient and derivative-free estimation of unknown parameters. Specifically in cases involving ill-posed inverse problems and high-dimensional parameter spaces, the scheme has shown promising success. However, in its general form, the EKI does not take constraints into account, which are essential and often stem from physical limitations or specific requirements. Based on a log-barrier approach, we suggest adapting the continuous-time formulation of EKI to incorporate convex inequality constraints. We underpin this adaptation with a theoretical analysis that provides lower and upper bounds on the ensemble collapse, as well as convergence to the constraint optimum for general nonlinear forward models. Finally, we showcase our results through two examples involving partial differential equations.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141942158","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 : 2024-07-30DOI: 10.1088/1361-6420/ad5eb4
Maximilian Dinkel, Carolin M Geitner, Gil Robalo Rei, Jonas Nitzler, Wolfgang A Wall
Solving inverse problems using Bayesian methods can become prohibitively expensive when likelihood evaluations involve complex and large scale numerical models. A common approach to circumvent this issue is to approximate the forward model or the likelihood function with a surrogate model. But also there, due to limited computational resources, only a few training points are available in many practically relevant cases. Thus, it can be advantageous to model the additional uncertainties of the surrogate in order to incorporate the epistemic uncertainty due to limited data. In this paper, we develop a novel approach to approximate the log likelihood by a constrained Gaussian process based on prior knowledge about its boundedness. This improves the accuracy of the surrogate approximation without increasing the number of training samples. Additionally, we introduce a formulation to integrate the epistemic uncertainty due to limited training points into the posterior density approximation. This is combined with a state of the art active learning strategy for selecting training points, which allows to approximate posterior densities in higher dimensions very efficiently. We demonstrate the fast convergence of our approach for a benchmark problem and infer a random field that is discretized by 30 parameters using only about 1000 model evaluations. In a practically relevant example, the parameters of a reduced lung model are calibrated based on flow observations over time and voltage measurements from a coupled electrical impedance tomography simulation.
{"title":"Solving Bayesian inverse problems with expensive likelihoods using constrained Gaussian processes and active learning","authors":"Maximilian Dinkel, Carolin M Geitner, Gil Robalo Rei, Jonas Nitzler, Wolfgang A Wall","doi":"10.1088/1361-6420/ad5eb4","DOIUrl":"https://doi.org/10.1088/1361-6420/ad5eb4","url":null,"abstract":"Solving inverse problems using Bayesian methods can become prohibitively expensive when likelihood evaluations involve complex and large scale numerical models. A common approach to circumvent this issue is to approximate the forward model or the likelihood function with a surrogate model. But also there, due to limited computational resources, only a few training points are available in many practically relevant cases. Thus, it can be advantageous to model the additional uncertainties of the surrogate in order to incorporate the epistemic uncertainty due to limited data. In this paper, we develop a novel approach to approximate the log likelihood by a constrained Gaussian process based on prior knowledge about its boundedness. This improves the accuracy of the surrogate approximation without increasing the number of training samples. Additionally, we introduce a formulation to integrate the epistemic uncertainty due to limited training points into the posterior density approximation. This is combined with a state of the art active learning strategy for selecting training points, which allows to approximate posterior densities in higher dimensions very efficiently. We demonstrate the fast convergence of our approach for a benchmark problem and infer a random field that is discretized by 30 parameters using only about 1000 model evaluations. In a practically relevant example, the parameters of a reduced lung model are calibrated based on flow observations over time and voltage measurements from a coupled electrical impedance tomography simulation.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866576","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}
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