Pub Date : 2024-02-23DOI: 10.1088/1361-6420/ad2696
A Bocchinfuso, D Calvetti, E Somersalo
We consider inverse problems estimating distributed parameters from indirect noisy observations through discretization of continuum models described by partial differential or integral equations. It is well understood that errors arising from the discretization can be detrimental for ill-posed inverse problems, as discretization error behaves as correlated noise. While this problem can be avoided with a discretization fine enough to decrease the modeling error level below that of the exogenous noise that is addressed, e.g. by regularization, the computational resources needed to deal with the additional degrees of freedom may increase so much as to require high performance computing environments. Following an earlier idea, we advocate the notion of the discretization as one of the unknowns of the inverse problem, which is updated iteratively together with the solution. In this approach, the discretization, defined in terms of an underlying metric, is refined selectively only where the representation power of the current mesh is insufficient. In this paper we allow the metrics and meshes to be anisotropic, and we show that this leads to significant reduction of memory allocation and computing time.
{"title":"Adaptive anisotropic Bayesian meshing for inverse problems","authors":"A Bocchinfuso, D Calvetti, E Somersalo","doi":"10.1088/1361-6420/ad2696","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2696","url":null,"abstract":"We consider inverse problems estimating distributed parameters from indirect noisy observations through discretization of continuum models described by partial differential or integral equations. It is well understood that errors arising from the discretization can be detrimental for ill-posed inverse problems, as discretization error behaves as correlated noise. While this problem can be avoided with a discretization fine enough to decrease the modeling error level below that of the exogenous noise that is addressed, e.g. by regularization, the computational resources needed to deal with the additional degrees of freedom may increase so much as to require high performance computing environments. Following an earlier idea, we advocate the notion of the discretization as one of the unknowns of the inverse problem, which is updated iteratively together with the solution. In this approach, the discretization, defined in terms of an underlying metric, is refined selectively only where the representation power of the current mesh is insufficient. In this paper we allow the metrics and meshes to be anisotropic, and we show that this leads to significant reduction of memory allocation and computing time.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140004917","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 : 2024-02-23DOI: 10.1088/1361-6420/ad2531
B M Afkham, K Knudsen, A K Rasmussen, T Tarvainen
This paper considers a Bayesian approach for inclusion detection in nonlinear inverse problems using two known and popular push-forward prior distributions: the star-shaped and level set prior distributions. We analyze the convergence of the corresponding posterior distributions in a small measurement noise limit. The methodology is general; it works for priors arising from any Hölder continuous transformation of Gaussian random fields and is applicable to a range of inverse problems. The level set and star-shaped prior distributions are examples of push-forward priors under Hölder continuous transformations that take advantage of the structure of inclusion detection problems. We show that the corresponding posterior mean converges to the ground truth in a proper probabilistic sense. Numerical tests on a two-dimensional quantitative photoacoustic tomography problem showcase the approach. The results highlight the convergence properties of the posterior distributions and the ability of the methodology to detect inclusions with sufficiently regular boundaries.
{"title":"A Bayesian approach for consistent reconstruction of inclusions","authors":"B M Afkham, K Knudsen, A K Rasmussen, T Tarvainen","doi":"10.1088/1361-6420/ad2531","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2531","url":null,"abstract":"This paper considers a Bayesian approach for inclusion detection in nonlinear inverse problems using two known and popular push-forward prior distributions: the star-shaped and level set prior distributions. We analyze the convergence of the corresponding posterior distributions in a small measurement noise limit. The methodology is general; it works for priors arising from any Hölder continuous transformation of Gaussian random fields and is applicable to a range of inverse problems. The level set and star-shaped prior distributions are examples of push-forward priors under Hölder continuous transformations that take advantage of the structure of inclusion detection problems. We show that the corresponding posterior mean converges to the ground truth in a proper probabilistic sense. Numerical tests on a two-dimensional quantitative photoacoustic tomography problem showcase the approach. The results highlight the convergence properties of the posterior distributions and the ability of the methodology to detect inclusions with sufficiently regular boundaries.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140004797","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 : 2024-02-16DOI: 10.1088/1361-6420/ad2a04
A. Skarlatos, R. Miorelli, C. Reboud, Frenk van den Berg
In this contribution, the magnetic characterisation of steel strips is studied using synthetic data of field-gradient transients, which have been produced via the finite integration technique (FIT). The material law is described and parametrized using the Jiles-Atherton (JA) model. The sensitivity of relevant magnetic indicators with respect to the material parameters is then analyzed using two global methods: Sobol indices and $delta$-sensitivity indices. In order to accelerate the evaluation of these quantities, a fast metamodel is built using machine learning techniques from a simulated dataset. The solution of the inverse problem based on a tailored learning framework is tested for the different proposed identifiers, and their suitability for the magnetic characterisation of the material in question is finally discussed.
{"title":"Magnetic characterisation of steel strips using transient field measurements: global sensitivity analysis and regression from a machine-learning perspective","authors":"A. Skarlatos, R. Miorelli, C. Reboud, Frenk van den Berg","doi":"10.1088/1361-6420/ad2a04","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2a04","url":null,"abstract":"\u0000 In this contribution, the magnetic characterisation of steel strips is studied using synthetic data of field-gradient transients, which have been produced via the finite integration technique (FIT). The material law is described and parametrized using the Jiles-Atherton (JA) model. The sensitivity of relevant magnetic indicators with respect to the material parameters is then analyzed using two global methods: Sobol indices and $delta$-sensitivity indices. In order to accelerate the evaluation of these quantities, a fast metamodel is built using machine learning techniques from a simulated dataset. The solution of the inverse problem based on a tailored learning framework is tested for the different proposed identifiers, and their suitability for the magnetic characterisation of the material in question is finally discussed.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139961207","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 : 2024-02-16DOI: 10.1088/1361-6420/ad2a03
De-Han Chen, Jingzhi Li, Ye Zhang
This paper features a study of statistical inference for linear inverse problems with Gaussian noise and priors in structured Banach spaces. Employing the tools of sectorial operators and Gaussian measures on Banach spaces, we overcome the theoretical difficulty of lacking the bias-variance decomposition in Banach spaces, characterize the posterior distribution of solution though its Radon-Nikodym derivative, and derive the optimal convergence rates of the corresponding square posterior contraction and the mean integrated square error. Our theoretical findings are applied to two scenarios, specifically a Volterra integral equation and an inverse source problem governed by an elliptic partial differential equation. Our investigation demonstrates the superiority of our approach over classical results. Notably, our method achieves same order of convergence rates for solutions with reduced smoothness even in a Hilbert setting.
{"title":"A posterior contraction for Bayesian inverse problems in Banach spaces","authors":"De-Han Chen, Jingzhi Li, Ye Zhang","doi":"10.1088/1361-6420/ad2a03","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2a03","url":null,"abstract":"\u0000 This paper features a study of statistical inference for linear inverse problems with Gaussian noise and priors in structured Banach spaces. Employing the tools of sectorial operators and Gaussian measures on Banach spaces, we overcome the theoretical difficulty of lacking the bias-variance decomposition in Banach spaces, characterize the posterior distribution of solution though its Radon-Nikodym derivative, and derive the optimal convergence rates of the corresponding square posterior contraction and the mean integrated square error. Our theoretical findings are applied to two scenarios, specifically a Volterra integral equation and an inverse source problem governed by an elliptic partial differential equation. Our investigation demonstrates the superiority of our approach over classical results. Notably, our method achieves same order of convergence rates for solutions with reduced smoothness even in a Hilbert setting.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139962112","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 : 2024-02-13DOI: 10.1088/1361-6420/ad22e9
Vincenzo Mottola, Antonio Corbo Esposito, Gianpaolo Piscitelli, Antonello Tamburrino
Inverse problems, which are related to Maxwell’s equations, in the presence of nonlinear materials is a quite new topic in the literature. The lack of contributions in this area can be ascribed to the significant challenges that such problems pose. Retrieving the spatial behavior of some unknown physical property, from boundary measurements, is a nonlinear and highly ill-posed problem even in the presence of linear materials. Furthermore, this complexity grows exponentially in the presence of nonlinear materials. In the tomography of linear materials, the Monotonicity Principle (MP) is the foundation of a class of non-iterative algorithms able to guarantee excellent performances and compatibility with real-time applications. Recently, the MP has been extended to nonlinear materials under very general assumptions. Starting from the theoretical background for this extension, we develop a first real-time inversion method for the inverse obstacle problem in the presence of nonlinear materials. The proposed method is intendend for all problems governed by the quasilinear Laplace equation, i.e. static problems involving nonlinear materials. In this paper, we provide some preliminary results which give the foundation of our method and some extended numerical examples.
{"title":"Imaging of nonlinear materials via the Monotonicity Principle","authors":"Vincenzo Mottola, Antonio Corbo Esposito, Gianpaolo Piscitelli, Antonello Tamburrino","doi":"10.1088/1361-6420/ad22e9","DOIUrl":"https://doi.org/10.1088/1361-6420/ad22e9","url":null,"abstract":"Inverse problems, which are related to Maxwell’s equations, in the presence of nonlinear materials is a quite new topic in the literature. The lack of contributions in this area can be ascribed to the significant challenges that such problems pose. Retrieving the spatial behavior of some unknown physical property, from boundary measurements, is a nonlinear and highly ill-posed problem even in the presence of linear materials. Furthermore, this complexity grows exponentially in the presence of nonlinear materials. In the tomography of linear materials, the Monotonicity Principle (MP) is the foundation of a class of non-iterative algorithms able to guarantee excellent performances and compatibility with real-time applications. Recently, the MP has been extended to nonlinear materials under very general assumptions. Starting from the theoretical background for this extension, we develop a first real-time inversion method for the inverse obstacle problem in the presence of nonlinear materials. The proposed method is intendend for all problems governed by the quasilinear Laplace equation, i.e. static problems involving nonlinear materials. In this paper, we provide some preliminary results which give the foundation of our method and some extended numerical examples.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140004658","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 : 2024-02-05DOI: 10.1088/1361-6420/ad1e2c
Tiangang Cui, Gianluca Detommaso, Robert Scheichl
We present a non-trivial integration of dimension-independent likelihood-informed (DILI) MCMC (Cui et al 2016) and the multilevel MCMC (Dodwell et al 2015) to explore the hierarchy of posterior distributions. This integration offers several advantages: First, DILI-MCMC employs an intrinsic likelihood-informed subspace (LIS) (Cui et al 2014)—which involves a number of forward and adjoint model simulations—to design accelerated operator-weighted proposals. By exploiting the multilevel structure of the discretised parameters and discretised forward models, we design a Rayleigh–Ritz procedure to significantly reduce the computational effort in building the LIS and operating with DILI proposals. Second, the resulting DILI-MCMC can drastically improve the sampling efficiency of MCMC at each level, and hence reduce the integration error of the multilevel algorithm for fixed CPU time. Numerical results confirm the improved computational efficiency of the multilevel DILI approach.
我们提出了一种与维度无关的似然信息(DILI)MCMC(Cui 等人,2016 年)和多级 MCMC(Dodwell 等人,2015 年)的非难整合,以探索后验分布的层次结构。这种整合具有几个优势:首先,DILI-MCMC 采用内在似然信息子空间(LIS)(Cui 等人,2014 年)--其中涉及大量前向和邻接模型模拟--来设计加速算子加权建议。通过利用离散参数和离散前向模型的多层次结构,我们设计了一种 Rayleigh-Ritz 程序,以显著减少构建 LIS 和使用 DILI 建议的计算量。其次,由此产生的 DILI-MCMC 可以大幅提高各层次 MCMC 的采样效率,从而在 CPU 时间固定的情况下降低多层次算法的积分误差。数值结果证实了多级 DILI 方法提高了计算效率。
{"title":"Multilevel dimension-independent likelihood-informed MCMC for large-scale inverse problems","authors":"Tiangang Cui, Gianluca Detommaso, Robert Scheichl","doi":"10.1088/1361-6420/ad1e2c","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1e2c","url":null,"abstract":"We present a non-trivial integration of dimension-independent likelihood-informed (DILI) MCMC (Cui <italic toggle=\"yes\">et al</italic> 2016) and the multilevel MCMC (Dodwell <italic toggle=\"yes\">et al</italic> 2015) to explore the hierarchy of posterior distributions. This integration offers several advantages: First, DILI-MCMC employs an intrinsic <italic toggle=\"yes\">likelihood-informed subspace</italic> (LIS) (Cui <italic toggle=\"yes\">et al</italic> 2014)—which involves a number of forward and adjoint model simulations—to design accelerated operator-weighted proposals. By exploiting the multilevel structure of the discretised parameters and discretised forward models, we design a <italic toggle=\"yes\">Rayleigh–Ritz procedure</italic> to significantly reduce the computational effort in building the LIS and operating with DILI proposals. Second, the resulting DILI-MCMC can drastically improve the sampling efficiency of MCMC at each level, and hence reduce the integration error of the multilevel algorithm for fixed CPU time. Numerical results confirm the improved computational efficiency of the multilevel DILI approach.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139762540","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 : 2024-02-01DOI: 10.1088/1361-6420/ad1fe5
Anna Fitzpatrick, Molly Folino, Andrea Arnold
Many real-world systems modeled using differential equations involve unknown or uncertain parameters. Standard approaches to address parameter estimation inverse problems in this setting typically focus on estimating constants; yet some unobservable system parameters may vary with time without known evolution models. In this work, we propose a novel approximation method inspired by the Fourier series to estimate time-varying parameters (TVPs) in deterministic dynamical systems modeled with ordinary differential equations. Using ensemble Kalman filtering in conjunction with Fourier series-based approximation models, we detail two possible implementation schemes for sequentially updating the time-varying parameter estimates given noisy observations of the system states. We demonstrate the capabilities of the proposed approach in estimating periodic parameters, both when the period is known and unknown, as well as non-periodic TVPs of different forms with several computed examples using a forced harmonic oscillator. Results emphasize the importance of the frequencies and number of approximation model terms on the time-varying parameter estimates and corresponding dynamical system predictions.
{"title":"Fourier series-based approximation of time-varying parameters in ordinary differential equations","authors":"Anna Fitzpatrick, Molly Folino, Andrea Arnold","doi":"10.1088/1361-6420/ad1fe5","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1fe5","url":null,"abstract":"Many real-world systems modeled using differential equations involve unknown or uncertain parameters. Standard approaches to address parameter estimation inverse problems in this setting typically focus on estimating constants; yet some unobservable system parameters may vary with time without known evolution models. In this work, we propose a novel approximation method inspired by the Fourier series to estimate time-varying parameters (TVPs) in deterministic dynamical systems modeled with ordinary differential equations. Using ensemble Kalman filtering in conjunction with Fourier series-based approximation models, we detail two possible implementation schemes for sequentially updating the time-varying parameter estimates given noisy observations of the system states. We demonstrate the capabilities of the proposed approach in estimating periodic parameters, both when the period is known and unknown, as well as non-periodic TVPs of different forms with several computed examples using a forced harmonic oscillator. Results emphasize the importance of the frequencies and number of approximation model terms on the time-varying parameter estimates and corresponding dynamical system predictions.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139762863","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}