Pub Date : 2024-03-15DOI: 10.1088/1361-6420/ad308a
Piermarco Cannarsa, Anna Doubova, Masahiro Yamamoto
We consider an inverse problem of reconstructing a degeneracy point in the diffusion coefficient in a one-dimensional parabolic equation by measuring the normal derivative on one side of the domain boundary. We analyze the sensitivity of the inverse problem to the initial data. We give sufficient conditions on the initial data for uniqueness and stability for the one-point measurement and show some examples of positive and negative results. On the other hand, we present more general uniqueness results, also for the identification of an initial data by measurements distributed over time. The proofs are based on an explicit form of the solution by means of Bessel functions of the first type. Finally, the theoretical results are supported by numerical experiments.
{"title":"Reconstruction of degenerate conductivity region for parabolic equations","authors":"Piermarco Cannarsa, Anna Doubova, Masahiro Yamamoto","doi":"10.1088/1361-6420/ad308a","DOIUrl":"https://doi.org/10.1088/1361-6420/ad308a","url":null,"abstract":"We consider an inverse problem of reconstructing a degeneracy point in the diffusion coefficient in a one-dimensional parabolic equation by measuring the normal derivative on one side of the domain boundary. We analyze the sensitivity of the inverse problem to the initial data. We give sufficient conditions on the initial data for uniqueness and stability for the one-point measurement and show some examples of positive and negative results. On the other hand, we present more general uniqueness results, also for the identification of an initial data by measurements distributed over time. The proofs are based on an explicit form of the solution by means of Bessel functions of the first type. Finally, the theoretical results are supported by numerical experiments.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"53 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315442","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 : 2024-03-15DOI: 10.1088/1361-6420/ad2d72
Zhonghua Liao, Qi Lü
Stochastic parabolic equations are widely used to model many random phenomena in natural sciences, such as the temperature distribution in a noisy medium, the dynamics of a chemical reaction in a noisy environment, or the evolution of the density of bacteria population. In many cases, the equation may involve an unknown moving boundary which could represent a change of phase, a reaction front, or an unknown population. In this paper, we focus on an inverse problem with the goal is to determine an unknown moving boundary based on data observed in a specific interior subdomain for the stochastic parabolic equation. The uniqueness of the solution of this problem is proved, and furthermore a stability estimate of log type is derived. This allows us, theoretically, to track and to monitor the behavior of the unknown boundary from observation in an arbitrary interior domain. The primary tool is a new Carleman estimate for stochastic parabolic equations. As a byproduct, we obtain a quantitative unique continuation property for stochastic parabolic equations.
{"title":"Stability estimate for an inverse stochastic parabolic problem of determining unknown time-varying boundary *","authors":"Zhonghua Liao, Qi Lü","doi":"10.1088/1361-6420/ad2d72","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2d72","url":null,"abstract":"Stochastic parabolic equations are widely used to model many random phenomena in natural sciences, such as the temperature distribution in a noisy medium, the dynamics of a chemical reaction in a noisy environment, or the evolution of the density of bacteria population. In many cases, the equation may involve an unknown moving boundary which could represent a change of phase, a reaction front, or an unknown population. In this paper, we focus on an inverse problem with the goal is to determine an unknown moving boundary based on data observed in a specific interior subdomain for the stochastic parabolic equation. The uniqueness of the solution of this problem is proved, and furthermore a stability estimate of log type is derived. This allows us, theoretically, to track and to monitor the behavior of the unknown boundary from observation in an arbitrary interior domain. The primary tool is a new Carleman estimate for stochastic parabolic equations. As a byproduct, we obtain a quantitative unique continuation property for stochastic parabolic equations.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"31 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315579","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 : 2024-03-13DOI: 10.1088/1361-6420/ad2ec9
Yasmina Zaky, Nicolas Fortino, Benoit Miramond, Jean-Yves Dauvignac
This study addresses the classification of objects using their electromagnetic signatures with convolutional neural networks (CNNs) trained on noiseless data. The singularity expansion method (SEM) was applied to establish a compact model that accurately represents the ultra-wideband scattered field (SF) of an object, independently of its orientation and observation angle. To perform the classification, we used a CNN associated with a noise-robust SEM technique to classify different objects based on their characteristic parameters. To validate this approach, we compared the performance of the classifier with and without SEM pre-processing of the SF for different noise levels and for object sizes not present in the training set. Moreover, we propose a procedure that determines the direction of the receiving antenna and orientation of an object based on the residues associated with each complex natural resonance. This classification procedure using pre-processed SEM data is promising and easy to train, especially when generalizing to object sizes not included in the training set.
本研究利用在无噪声数据上训练的卷积神经网络（CNN），利用物体的电磁特征对其进行分类。应用奇异性扩展法（SEM）建立了一个紧凑的模型，该模型能准确地表示物体的超宽带散射场（SF），不受其方向和观测角度的影响。为了进行分类，我们使用了与噪声抑制 SEM 技术相关的 CNN，根据不同物体的特征参数对其进行分类。为了验证这种方法，我们比较了分类器在对 SF 进行 SEM 预处理和未进行 SEM 预处理的情况下，针对不同噪声水平和训练集中不存在的物体大小的性能。此外，我们还提出了一种程序，可根据与每个复杂自然共振相关的残差来确定接收天线的方向和物体的方位。这种使用预处理 SEM 数据的分类程序前景广阔且易于训练，尤其是在对训练集中未包含的物体尺寸进行泛化时。
{"title":"Shape and orientation classification of objects based on their electromagnetic signatures using convolutional neural networks","authors":"Yasmina Zaky, Nicolas Fortino, Benoit Miramond, Jean-Yves Dauvignac","doi":"10.1088/1361-6420/ad2ec9","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2ec9","url":null,"abstract":"This study addresses the classification of objects using their electromagnetic signatures with convolutional neural networks (CNNs) trained on noiseless data. The singularity expansion method (SEM) was applied to establish a compact model that accurately represents the ultra-wideband scattered field (SF) of an object, independently of its orientation and observation angle. To perform the classification, we used a CNN associated with a noise-robust SEM technique to classify different objects based on their characteristic parameters. To validate this approach, we compared the performance of the classifier with and without SEM pre-processing of the SF for different noise levels and for object sizes not present in the training set. Moreover, we propose a procedure that determines the direction of the receiving antenna and orientation of an object based on the residues associated with each complex natural resonance. This classification procedure using pre-processed SEM data is promising and easy to train, especially when generalizing to object sizes not included in the training set.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"53 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315598","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 : 2024-03-12DOI: 10.1088/1361-6420/ad2ecb
Rohit Kumar Mishra, Chandni Thakkar
In this paper, a restricted transverse ray transform acting on vector and symmetric m-tensor fields is studied. We developed inversion algorithms using restricted transverse ray transform data to recover symmetric m-tensor fields in R3 and vector fields in Rn. We restrict the transverse ray transform to all lines going through a fixed curve γ that satisfies the Kirillov–Tuy condition. We show that the known restricted data can be used to reconstruct a specific weighted Radon transform of the unknown vector/tensor field’s components, which we then use to explicitly recover the unknown field.
本文研究了作用于矢量场和对称 m 张量场的受限横向射线变换。我们利用受限横射线变换数据开发了反演算法，以恢复 R3 中的对称 m 张量场和 Rn 中的矢量场。我们将横向射线变换限制为通过满足基里洛夫-图伊条件的固定曲线 γ 的所有线段。我们证明，已知的受限数据可用来重建未知矢量/张量场分量的特定加权拉顿变换，然后用它来明确恢复未知场。
{"title":"Inversion of a restricted transverse ray transform with sources on a curve","authors":"Rohit Kumar Mishra, Chandni Thakkar","doi":"10.1088/1361-6420/ad2ecb","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2ecb","url":null,"abstract":"In this paper, a restricted transverse ray transform acting on vector and symmetric <italic toggle=\"yes\">m</italic>-tensor fields is studied. We developed inversion algorithms using restricted transverse ray transform data to recover symmetric <italic toggle=\"yes\">m</italic>-tensor fields in <inline-formula>\u0000<tex-math><?CDATA $mathbb{R}^3$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math>\u0000<inline-graphic xlink:href=\"ipad2ecbieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> and vector fields in <inline-formula>\u0000<tex-math><?CDATA $mathbb{R}^n$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math>\u0000<inline-graphic xlink:href=\"ipad2ecbieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>. We restrict the transverse ray transform to all lines going through a fixed curve <italic toggle=\"yes\">γ</italic> that satisfies the Kirillov–Tuy condition. We show that the known restricted data can be used to reconstruct a specific weighted Radon transform of the unknown vector/tensor field’s components, which we then use to explicitly recover the unknown field.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"33 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315577","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 : 2024-03-12DOI: 10.1088/1361-6420/ad2aa9
Alexander V Goncharsky, Sergey Y Romanov, Sergey Y Seryozhnikov
This paper proves the theorem of uniqueness for the solution of a coefficient inverse problem for the wave equation in (with two unknown coefficients: speed of sound and absorption. The original nonlinear coefficient inverse problem is reduced to an equivalent system of two uniquely solvable linear integral equations of the first kind with respect to the sound speed and absorption coefficients. Estimates are made, substantiating the multistage method for two unknown coefficients. These estimates show that given sufficiently low frequencies and small inhomogeneities, the residual functional for the nonlinear inverse problem approaches a convex one. This solution method for nonlinear coefficient inverse problems is not linked to the limit approach as frequency tends to zero, but assumes solving the inverse problem using sufficiently low, but not zero, frequencies at the first stage. For small inhomogeneities that are typical, for instance, for medical tasks, carrying out real experiments at such frequencies does not present major difficulties. The capabilities of the method are demonstrated on a model inverse problem with unknown sound speed and absorption coefficients. The method effectively solves the nonlinear problem with parameter values typical for tomographic diagnostics of soft tissues in medicine. A resolution of approximately 2 mm was achieved using an average sounding pulse wavelength of 5 mm.
{"title":"On mathematical problems of two-coefficient inverse problems of ultrasonic tomography","authors":"Alexander V Goncharsky, Sergey Y Romanov, Sergey Y Seryozhnikov","doi":"10.1088/1361-6420/ad2aa9","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2aa9","url":null,"abstract":"This paper proves the theorem of uniqueness for the solution of a coefficient inverse problem for the wave equation in (with two unknown coefficients: speed of sound and absorption. The original nonlinear coefficient inverse problem is reduced to an equivalent system of two uniquely solvable linear integral equations of the first kind with respect to the sound speed and absorption coefficients. Estimates are made, substantiating the multistage method for two unknown coefficients. These estimates show that given sufficiently low frequencies and small inhomogeneities, the residual functional for the nonlinear inverse problem approaches a convex one. This solution method for nonlinear coefficient inverse problems is not linked to the limit approach as frequency tends to zero, but assumes solving the inverse problem using sufficiently low, but not zero, frequencies at the first stage. For small inhomogeneities that are typical, for instance, for medical tasks, carrying out real experiments at such frequencies does not present major difficulties. The capabilities of the method are demonstrated on a model inverse problem with unknown sound speed and absorption coefficients. The method effectively solves the nonlinear problem with parameter values typical for tomographic diagnostics of soft tissues in medicine. A resolution of approximately 2 mm was achieved using an average sounding pulse wavelength of 5 mm.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"63 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315428","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 : 2024-03-11DOI: 10.1088/1361-6420/ad2cf7
Zhiyao Tian, Anthony Lee, Shunhua Zhou
Bayesian curve fitting plays an important role in inverse problems, and is often addressed using the reversible jump Markov chain Monte Carlo (RJMCMC) algorithm. However, this algorithm can be computationally inefficient without appropriately tuned proposals. As a remedy, we present an adaptive RJMCMC algorithm for the curve fitting problems by extending the adaptive Metropolis sampler from a fixed-dimensional to a trans-dimensional case. In this presented algorithm, both the size and orientation of the proposal function can be automatically adjusted in the sampling process. Specifically, the curve fitting setting allows for the approximation of the posterior covariance of the a priori unknown function on a representative grid of points. This approximation facilitates the definition of efficient proposals. In addition, we introduce an auxiliary-tempered version of this algorithm via non-reversible parallel tempering. To evaluate the algorithms, we conduct numerical tests involving a series of controlled experiments. The results demonstrate that the adaptive algorithms exhibit significantly higher efficiency compared to the conventional ones. Even in cases where the posterior distribution is highly complex, leading to ineffective convergence in the auxiliary-tempered conventional RJMCMC, the proposed auxiliary-tempered adaptive RJMCMC performs satisfactorily. Furthermore, we present a realistic inverse example to test the algorithms. The successful application of the adaptive algorithm distinguishes it again from the conventional one that fails to converge effectively even after millions of iterations.
{"title":"Adaptive tempered reversible jump algorithm for Bayesian curve fitting","authors":"Zhiyao Tian, Anthony Lee, Shunhua Zhou","doi":"10.1088/1361-6420/ad2cf7","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2cf7","url":null,"abstract":"Bayesian curve fitting plays an important role in inverse problems, and is often addressed using the reversible jump Markov chain Monte Carlo (RJMCMC) algorithm. However, this algorithm can be computationally inefficient without appropriately tuned proposals. As a remedy, we present an adaptive RJMCMC algorithm for the curve fitting problems by extending the adaptive Metropolis sampler from a fixed-dimensional to a trans-dimensional case. In this presented algorithm, both the size and orientation of the proposal function can be automatically adjusted in the sampling process. Specifically, the curve fitting setting allows for the approximation of the posterior covariance of the <italic toggle=\"yes\">a priori</italic> unknown function on a representative grid of points. This approximation facilitates the definition of efficient proposals. In addition, we introduce an auxiliary-tempered version of this algorithm via non-reversible parallel tempering. To evaluate the algorithms, we conduct numerical tests involving a series of controlled experiments. The results demonstrate that the adaptive algorithms exhibit significantly higher efficiency compared to the conventional ones. Even in cases where the posterior distribution is highly complex, leading to ineffective convergence in the auxiliary-tempered conventional RJMCMC, the proposed auxiliary-tempered adaptive RJMCMC performs satisfactorily. Furthermore, we present a realistic inverse example to test the algorithms. The successful application of the adaptive algorithm distinguishes it again from the conventional one that fails to converge effectively even after millions of iterations.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"5 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315617","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 : 2024-03-08DOI: 10.1088/1361-6420/ad2cf9
Kui Ren, Nathan Soedjak
We study an inverse problem for a coupled system of semilinear Helmholtz equations where we are interested in reconstructing multiple coefficients in the system from internal data measured in applications such as thermoacoustic imaging. The system serves as a simplified model of the second harmonic generation process in a heterogeneous medium. We derive results on the uniqueness and stability of the inverse problem in the case of small boundary data based on the technique of first- and higher-order linearization. Numerical simulations are provided to illustrate the quality of reconstructions that can be expected from noisy data.
{"title":"Recovering coefficients in a system of semilinear Helmholtz equations from internal data","authors":"Kui Ren, Nathan Soedjak","doi":"10.1088/1361-6420/ad2cf9","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2cf9","url":null,"abstract":"We study an inverse problem for a coupled system of semilinear Helmholtz equations where we are interested in reconstructing multiple coefficients in the system from internal data measured in applications such as thermoacoustic imaging. The system serves as a simplified model of the second harmonic generation process in a heterogeneous medium. We derive results on the uniqueness and stability of the inverse problem in the case of small boundary data based on the technique of first- and higher-order linearization. Numerical simulations are provided to illustrate the quality of reconstructions that can be expected from noisy data.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"27 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315620","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 : 2024-03-07DOI: 10.1088/1361-6420/ad2c31
Florian Mannel, Hari Om Aggrawal, Jan Modersitzki
Many inverse problems are phrased as optimization problems in which the objective function is the sum of a data-fidelity term and a regularization. Often, the Hessian of the fidelity term is computationally unavailable while the Hessian of the regularizer allows for cheap matrix-vector products. In this paper, we study an L-BFGS method that takes advantage of this structure. We show that the method converges globally without convexity assumptions and that the convergence is linear under a Kurdyka–Łojasiewicz-type inequality. In addition, we prove linear convergence to cluster points near which the objective function is strongly convex. To the best of our knowledge, this is the first time that linear convergence of an L-BFGS method is established in a non-convex setting. The convergence analysis is carried out in infinite dimensional Hilbert space, which is appropriate for inverse problems but has not been done before. Numerical results show that the new method outperforms other structured L-BFGS methods and classical L-BFGS on non-convex real-life problems from medical image registration. It also compares favorably with classical L-BFGS on ill-conditioned quadratic model problems. An implementation of the method is freely available.
{"title":"A structured L-BFGS method and its application to inverse problems","authors":"Florian Mannel, Hari Om Aggrawal, Jan Modersitzki","doi":"10.1088/1361-6420/ad2c31","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2c31","url":null,"abstract":"Many inverse problems are phrased as optimization problems in which the objective function is the sum of a data-fidelity term and a regularization. Often, the Hessian of the fidelity term is computationally unavailable while the Hessian of the regularizer allows for cheap matrix-vector products. In this paper, we study an L-BFGS method that takes advantage of this structure. We show that the method converges globally without convexity assumptions and that the convergence is linear under a Kurdyka–Łojasiewicz-type inequality. In addition, we prove linear convergence to cluster points near which the objective function is strongly convex. To the best of our knowledge, this is the first time that linear convergence of an L-BFGS method is established in a non-convex setting. The convergence analysis is carried out in infinite dimensional Hilbert space, which is appropriate for inverse problems but has not been done before. Numerical results show that the new method outperforms other structured L-BFGS methods and classical L-BFGS on non-convex real-life problems from medical image registration. It also compares favorably with classical L-BFGS on ill-conditioned quadratic model problems. An implementation of the method is freely available.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"19 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315621","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 : 2024-03-06DOI: 10.1088/1361-6420/ad2905
Tram Thi Ngoc Nguyen
We investigate the ill-posed inverse problem of recovering unknown spatially dependent parameters in nonlinear evolution partial differential equations (PDEs). We propose a bi-level Landweber scheme, where the upper-level parameter reconstruction embeds a lower-level state approximation. This can be seen as combining the classical reduced setting and the newer all-at-once setting, allowing us to, respectively, utilize well-posedness of the parameter-to-state map, and to bypass having to solve nonlinear PDEs exactly. Using this, we derive stopping rules for lower- and upper-level iterations and convergence of the bi-level method. We discuss application to parameter identification for the Landau–Lifshitz–Gilbert equation in magnetic particle imaging.
{"title":"Bi-level iterative regularization for inverse problems in nonlinear PDEs","authors":"Tram Thi Ngoc Nguyen","doi":"10.1088/1361-6420/ad2905","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2905","url":null,"abstract":"We investigate the ill-posed inverse problem of recovering unknown spatially dependent parameters in nonlinear evolution partial differential equations (PDEs). We propose a bi-level Landweber scheme, where the upper-level parameter reconstruction embeds a lower-level state approximation. This can be seen as combining the classical reduced setting and the newer all-at-once setting, allowing us to, respectively, utilize well-posedness of the parameter-to-state map, and to bypass having to solve nonlinear PDEs exactly. Using this, we derive stopping rules for lower- and upper-level iterations and convergence of the bi-level method. We discuss application to parameter identification for the Landau–Lifshitz–Gilbert equation in magnetic particle imaging.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"749 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315290","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 : 2024-03-06DOI: 10.1088/1361-6420/ad2aaa
Clemens Arndt, Sören Dittmer, Nick Heilenkötter, Meira Iske, Tobias Kluth, Judith Nickel
Learning-based methods for inverse problems, adapting to the data’s inherent structure, have become ubiquitous in the last decade. Besides empirical investigations of their often remarkable performance, an increasing number of works address the issue of theoretical guarantees. Recently, Arndt et al (2023 Inverse Problems39 125018) exploited invertible residual networks (iResNets) to learn provably convergent regularizations given reasonable assumptions. They enforced these guarantees by approximating the linear forward operator with an iResNet. Supervised training on relevant samples introduces data dependency into the approach. An open question in this context is to which extent the data’s inherent structure influences the training outcome, i.e. the learned reconstruction scheme. Here, we address this delicate interplay of training design and data dependency from a Bayesian perspective and shed light on opportunities and limitations. We resolve these limitations by analyzing reconstruction-based training of the inverses of iResNets, where we show that this optimization strategy introduces a level of data-dependency that cannot be achieved by approximation training. We further provide and discuss a series of numerical experiments underpinning and extending the theoretical findings.
{"title":"Bayesian view on the training of invertible residual networks for solving linear inverse problems *","authors":"Clemens Arndt, Sören Dittmer, Nick Heilenkötter, Meira Iske, Tobias Kluth, Judith Nickel","doi":"10.1088/1361-6420/ad2aaa","DOIUrl":"https://doi.org/10.1088/1361-6420/ad2aaa","url":null,"abstract":"Learning-based methods for inverse problems, adapting to the data’s inherent structure, have become ubiquitous in the last decade. Besides empirical investigations of their often remarkable performance, an increasing number of works address the issue of theoretical guarantees. Recently, Arndt <italic toggle=\"yes\">et al</italic> (2023 <italic toggle=\"yes\">Inverse Problems</italic>\u0000<bold>39</bold> 125018) exploited invertible residual networks (iResNets) to learn provably convergent regularizations given reasonable assumptions. They enforced these guarantees by approximating the linear forward operator with an iResNet. Supervised training on relevant samples introduces data dependency into the approach. An open question in this context is to which extent the data’s inherent structure influences the training outcome, i.e. the learned reconstruction scheme. Here, we address this delicate interplay of training design and data dependency from a Bayesian perspective and shed light on opportunities and limitations. We resolve these limitations by analyzing reconstruction-based training of the inverses of iResNets, where we show that this optimization strategy introduces a level of data-dependency that cannot be achieved by approximation training. We further provide and discuss a series of numerical experiments underpinning and extending the theoretical findings.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"107 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315293","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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