Pub Date : 2023-12-22DOI: 10.1088/1361-6420/ad149d
Jörn Zimmerling, Vladimir Druskin, Murthy Guddati, Elena Cherkaev, Rob Remis
We present a reduced-order model (ROM) methodology for inverse scattering problems in which the ROMs are data-driven, i.e. they are constructed directly from data gathered by sensors. Moreover, the entries of the ROM contain localised information about the coefficients of the wave equation. We solve the inverse problem by embedding the ROM in physical space. Such an approach is also followed in the theory of ‘optimal grids,’ where the ROMs are interpreted as two-point finite-difference discretisations of an underlying set of equations of a first-order continuous system on this special grid. Here, we extend this line of work to wave equations and introduce a new embedding technique, which we call Krein embedding, since it is inspired by Krein’s seminal work on vibrations of a string. In this embedding approach, an adaptive grid and a set of medium parameters can be directly extracted from a ROM and we show that several limitations of optimal grid embeddings can be avoided. Furthermore, we show how Krein embedding is connected to classical optimal grid embedding and that convergence results for optimal grids can be extended to this novel embedding approach. Finally, we also briefly discuss Krein embedding for open domains, that is, semi-infinite domains that extend to infinity in one direction.
我们提出了一种用于反向散射问题的降阶模型（ROM）方法，其中的 ROM 由数据驱动，即直接从传感器收集的数据中构建。此外，ROM 的条目包含波方程系数的局部信息。我们通过将 ROM 嵌入物理空间来解决逆问题。这种方法在 "最优网格 "理论中也有应用，在这种特殊网格上，ROM 被解释为一阶连续系统底层方程组的两点有限差分离散。在这里，我们将这一研究思路扩展到波方程，并引入了一种新的嵌入技术，我们称之为 Krein 嵌入，因为它受到了 Krein 在弦振动方面的开创性工作的启发。在这种嵌入方法中，自适应网格和介质参数集可以直接从 ROM 中提取，而且我们证明可以避免最优网格嵌入的一些限制。此外，我们还展示了 Krein 嵌入与经典最优网格嵌入之间的联系，以及最优网格的收敛结果可以扩展到这种新颖的嵌入方法。最后，我们还简要讨论了开放域的 Krein 嵌入，即在一个方向上延伸到无穷大的半无限域。
{"title":"Solving inverse scattering problems via reduced-order model embedding procedures","authors":"Jörn Zimmerling, Vladimir Druskin, Murthy Guddati, Elena Cherkaev, Rob Remis","doi":"10.1088/1361-6420/ad149d","DOIUrl":"https://doi.org/10.1088/1361-6420/ad149d","url":null,"abstract":"We present a reduced-order model (ROM) methodology for inverse scattering problems in which the ROMs are data-driven, i.e. they are constructed directly from data gathered by sensors. Moreover, the entries of the ROM contain localised information about the coefficients of the wave equation. We solve the inverse problem by embedding the ROM in physical space. Such an approach is also followed in the theory of ‘optimal grids,’ where the ROMs are interpreted as two-point finite-difference discretisations of an underlying set of equations of a first-order continuous system on this special grid. Here, we extend this line of work to wave equations and introduce a new embedding technique, which we call <italic toggle=\"yes\">Krein embedding</italic>, since it is inspired by Krein’s seminal work on vibrations of a string. In this embedding approach, an adaptive grid and a set of medium parameters can be directly extracted from a ROM and we show that several limitations of optimal grid embeddings can be avoided. Furthermore, we show how Krein embedding is connected to classical optimal grid embedding and that convergence results for optimal grids can be extended to this novel embedding approach. Finally, we also briefly discuss Krein embedding for open domains, that is, semi-infinite domains that extend to infinity in one direction.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"10 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139055341","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 : 2023-12-22DOI: 10.1088/1361-6420/ad141f
P Héas, F Cérou, M Rousset
Atmospheric motion vectors (AMVs) extracted from satellite imagery are the only wind observations with good global coverage. They are important features for feeding numerical weather prediction (NWP) models. Several Bayesian models have been proposed to estimate AMVs. Although critical for correct assimilation into NWP models, very few methods provide a thorough characterization of the estimation errors. The difficulty of estimating errors stems from the specificity of the posterior distribution, which is both very high dimensional, and highly ill-conditioned due to a singular likelihood, which becomes critical in particular in the case of missing data (unobserved pixels). Motivated by this difficult inverse problem, this work studies the evaluation of the (expected) estimation errors using gradient-based Markov chain Monte Carlo (MCMC) algorithms. The main contribution is to propose a general strategy, called here ‘chilling’, which amounts to sampling a local approximation of the posterior distribution in the neighborhood of a point estimate. From a theoretical point of view, we show that under regularity assumptions, the family of chilled posterior distributions converges in distribution as temperature decreases to an optimal Gaussian approximation at a point estimate given by the maximum a posteriori, also known as the Laplace approximation. Chilled sampling therefore provides access to this approximation generally out of reach in such high-dimensional nonlinear contexts. From an empirical perspective, we evaluate the proposed approach based on some quantitative Bayesian criteria. Our numerical simulations are performed on synthetic and real meteorological data. They reveal that not only the proposed chilling exhibits a significant gain in terms of accuracy of the AMV point estimates and of their associated expected error estimates, but also a substantial acceleration in the convergence speed of the MCMC algorithms.
{"title":"Chilled sampling for uncertainty quantification: a motivation from a meteorological inverse problem *","authors":"P Héas, F Cérou, M Rousset","doi":"10.1088/1361-6420/ad141f","DOIUrl":"https://doi.org/10.1088/1361-6420/ad141f","url":null,"abstract":"Atmospheric motion vectors (AMVs) extracted from satellite imagery are the only wind observations with good global coverage. They are important features for feeding numerical weather prediction (NWP) models. Several Bayesian models have been proposed to estimate AMVs. Although critical for correct assimilation into NWP models, very few methods provide a thorough characterization of the estimation errors. The difficulty of estimating errors stems from the specificity of the posterior distribution, which is both very high dimensional, and highly ill-conditioned due to a singular likelihood, which becomes critical in particular in the case of missing data (unobserved pixels). Motivated by this difficult inverse problem, this work studies the evaluation of the (expected) estimation errors using gradient-based Markov chain Monte Carlo (MCMC) algorithms. The main contribution is to propose a general strategy, called here ‘chilling’, which amounts to sampling a local approximation of the posterior distribution in the neighborhood of a point estimate. From a theoretical point of view, we show that under regularity assumptions, the family of chilled posterior distributions converges in distribution as temperature decreases to an optimal Gaussian approximation at a point estimate given by the maximum <italic toggle=\"yes\">a posteriori</italic>, also known as the Laplace approximation. Chilled sampling therefore provides access to this approximation generally out of reach in such high-dimensional nonlinear contexts. From an empirical perspective, we evaluate the proposed approach based on some quantitative Bayesian criteria. Our numerical simulations are performed on synthetic and real meteorological data. They reveal that not only the proposed chilling exhibits a significant gain in terms of accuracy of the AMV point estimates and of their associated expected error estimates, but also a substantial acceleration in the convergence speed of the MCMC algorithms.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"44 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139055266","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 : 2023-12-19DOI: 10.1088/1361-6420/ad0fab
Leopold Veselka, Peter Elbau, Leonidas Mindrinos, Lisa Krainz, Wolfgang Drexler
Quantitative tissue information, like the light scattering properties, is considered as a key player in the detection of cancerous cells in medical diagnosis. A promising method to obtain these data is optical coherence tomography (OCT). In this article, we will therefore discuss the refractive index reconstruction from OCT data, employing a Gaussian beam based forward model. We consider in particular samples with a layered structure, meaning that the refractive index as a function of depth is well approximated by a piecewise constant function. For the reconstruction, we present a layer-by-layer method where in every step the refractive index is obtained via a discretized least squares minimization. For an approximated form of the minimization problem, we present an existence and uniqueness result. The applicability of the proposed method is then verified by reconstructing refractive indices of layered media from both simulated and experimental OCT data.
在医学诊断中，组织的定量信息（如光散射特性）被认为是检测癌细胞的关键因素。光学相干断层扫描（OCT）是获得这些数据的一种很有前途的方法。因此，在本文中，我们将采用基于高斯光束的前向模型，讨论从 OCT 数据重建折射率的问题。我们特别考虑了具有分层结构的样本，这意味着折射率与深度的函数关系可以很好地近似为片状常数函数。为了重构，我们提出了一种逐层方法，在每一步中，折射率都是通过离散最小二乘法最小化得到的。对于最小化问题的近似形式，我们提出了存在性和唯一性结果。然后，我们从模拟和实验 OCT 数据中重建了层介质的折射率，从而验证了所提方法的适用性。
{"title":"Quantitative parameter reconstruction from optical coherence tomographic data","authors":"Leopold Veselka, Peter Elbau, Leonidas Mindrinos, Lisa Krainz, Wolfgang Drexler","doi":"10.1088/1361-6420/ad0fab","DOIUrl":"https://doi.org/10.1088/1361-6420/ad0fab","url":null,"abstract":"Quantitative tissue information, like the light scattering properties, is considered as a key player in the detection of cancerous cells in medical diagnosis. A promising method to obtain these data is optical coherence tomography (OCT). In this article, we will therefore discuss the refractive index reconstruction from OCT data, employing a Gaussian beam based forward model. We consider in particular samples with a layered structure, meaning that the refractive index as a function of depth is well approximated by a piecewise constant function. For the reconstruction, we present a layer-by-layer method where in every step the refractive index is obtained via a discretized least squares minimization. For an approximated form of the minimization problem, we present an existence and uniqueness result. The applicability of the proposed method is then verified by reconstructing refractive indices of layered media from both simulated and experimental OCT data.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"38 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139051400","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 : 2023-12-14DOI: 10.1088/1361-6420/ad11a9
Albert Fannjiang
This paper develops uniqueness theory for 3D phase retrieval with finite, discrete measurement data for strong phase objects and weak phase objects, including: (i) Unique determination of (phase) projections from diffraction patterns—General measurement schemes with coded and uncoded apertures are proposed and shown to ensure unique reduction of diffraction patterns to the phase projection for a strong phase object (respectively, the projection for a weak phase object) in each direction separately without the knowledge of relative orientations and locations. (ii) Uniqueness for 3D phase unwrapping—General conditions for unique determination of a 3D strong phase object from its phase projection data are established, including, but not limited to, random tilt schemes densely sampled from a spherical triangle of vertexes in three orthogonal directions and other deterministic tilt schemes. (iii) Uniqueness for projection tomography—Unique determination of an object of n3 voxels from generic n projections or n + 1 coded diffraction patterns is proved. This approach of reducing 3D phase retrieval to the problem of (phase) projection tomography has the practical implication of enabling classification and alignment, when relative orientations are unknown, to be carried out in terms of (phase) projections, instead of diffraction patterns. The applications with the measurement schemes such as single-axis tilt, conical tilt, dual-axis tilt, random conical tilt and general random tilt are discussed.
本文提出了利用有限、离散测量数据对强相位物体和弱相位物体进行三维相位检索的唯一性理论，包括：(i) 从衍射图样唯一确定（相位）投影--本文提出并展示了具有编码和非编码孔径的通用测量方案，以确保在不知道相对方向和位置的情况下，将衍射图样分别在每个方向上唯一还原为强相位物体的相位投影（分别为弱相位物体的投影）。(ii) 三维相位解包的唯一性--建立了从相位投影数据唯一确定三维强相位对象的一般条件，包括但不限于从三个正交方向的球面三角形顶点密集采样的随机倾斜方案和其他确定性倾斜方案。(iii) 投影层析成像的唯一性--证明了从一般 n 个投影或 n + 1 个编码衍射图样中确定 n3 个体素对象的唯一性。这种将三维相位检索简化为（相位）投影层析成像问题的方法具有实际意义，即在相对方向未知的情况下，可以根据（相位）投影而不是衍射图样进行分类和配准。本文讨论了单轴倾斜、锥形倾斜、双轴倾斜、随机锥形倾斜和一般随机倾斜等测量方案的应用。
{"title":"3D tomographic phase retrieval and unwrapping","authors":"Albert Fannjiang","doi":"10.1088/1361-6420/ad11a9","DOIUrl":"https://doi.org/10.1088/1361-6420/ad11a9","url":null,"abstract":"This paper develops uniqueness theory for 3D phase retrieval with finite, discrete measurement data for strong phase objects and weak phase objects, including: (i) <italic toggle=\"yes\">Unique determination of (phase) projections from diffraction patterns</italic>—General measurement schemes with coded and uncoded apertures are proposed and shown to ensure unique reduction of diffraction patterns to the phase projection for a strong phase object (respectively, the projection for a weak phase object) in each direction separately without the knowledge of relative orientations and locations. (ii) <italic toggle=\"yes\">Uniqueness for 3D phase unwrapping</italic>—General conditions for unique determination of a 3D strong phase object from its phase projection data are established, including, but not limited to, random tilt schemes densely sampled from a spherical triangle of vertexes in three orthogonal directions and other deterministic tilt schemes. (iii) <italic toggle=\"yes\">Uniqueness for projection tomography</italic>—Unique determination of an object of <italic toggle=\"yes\">n</italic>\u0000<sup>3</sup> voxels from generic <italic toggle=\"yes\">n</italic> projections or <italic toggle=\"yes\">n</italic> + 1 coded diffraction patterns is proved. This approach of reducing 3D phase retrieval to the problem of (phase) projection tomography has the practical implication of enabling classification and alignment, when relative orientations are unknown, to be carried out in terms of (phase) projections, instead of diffraction patterns. The applications with the measurement schemes such as single-axis tilt, conical tilt, dual-axis tilt, random conical tilt and general random tilt are discussed.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"204 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138687947","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 : 2023-12-12DOI: 10.1088/1361-6420/ad10c8
Deyue Zhang, Yan Chang, Yukun Guo
A numerical method is developed for recovering both the source locations and the obstacle from the scattered Cauchy data of the time-harmonic acoustic field. First of all, the incident and scattered components are decomposed from the coupled Cauchy data by the representation of the single-layer potentials and the solution to the resulting linear integral system. As a consequence of this decomposition, the original problem of joint inversion is reformulated into two decoupled subproblems: an inverse source problem and an inverse obstacle scattering problem. Then, two sampling-type schemes are proposed to recover the shape of the obstacle and the source locations, respectively. The sampling methods rely on the specific indicator functions defined on target-oriented probing domains of circular shape. The error estimates of the decoupling procedure are established and the asymptotic behaviors of the indicator functions are analyzed. Extensive numerical experiments are also conducted to verify the performance of the sampling schemes.
{"title":"Jointly determining the point sources and obstacle from Cauchy data","authors":"Deyue Zhang, Yan Chang, Yukun Guo","doi":"10.1088/1361-6420/ad10c8","DOIUrl":"https://doi.org/10.1088/1361-6420/ad10c8","url":null,"abstract":"A numerical method is developed for recovering both the source locations and the obstacle from the scattered Cauchy data of the time-harmonic acoustic field. First of all, the incident and scattered components are decomposed from the coupled Cauchy data by the representation of the single-layer potentials and the solution to the resulting linear integral system. As a consequence of this decomposition, the original problem of joint inversion is reformulated into two decoupled subproblems: an inverse source problem and an inverse obstacle scattering problem. Then, two sampling-type schemes are proposed to recover the shape of the obstacle and the source locations, respectively. The sampling methods rely on the specific indicator functions defined on target-oriented probing domains of circular shape. The error estimates of the decoupling procedure are established and the asymptotic behaviors of the indicator functions are analyzed. Extensive numerical experiments are also conducted to verify the performance of the sampling schemes.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"6 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688216","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 : 2023-12-12DOI: 10.1088/1361-6420/ad149e
Ye Zhang, Chuchu Chen
The a posteriori stopping rule plays a significant role in the design of efficient stochastic algorithms for various tasks in computational mathematics, such as inverse problems, optimization, and machine learning. Through the lens of classical regularization theory, this paper describes a novel analysis of Morozov’s discrepancy principle for the stochastic generalized Landweber iteration and its continuous analog of generalized stochastic asymptotical regularization. Unlike existing results relating to convergence in probability, we prove the strong convergence of the regularization error using tools from stochastic analysis, namely the theory of martingales. Numerical experiments are conducted to verify the convergence of the discrepancy principle and demonstrate two new capabilities of stochastic generalized Landweber iteration, which should also be valid for other stochastic/statistical approaches: improved accuracy by selecting the optimal path and the identification of multi-solutions by clustering samples of obtained approximate solutions.
{"title":"Stochastic linear regularization methods: random discrepancy principle and applications","authors":"Ye Zhang, Chuchu Chen","doi":"10.1088/1361-6420/ad149e","DOIUrl":"https://doi.org/10.1088/1361-6420/ad149e","url":null,"abstract":"\u0000 The a posteriori stopping rule plays a significant role in the design of efficient stochastic algorithms for various tasks in computational mathematics, such as inverse problems, optimization, and machine learning. Through the lens of classical regularization theory, this paper describes a novel analysis of Morozov’s discrepancy principle for the stochastic generalized Landweber iteration and its continuous analog of generalized stochastic asymptotical regularization. Unlike existing results relating to convergence in probability, we prove the strong convergence of the regularization error using tools from stochastic analysis, namely the theory of martingales. Numerical experiments are conducted to verify the convergence of the discrepancy principle and demonstrate two new capabilities of stochastic generalized Landweber iteration, which should also be valid for other stochastic/statistical approaches: improved accuracy by selecting the optimal path and the identification of multi-solutions by clustering samples of obtained approximate solutions.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"15 11","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139009579","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 : 2023-12-12DOI: 10.1088/1361-6420/ad14a1
Lorenzo Della Cioppa, Michela Tartaglione, Annalisa Pascarella, F. Pitolli
Electroencephalography (EEG) source imaging aims to reconstruct brain activity maps from the neuroelectric potential difference measured on the skull. To obtain the brain activity map, we need to solve an ill-posed and ill-conditioned inverse problem that requires regularization techniques to make the solution viable. When dealing with real-time applications, dimensionality reduction techniques can be used to reduce the computational load required to evaluate the numerical solution of the EEG inverse problem. To this end, in this paper we use the random dipole sampling method, in which a Monte Carlo technique is used to reduce the number of neural sources. This is equivalent to reducing the number of the unknowns in the inverse problem and can be seen as a first regularization step. Then, we solve the reduced EEG inverse problem with two popular inversion methods, the weighted Minimum Norm Estimate (wMNE) and the standardized LOw Resolution brain Electromagnetic TomogrAphy (sLORETA). The main result of this paper is the error estimates of the reconstructed activity map obtained with the randomized version of wMNE and sLORETA. Numerical experiments on synthetic EEG data demonstrate the effectiveness of the random dipole sampling method.
{"title":"Solution of the EEG inverse problem by random dipole sampling","authors":"Lorenzo Della Cioppa, Michela Tartaglione, Annalisa Pascarella, F. Pitolli","doi":"10.1088/1361-6420/ad14a1","DOIUrl":"https://doi.org/10.1088/1361-6420/ad14a1","url":null,"abstract":"\u0000 Electroencephalography (EEG) source imaging aims to reconstruct brain activity maps from the neuroelectric potential difference measured on the skull. To obtain the brain activity map, we need to solve an ill-posed and ill-conditioned inverse problem that requires regularization techniques to make the solution viable. When dealing with real-time applications, dimensionality reduction techniques can be used to reduce the computational load required to evaluate the numerical solution of the EEG inverse problem. To this end, in this paper we use the random dipole sampling method, in which a Monte Carlo technique is used to reduce the number of neural sources. This is equivalent to reducing the number of the unknowns in the inverse problem and can be seen as a first regularization step. Then, we solve the reduced EEG inverse problem with two popular inversion methods, the weighted Minimum Norm Estimate (wMNE) and the standardized LOw Resolution brain Electromagnetic TomogrAphy (sLORETA). The main result of this paper is the error estimates of the reconstructed activity map obtained with the randomized version of wMNE and sLORETA. Numerical experiments on synthetic EEG data demonstrate the effectiveness of the random dipole sampling method.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"80 2","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139008079","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 : 2023-12-11DOI: 10.1088/1361-6420/ad0fad
Xiaofan Lu, Linan Zhang, Hongjin He
Automated model selection is an important application in science and engineering. In this work, we develop a learning approach for identifying structured dynamical systems from undersampled and noisy spatiotemporal data. The learning is performed by a sparse least-squares fitting over a large set of candidate functions via a nonconvex ℓ1−ℓ2 sparse optimization solved by the alternating direction method of multipliers. We show that if the set of candidate functions forms a structured random sampling matrix of a bounded orthogonal system, the recovery is stable and the error is bounded. The learning approach is validated on synthetic data generated by the viscous Burgers’ equation and two reaction–diffusion equations. The computational results demonstrate the theoretical guarantees of success and the efficiency with respect to the number of candidate functions.
{"title":"Structured model selection via ℓ1−ℓ2 optimization","authors":"Xiaofan Lu, Linan Zhang, Hongjin He","doi":"10.1088/1361-6420/ad0fad","DOIUrl":"https://doi.org/10.1088/1361-6420/ad0fad","url":null,"abstract":"Automated model selection is an important application in science and engineering. In this work, we develop a learning approach for identifying structured dynamical systems from undersampled and noisy spatiotemporal data. The learning is performed by a sparse least-squares fitting over a large set of candidate functions via a nonconvex <inline-formula>\u0000<tex-math><?CDATA $ell_1-ell_2$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msub><mml:mi>ℓ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>ℓ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>\u0000<inline-graphic xlink:href=\"ipad0fadieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> sparse optimization solved by the alternating direction method of multipliers. We show that if the set of candidate functions forms a structured random sampling matrix of a bounded orthogonal system, the recovery is stable and the error is bounded. The learning approach is validated on synthetic data generated by the viscous Burgers’ equation and two reaction–diffusion equations. The computational results demonstrate the theoretical guarantees of success and the efficiency with respect to the number of candidate functions.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"24 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688101","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 : 2023-12-11DOI: 10.1088/1361-6420/ad1132
Remo Kretschmann, Daniel Wachsmuth, Frank Werner
Testing of hypotheses is a well studied topic in mathematical statistics. Recently, this issue has also been addressed in the context of inverse problems, where the quantity of interest is not directly accessible but only after the inversion of a (potentially) ill-posed operator. In this study, we propose a regularized approach to hypothesis testing in inverse problems in the sense that the underlying estimators (or test statistics) are allowed to be biased. Under mild source-condition type assumptions, we derive a family of tests with prescribed level α and subsequently analyze how to choose the test with maximal power out of this family. As one major result we prove that regularized testing is always at least as good as (classical) unregularized testing. Furthermore, using tools from convex optimization, we provide an adaptive test by maximizing the power functional, which then outperforms previous unregularized tests in numerical simulations by several orders of magnitude.
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Pub Date : 2023-12-11DOI: 10.1088/1361-6420/ad1131
Jun-Liang Fu, Jijun Liu
We propose a double-parameter regularization scheme for dealing with the backward diffusion process. Considering the smoothing effect of Yosida approximation for PDE, we propose to regularize this ill-posed problem by modifying original governed system in terms of a pseudoparabolic equation together with a quasi-boundary condition simultaneously, which consequently contains two regularizing parameters. Theoretically, we establish the optimal error estimates between the regularizing solution and the exact one in terms of suitable choice strategy for the regularizing parameters, under a-priori regularity assumptions on the exact solution. The a-posteriori choice strategy for the regularizing parameters based on the discrepancy principle is also studied. To weaken the heavy computational cost for solving the discrete nonsymmetric linear regularizing system by finite difference scheme, especially in higher spatial dimensional cases, the block divide-and-conquer method together with the properties of the Schur complement is applied to decompose the linear system into two half-size linear systems, one of which can be solved by the diagonalization technique, and consequently an efficient parallel-in-time algorithm originally developed for direct problem is applicable. Our proposed method is of much lower complexity than the standard solver for the corresponding linear system. Finally, some numerical examples are presented to verify the efficiency of our proposed method.
{"title":"Double-parameter regularization for solving the backward diffusion problem with parallel-in-time algorithm","authors":"Jun-Liang Fu, Jijun Liu","doi":"10.1088/1361-6420/ad1131","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1131","url":null,"abstract":"We propose a double-parameter regularization scheme for dealing with the backward diffusion process. Considering the smoothing effect of Yosida approximation for PDE, we propose to regularize this ill-posed problem by modifying original governed system in terms of a pseudoparabolic equation together with a quasi-boundary condition simultaneously, which consequently contains two regularizing parameters. Theoretically, we establish the optimal error estimates between the regularizing solution and the exact one in terms of suitable choice strategy for the regularizing parameters, under <italic toggle=\"yes\">a-priori</italic> regularity assumptions on the exact solution. The <italic toggle=\"yes\">a-posteriori</italic> choice strategy for the regularizing parameters based on the discrepancy principle is also studied. To weaken the heavy computational cost for solving the discrete nonsymmetric linear regularizing system by finite difference scheme, especially in higher spatial dimensional cases, the block divide-and-conquer method together with the properties of the Schur complement is applied to decompose the linear system into two half-size linear systems, one of which can be solved by the diagonalization technique, and consequently an efficient parallel-in-time algorithm originally developed for direct problem is applicable. Our proposed method is of much lower complexity than the standard solver for the corresponding linear system. Finally, some numerical examples are presented to verify the efficiency of our proposed method.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"42 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138688099","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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