In this article, we introduce and study various V-line transforms (VLTs) defined on symmetric 2-tensor fields in R2. The operators of interest include the longitudinal, transverse, and mixed VLTs, their integral moments, and the star transform. With the exception of the star transform, all these operators are natural generalizations to the broken-ray trajectories of the corresponding well-studied concepts defined for straight-line paths of integration. We characterize the kernels of the VLTs and derive exact formulas for reconstruction of tensor fields from various combinations of these transforms. The star transform on tensor fields is an extension of the corresponding concepts that have been previously studied on vector fields and scalar fields (functions). We describe all injective configurations of the star transform on symmetric 2-tensor fields and derive an exact, closed-form inversion formula for that operator.
{"title":"V-line 2-tensor tomography in the plane","authors":"Gaik Ambartsoumian, Rohit Kumar Mishra, Indrani Zamindar","doi":"10.1088/1361-6420/ad1f83","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1f83","url":null,"abstract":"In this article, we introduce and study various V-line transforms (VLTs) defined on symmetric 2-tensor fields in <inline-formula>\u0000<tex-math><?CDATA $mathbb{R}^2$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math>\u0000<inline-graphic xlink:href=\"ipad1f83ieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>. The operators of interest include the longitudinal, transverse, and mixed VLTs, their integral moments, and the star transform. With the exception of the star transform, all these operators are natural generalizations to the broken-ray trajectories of the corresponding well-studied concepts defined for straight-line paths of integration. We characterize the kernels of the VLTs and derive exact formulas for reconstruction of tensor fields from various combinations of these transforms. The star transform on tensor fields is an extension of the corresponding concepts that have been previously studied on vector fields and scalar fields (functions). We describe all injective configurations of the star transform on symmetric 2-tensor fields and derive an exact, closed-form inversion formula for that operator.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"8 2 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139762535","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 : 2024-01-16DOI: 10.1088/1361-6420/ad1609
Pierre Maréchal, Faouzi Triki, Walter C Simo Tao Lee
In this paper we study the inverse Laplace transform. We first derive a new global logarithmic stability estimate that shows that the inversion is severely ill-posed. Then we propose a regularization method to compute the inverse Laplace transform using the concept of mollification. Taking into account the exponential instability we derive a criterion for selection of the regularization parameter. We show that by taking the optimal value of this parameter we improve significantly the convergence of the method. Finally, making use of the holomorphic extension of the Laplace transform, we suggest a new PDEs based numerical method for the computation of the solution. The effectiveness of the proposed regularization method is demonstrated through several numerical examples.
{"title":"Regularization of the inverse Laplace transform by mollification","authors":"Pierre Maréchal, Faouzi Triki, Walter C Simo Tao Lee","doi":"10.1088/1361-6420/ad1609","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1609","url":null,"abstract":"In this paper we study the inverse Laplace transform. We first derive a new global logarithmic stability estimate that shows that the inversion is severely ill-posed. Then we propose a regularization method to compute the inverse Laplace transform using the concept of mollification. Taking into account the exponential instability we derive a criterion for selection of the regularization parameter. We show that by taking the optimal value of this parameter we improve significantly the convergence of the method. Finally, making use of the holomorphic extension of the Laplace transform, we suggest a new PDEs based numerical method for the computation of the solution. The effectiveness of the proposed regularization method is demonstrated through several numerical examples.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139506590","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 : 2024-01-08DOI: 10.1088/1361-6420/ad149f
Hongyu Liu, Catharine W K Lo
In this paper, we consider the inverse problem of determining some coefficients within a coupled nonlinear parabolic system, through boundary observation of its non-negative solutions. In the physical setup, the non-negative solutions represent certain probability densities in different contexts. We innovate the successive linearisation method by further developing a high-order variation scheme which can both ensure the positivity of the solutions and effectively tackle the nonlinear inverse problem. This enables us to establish several novel unique identifiability results for the inverse problem in a rather general setup. For a theoretical perspective, our study addresses an important topic in partial differential equation (PDE) analysis on how to characterise the function spaces generated by the products of non-positive solutions of parabolic PDEs. As a typical and practically interesting application, we apply our general results to inverse problems for ecological population models, where the positive solutions signify the population densities.
{"title":"Determining a parabolic system by boundary observation of its non-negative solutions with biological applications","authors":"Hongyu Liu, Catharine W K Lo","doi":"10.1088/1361-6420/ad149f","DOIUrl":"https://doi.org/10.1088/1361-6420/ad149f","url":null,"abstract":"In this paper, we consider the inverse problem of determining some coefficients within a coupled nonlinear parabolic system, through boundary observation of its non-negative solutions. In the physical setup, the non-negative solutions represent certain probability densities in different contexts. We innovate the successive linearisation method by further developing a high-order variation scheme which can both ensure the positivity of the solutions and effectively tackle the nonlinear inverse problem. This enables us to establish several novel unique identifiability results for the inverse problem in a rather general setup. For a theoretical perspective, our study addresses an important topic in partial differential equation (PDE) analysis on how to characterise the function spaces generated by the products of non-positive solutions of parabolic PDEs. As a typical and practically interesting application, we apply our general results to inverse problems for ecological population models, where the positive solutions signify the population densities.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139506406","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 : 2024-01-03DOI: 10.1088/1361-6420/ad1a3c
Zhuoxu Cui, Qingyong Zhu, Jing Cheng, Bo Zhang, Dong Liang
Deep unfolding methods have gained significant popularity in the field of inverse problems as they have driven the design of deep neural networks (DNNs) using iterative algorithms. In contrast to general DNNs, unfolding methods offer improved interpretability and performance. However, their theoretical stability or regularity in solving inverse problems remains subject to certain limitations. To address this, we reevaluate unfolded DNNs and observe that their algorithmically-driven cascading structure exhibits a closer resemblance to iterative regularization. Recognizing this, we propose a modified training approach and configure termination criteria for unfolded DNNs, thereby establishing the unfolding method as an iterative regularization technique. Specifically, our method involves the joint learning of a convex penalty function using an input-convex neural network (ICNN) to quantify distance to a real data manifold. Then, we train a DNN unfolded from the proximal gradient descent algorithm, incorporating this learned penalty. Additionally, we introduce a new termination criterion for the unfolded DNN. Under the assumption that the real data manifold intersects the solutions of the inverse problem with a unique real solution, even when measurements contain perturbations, we provide a theoretical proof of the stable convergence of the unfolded DNN to this solution. Furthermore, we demonstrate with an example of MRI reconstruction that the proposed method outperforms original unfolding methods and traditional regularization methods in terms of reconstruction quality, stability, and convergence speed.
{"title":"Deep unfolding as iterative regularization for imaging inverse problems","authors":"Zhuoxu Cui, Qingyong Zhu, Jing Cheng, Bo Zhang, Dong Liang","doi":"10.1088/1361-6420/ad1a3c","DOIUrl":"https://doi.org/10.1088/1361-6420/ad1a3c","url":null,"abstract":"\u0000 Deep unfolding methods have gained significant popularity in the field of inverse problems as they have driven the design of deep neural networks (DNNs) using iterative algorithms. In contrast to general DNNs, unfolding methods offer improved interpretability and performance. However, their theoretical stability or regularity in solving inverse problems remains subject to certain limitations. To address this, we reevaluate unfolded DNNs and observe that their algorithmically-driven cascading structure exhibits a closer resemblance to iterative regularization. Recognizing this, we propose a modified training approach and configure termination criteria for unfolded DNNs, thereby establishing the unfolding method as an iterative regularization technique. Specifically, our method involves the joint learning of a convex penalty function using an input-convex neural network (ICNN) to quantify distance to a real data manifold. Then, we train a DNN unfolded from the proximal gradient descent algorithm, incorporating this learned penalty. Additionally, we introduce a new termination criterion for the unfolded DNN. Under the assumption that the real data manifold intersects the solutions of the inverse problem with a unique real solution, even when measurements contain perturbations, we provide a theoretical proof of the stable convergence of the unfolded DNN to this solution. Furthermore, we demonstrate with an example of MRI reconstruction that the proposed method outperforms original unfolding methods and traditional regularization methods in terms of reconstruction quality, stability, and convergence speed.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"14 6","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139451521","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 : 2023-12-28DOI: 10.1088/1361-6420/ad14a0
Bangti Jin, Kwancheol Shin, Zhi Zhou
In this work we investigate an inverse problem of recovering a time-dependent potential in a semilinear subdiffusion model from an integral measurement of the solution over the domain. The model involves the Djrbashian–Caputo fractional derivative in time. Theoretically, we prove a novel conditional Lipschitz stability result, and numerically, we develop an easy-to-implement fixed point iteration for recovering the unknown coefficient. In addition, we establish rigorous error bounds on the discrete approximation. These results are obtained by crucially using smoothing properties of the solution operators and suitable choice of a weighted Lp(0,T) norm. The efficiency and accuracy of the scheme are showcased on several numerical experiments in one- and two-dimensions.
{"title":"Numerical recovery of a time-dependent potential in subdiffusion *","authors":"Bangti Jin, Kwancheol Shin, Zhi Zhou","doi":"10.1088/1361-6420/ad14a0","DOIUrl":"https://doi.org/10.1088/1361-6420/ad14a0","url":null,"abstract":"In this work we investigate an inverse problem of recovering a time-dependent potential in a semilinear subdiffusion model from an integral measurement of the solution over the domain. The model involves the Djrbashian–Caputo fractional derivative in time. Theoretically, we prove a novel conditional Lipschitz stability result, and numerically, we develop an easy-to-implement fixed point iteration for recovering the unknown coefficient. In addition, we establish rigorous error bounds on the discrete approximation. These results are obtained by crucially using smoothing properties of the solution operators and suitable choice of a weighted <inline-formula>\u0000<tex-math><?CDATA $L^p(0,T)$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msup><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math>\u0000<inline-graphic xlink:href=\"ipad14a0ieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> norm. The efficiency and accuracy of the scheme are showcased on several numerical experiments in one- and two-dimensions.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"30 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139095803","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 : 2023-12-27DOI: 10.1088/1361-6420/ad12e0
Housen Li, Frank Werner
We consider statistical linear inverse problems in separable Hilbert spaces and filter-based reconstruction methods of the form <inline-formula> <tex-math><?CDATA $widehat f_alpha = q_alpha left(T,^*Tright)T,^*Y$?></tex-math> <mml:math overflow="scroll"><mml:msub><mml:mover><mml:mi>f</mml:mi><mml:mo>ˆ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mfenced close=")" open="("><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mi>T</mml:mi></mml:mrow></mml:mfenced><mml:msup><mml:mi>T</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mi>Y</mml:mi></mml:math> <inline-graphic xlink:href="ipad12e0ieqn1.gif" xlink:type="simple"></inline-graphic> </inline-formula>, where <italic toggle="yes">Y</italic> is the available data, <italic toggle="yes">T</italic> the forward operator, <inline-formula> <tex-math><?CDATA $left(q_alpharight)_{alpha in mathcal A}$?></tex-math> <mml:math overflow="scroll"><mml:msub><mml:mfenced close=")" open="("><mml:msub><mml:mi>q</mml:mi><mml:mi>α</mml:mi></mml:msub></mml:mfenced><mml:mrow><mml:mi>α</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math> <inline-graphic xlink:href="ipad12e0ieqn2.gif" xlink:type="simple"></inline-graphic> </inline-formula> an ordered filter, and <italic toggle="yes">α</italic> > 0 a regularization parameter. Whenever such a method is used in practice, <italic toggle="yes">α</italic> has to be appropriately chosen. Typically, the aim is to find or at least approximate the best possible <italic toggle="yes">α</italic> in the sense that mean squared error (MSE) <inline-formula> <tex-math><?CDATA $mathbb{E} [Vert widehat f_alpha - f^daggerVert^2]$?></tex-math> <mml:math overflow="scroll"><mml:mrow><mml:mi mathvariant="double-struck">E</mml:mi></mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mo>∥</mml:mo><mml:msub><mml:mover><mml:mi>f</mml:mi><mml:mo>ˆ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msup><mml:mi>f</mml:mi><mml:mo>†</mml:mo></mml:msup><mml:mrow><mml:msup><mml:mo>∥</mml:mo><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:math> <inline-graphic xlink:href="ipad12e0ieqn3.gif" xlink:type="simple"></inline-graphic> </inline-formula> w.r.t. the true solution <inline-formula> <tex-math><?CDATA $f^dagger$?></tex-math> <mml:math overflow="scroll"><mml:msup><mml:mi>f</mml:mi><mml:mo>†</mml:mo></mml:msup></mml:math> <inline-graphic xlink:href="ipad12e0ieqn4.gif" xlink:type="simple"></inline-graphic> </inline-formula> is minimized. In this paper, we introduce the Sharp Optimal Lepskiĭ-Inspired Tuning (SOLIT) method, which yields an <italic toggle="yes">a posteriori</italic> parameter choice rule ensuring adaptive minimax rates of convergence. It depends only on <italic toggle="yes">Y</italic> and the noise level <italic toggle="yes">σ</italic> as well as the operator <italic toggle="yes">T</italic> and th
{"title":"Adaptive minimax optimality in statistical inverse problems via SOLIT—Sharp Optimal Lepskiĭ-Inspired Tuning","authors":"Housen Li, Frank Werner","doi":"10.1088/1361-6420/ad12e0","DOIUrl":"https://doi.org/10.1088/1361-6420/ad12e0","url":null,"abstract":"We consider statistical linear inverse problems in separable Hilbert spaces and filter-based reconstruction methods of the form <inline-formula>\u0000<tex-math><?CDATA $widehat f_alpha = q_alpha left(T,^*Tright)T,^*Y$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msub><mml:mover><mml:mi>f</mml:mi><mml:mo>ˆ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mi>T</mml:mi></mml:mrow></mml:mfenced><mml:msup><mml:mi>T</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mi>Y</mml:mi></mml:math>\u0000<inline-graphic xlink:href=\"ipad12e0ieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula>, where <italic toggle=\"yes\">Y</italic> is the available data, <italic toggle=\"yes\">T</italic> the forward operator, <inline-formula>\u0000<tex-math><?CDATA $left(q_alpharight)_{alpha in mathcal A}$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>q</mml:mi><mml:mi>α</mml:mi></mml:msub></mml:mfenced><mml:mrow><mml:mi>α</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math>\u0000<inline-graphic xlink:href=\"ipad12e0ieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> an ordered filter, and <italic toggle=\"yes\">α</italic> > 0 a regularization parameter. Whenever such a method is used in practice, <italic toggle=\"yes\">α</italic> has to be appropriately chosen. Typically, the aim is to find or at least approximate the best possible <italic toggle=\"yes\">α</italic> in the sense that mean squared error (MSE) <inline-formula>\u0000<tex-math><?CDATA $mathbb{E} [Vert widehat f_alpha - f^daggerVert^2]$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:mrow><mml:mi mathvariant=\"double-struck\">E</mml:mi></mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>∥</mml:mo><mml:msub><mml:mover><mml:mi>f</mml:mi><mml:mo>ˆ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msup><mml:mi>f</mml:mi><mml:mo>†</mml:mo></mml:msup><mml:mrow><mml:msup><mml:mo>∥</mml:mo><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:math>\u0000<inline-graphic xlink:href=\"ipad12e0ieqn3.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> w.r.t. the true solution <inline-formula>\u0000<tex-math><?CDATA $f^dagger$?></tex-math>\u0000<mml:math overflow=\"scroll\"><mml:msup><mml:mi>f</mml:mi><mml:mo>†</mml:mo></mml:msup></mml:math>\u0000<inline-graphic xlink:href=\"ipad12e0ieqn4.gif\" xlink:type=\"simple\"></inline-graphic>\u0000</inline-formula> is minimized. In this paper, we introduce the Sharp Optimal Lepskiĭ-Inspired Tuning (SOLIT) method, which yields an <italic toggle=\"yes\">a posteriori</italic> parameter choice rule ensuring adaptive minimax rates of convergence. It depends only on <italic toggle=\"yes\">Y</italic> and the noise level <italic toggle=\"yes\">σ</italic> as well as the operator <italic toggle=\"yes\">T</italic> and th","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":"43 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139051602","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/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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：