Martin Benning, Tatiana A Bubba, Luca Ratti, Danilo Riccio
Source conditions are a key tool in regularisation theory that are needed to derive error estimates and convergence rates for ill-posed inverse problems. In this paper, we provide a recipe to practically compute source condition elements as the solution of convex minimisation problems that can be solved with first-order algorithms. We demonstrate the validity of our approach by testing it on two inverse problem case studies in machine learning and image processing: sparse coefficient estimation of a polynomial via LASSO regression and recovering an image from a subset of the coefficients of its discrete Fourier transform. We further demonstrate that the proposed approach can easily be modified to solve the machine learning task of identifying the optimal sampling pattern in the Fourier domain for a given image and variational regularisation method, which has applications in the context of sparsity promoting reconstruction from magnetic resonance imaging data.
{"title":"Trust your source: quantifying source condition elements for variational regularisation methods","authors":"Martin Benning, Tatiana A Bubba, Luca Ratti, Danilo Riccio","doi":"10.1093/imamat/hxae008","DOIUrl":"https://doi.org/10.1093/imamat/hxae008","url":null,"abstract":"Source conditions are a key tool in regularisation theory that are needed to derive error estimates and convergence rates for ill-posed inverse problems. In this paper, we provide a recipe to practically compute source condition elements as the solution of convex minimisation problems that can be solved with first-order algorithms. We demonstrate the validity of our approach by testing it on two inverse problem case studies in machine learning and image processing: sparse coefficient estimation of a polynomial via LASSO regression and recovering an image from a subset of the coefficients of its discrete Fourier transform. We further demonstrate that the proposed approach can easily be modified to solve the machine learning task of identifying the optimal sampling pattern in the Fourier domain for a given image and variational regularisation method, which has applications in the context of sparsity promoting reconstruction from magnetic resonance imaging data.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140117449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper investigates a two-species chemotaxis-fluid system with indirect pursuit-evasion interaction in a bounded domain with smooth boundary. Under suitably regular initial data and no-flux/no-flux/no-flux/no-flux/Dirichlet boundary conditions, we prove that the system possesses a global bounded classical solution in the two-dimensional and three-dimensional cases. Our results extend the result obtained in previously known ones and partly result is new.
{"title":"Global existence and boundedness in a two-species chemotaxis-fluid system with indirect pursuit-evasion interaction","authors":"Chao Liu, Bin Liu","doi":"10.1093/imamat/hxae009","DOIUrl":"https://doi.org/10.1093/imamat/hxae009","url":null,"abstract":"This paper investigates a two-species chemotaxis-fluid system with indirect pursuit-evasion interaction in a bounded domain with smooth boundary. Under suitably regular initial data and no-flux/no-flux/no-flux/no-flux/Dirichlet boundary conditions, we prove that the system possesses a global bounded classical solution in the two-dimensional and three-dimensional cases. Our results extend the result obtained in previously known ones and partly result is new.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140026269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A comparative analysis is performed on the effects of heterogeneity in both linear and nonlinear material characteristics of half-space on the propagation of bright and dark solitary Love waves in a nonlinear layered half-space. The layer is assumed to be homogeneous, nonlinear, elastic while the half-space is vertically heterogeneous. The problem is formulated for two types of elastic materials, incompressible and generalized neo-Hookean materials, and the differences caused by the two materials in the formulation are revealed. For the media composed of generalized neo-Hookean materials, a nonlinear Schrödinger (NLS) equation describing the self interaction of Love waves is obtained by utilizing multiple scales method. The differences in the effects of linear and nonlinear material properties of the half-space on both the existence and nonlinear evolution of bright and dark solitary Love waves are compared graphically.
{"title":"The comparison between effects of heterogeneous and homogeneous half-spaces underlying homogeneous layer on solitary Love waves","authors":"Ekin Deliktas-Ozdemir","doi":"10.1093/imamat/hxae007","DOIUrl":"https://doi.org/10.1093/imamat/hxae007","url":null,"abstract":"A comparative analysis is performed on the effects of heterogeneity in both linear and nonlinear material characteristics of half-space on the propagation of bright and dark solitary Love waves in a nonlinear layered half-space. The layer is assumed to be homogeneous, nonlinear, elastic while the half-space is vertically heterogeneous. The problem is formulated for two types of elastic materials, incompressible and generalized neo-Hookean materials, and the differences caused by the two materials in the formulation are revealed. For the media composed of generalized neo-Hookean materials, a nonlinear Schrödinger (NLS) equation describing the self interaction of Love waves is obtained by utilizing multiple scales method. The differences in the effects of linear and nonlinear material properties of the half-space on both the existence and nonlinear evolution of bright and dark solitary Love waves are compared graphically.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139954311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider gradient flows of surface energies which depend on the surface by a parameterization and on a tangential tensor field. The flow allows for dissipation by evolving the parameterization and the tensor field simultaneously. This requires the choice of a notation for independence. We introduce different gauges of surface independence and show their consequences for the evolution. In order to guarantee a decrease in energy, the gauge of surface independence and the time derivative have to be chosen consistently. We demonstrate the results for a surface Frank-Oseen-Hilfrich energy.
{"title":"Tangential Tensor Fields on Deformable Surfaces – How to Derive Consistent L2-Gradient Flows","authors":"Ingo Nitschke, Souhayl Sadik, Axel Voigt","doi":"10.1093/imamat/hxae006","DOIUrl":"https://doi.org/10.1093/imamat/hxae006","url":null,"abstract":"We consider gradient flows of surface energies which depend on the surface by a parameterization and on a tangential tensor field. The flow allows for dissipation by evolving the parameterization and the tensor field simultaneously. This requires the choice of a notation for independence. We introduce different gauges of surface independence and show their consequences for the evolution. In order to guarantee a decrease in energy, the gauge of surface independence and the time derivative have to be chosen consistently. We demonstrate the results for a surface Frank-Oseen-Hilfrich energy.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139926822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Siiri Rautio, Rashmi Murthy, Tatiana A Bubba, Matti Lassas, Samuli Siltanen
Limited-angle tomography is a highly ill-posed linear inverse problem. It arises in many applications, such as digital breast tomosynthesis. Reconstructions from limited-angle data typically suffer from severe stretching of features along the central direction of projections, leading to poor separation between slices perpendicular to the central direction. In this paper, a new method is introduced, based on machine learning and geometry, producing an estimate for interfaces between regions of different X-ray attenuation. The estimate can be presented on top of the reconstruction, indicating more reliably the separation between features. The method uses directional edge detection, implemented using complex wavelets and enhanced with morphological operations. By using convolutional neural networks, the visible part of the singular support is first extracted and then extended to the full domain, filling in the parts of the singular support that would otherwise be hidden due to the lack of measurement directions.
限角断层扫描是一个高难度线性逆问题。它出现在许多应用中，如数字乳腺断层合成。有限角度数据的重建通常会受到沿投影中心方向特征严重拉伸的影响，导致垂直于中心方向的切片之间分离不佳。本文介绍了一种基于机器学习和几何学的新方法，可对不同 X 射线衰减区域之间的界面进行估计。该估计值可以在重建的基础上显示，从而更可靠地显示特征之间的分离情况。该方法使用复杂小波实现定向边缘检测，并通过形态学操作进行增强。通过使用卷积神经网络，首先提取奇异支撑的可见部分，然后扩展到全域，填补奇异支撑中由于缺乏测量方向而被隐藏的部分。
{"title":"Learning a microlocal prior for limited-angle tomography","authors":"Siiri Rautio, Rashmi Murthy, Tatiana A Bubba, Matti Lassas, Samuli Siltanen","doi":"10.1093/imamat/hxae005","DOIUrl":"https://doi.org/10.1093/imamat/hxae005","url":null,"abstract":"Limited-angle tomography is a highly ill-posed linear inverse problem. It arises in many applications, such as digital breast tomosynthesis. Reconstructions from limited-angle data typically suffer from severe stretching of features along the central direction of projections, leading to poor separation between slices perpendicular to the central direction. In this paper, a new method is introduced, based on machine learning and geometry, producing an estimate for interfaces between regions of different X-ray attenuation. The estimate can be presented on top of the reconstruction, indicating more reliably the separation between features. The method uses directional edge detection, implemented using complex wavelets and enhanced with morphological operations. By using convolutional neural networks, the visible part of the singular support is first extracted and then extended to the full domain, filling in the parts of the singular support that would otherwise be hidden due to the lack of measurement directions.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139753193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lyndon Koens, Rohan Vernekar, Timm Krüger, Maciej Lisicki, David W Inglis
The slow viscous flow through a doubly-periodic array of cylinders does not have an analytical solution. However, as a reduced model for the flow within fibrous porous media and microfluidic arrays, this solution is important for many real-world systems. We asymptotically determine the flow around a general rectangular doubly-periodic array of infinite slender cylinders, extending the existing asymptotic solution for square arrays. The flow in the cell is represented by a collection of doubly-periodic, rapidly-convergent two-dimensional singularity solutions, and the boundary condition on the surface of the cylinder is solved asymptotically in powers of the cylinder radius. The asymptotic solution provides an easily computed closed-form estimate for the flow and forces as a function of the radius and the dimensions of the cell. The force is compared to results from lattice-Boltzmann simulations of low-Reynolds-number flows in the same geometry, and the accuracy of the no-slip condition on the surface of the cylinder, predicted by the asymptotic theory, is assessed. Finally, the behaviour of the flow, flux, force and effective permeability of the cell is investigated as a function of the geometric parameters. The structure of the asymptotic permeability is consistent with previous single-geometry predictions but provides a closed-form estimate for how the aspect ratio of the cell changes the leading-order behaviour. These models could be used to help understand the flows within porous systems composed of fibres and systems involving periodic arrays such as systems based on deterministic lateral displacement.
{"title":"The slow viscous flow around a general rectangular doubly-periodic arrays of infinite slender cylinders","authors":"Lyndon Koens, Rohan Vernekar, Timm Krüger, Maciej Lisicki, David W Inglis","doi":"10.1093/imamat/hxae003","DOIUrl":"https://doi.org/10.1093/imamat/hxae003","url":null,"abstract":"The slow viscous flow through a doubly-periodic array of cylinders does not have an analytical solution. However, as a reduced model for the flow within fibrous porous media and microfluidic arrays, this solution is important for many real-world systems. We asymptotically determine the flow around a general rectangular doubly-periodic array of infinite slender cylinders, extending the existing asymptotic solution for square arrays. The flow in the cell is represented by a collection of doubly-periodic, rapidly-convergent two-dimensional singularity solutions, and the boundary condition on the surface of the cylinder is solved asymptotically in powers of the cylinder radius. The asymptotic solution provides an easily computed closed-form estimate for the flow and forces as a function of the radius and the dimensions of the cell. The force is compared to results from lattice-Boltzmann simulations of low-Reynolds-number flows in the same geometry, and the accuracy of the no-slip condition on the surface of the cylinder, predicted by the asymptotic theory, is assessed. Finally, the behaviour of the flow, flux, force and effective permeability of the cell is investigated as a function of the geometric parameters. The structure of the asymptotic permeability is consistent with previous single-geometry predictions but provides a closed-form estimate for how the aspect ratio of the cell changes the leading-order behaviour. These models could be used to help understand the flows within porous systems composed of fibres and systems involving periodic arrays such as systems based on deterministic lateral displacement.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139661342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
When a hazardous chemical soaks into a porous material such as a concrete floor, it can be difficult to remove. One approach is chemical decontamination, where a cleanser is added to react with and neutralise the contaminating agent. The goal of this paper is to investigate the reaction dynamics and the factors that affect the efficacy of the decontamination procedure. We consider a one-dimensional porous medium initially saturated with an oil-based agent. An aqueous cleanser is applied at the surface, so the two chemicals are immiscible and a boundary forms between them. A neutralising reaction takes place at this boundary in which cleanser and agent are consumed and reaction products are created. This is a Stefan problem, and the boundary between the cleanser and agent moves as the reaction proceeds. Reaction products formed at the interface may dissolve in one or both liquids. This may temporarily prevent cleanser and/or agent from reaching the reaction site, so diffusion of the chemical species, in particular the diffusion of product from the interface, plays a key role. The scenario described above was considered previously by ?? in the limit where the depth of the porous medium is large compared to the length scale over which concentrations vary inside the medium. Here, we present results that are valid for any ratio between these length scales and an analysis of agent removal times for various dimensionless parameter regimes. We also highlight the emergence of a boundary layer associated with diffusion in the oil phase for early times, where the thickness of the boundary layer is directly proportional to the square root of the time variable.
{"title":"Reaction dynamics and early-time behaviour of chemical decontamination","authors":"S Murphy, M Vynnycky, S L Mitchell, D O’Kiely","doi":"10.1093/imamat/hxae001","DOIUrl":"https://doi.org/10.1093/imamat/hxae001","url":null,"abstract":"When a hazardous chemical soaks into a porous material such as a concrete floor, it can be difficult to remove. One approach is chemical decontamination, where a cleanser is added to react with and neutralise the contaminating agent. The goal of this paper is to investigate the reaction dynamics and the factors that affect the efficacy of the decontamination procedure. We consider a one-dimensional porous medium initially saturated with an oil-based agent. An aqueous cleanser is applied at the surface, so the two chemicals are immiscible and a boundary forms between them. A neutralising reaction takes place at this boundary in which cleanser and agent are consumed and reaction products are created. This is a Stefan problem, and the boundary between the cleanser and agent moves as the reaction proceeds. Reaction products formed at the interface may dissolve in one or both liquids. This may temporarily prevent cleanser and/or agent from reaching the reaction site, so diffusion of the chemical species, in particular the diffusion of product from the interface, plays a key role. The scenario described above was considered previously by ?? in the limit where the depth of the porous medium is large compared to the length scale over which concentrations vary inside the medium. Here, we present results that are valid for any ratio between these length scales and an analysis of agent removal times for various dimensionless parameter regimes. We also highlight the emergence of a boundary layer associated with diffusion in the oil phase for early times, where the thickness of the boundary layer is directly proportional to the square root of the time variable.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139584709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Zilong Song, Robert Eisenberg, Shixin Xu, Huaxiong Huang
Voltage-gated K$_{mathrm{v}}$ channels play fundamental roles in many biological processes, such as the generation of the action potential. The gating mechanism of K$_{mathrm{v}}$ channels is characterized experimentally by single-channel recordings and ensemble properties of the channel currents. In this work, we propose a bubble model coupled with a Poisson-Nernst-Planck (PNP) system to capture the key characteristics, particularly the delay in the opening of channels. The coupled PNP system is solved numerically by a finite-difference method and the solution is compared with an analytical approximation. We hypothesize that the stochastic behaviour of the gating phenomenon is due to randomness of the bubble and channel sizes. The predicted ensemble average of the currents under various applied voltage across the channels is consistent with experimental observations, and the Cole-Moore delay is captured by varying the holding potential.
{"title":"A Bubble Model for the Gating of Kv Channels","authors":"Zilong Song, Robert Eisenberg, Shixin Xu, Huaxiong Huang","doi":"10.1093/imamat/hxae002","DOIUrl":"https://doi.org/10.1093/imamat/hxae002","url":null,"abstract":"Voltage-gated K$_{mathrm{v}}$ channels play fundamental roles in many biological processes, such as the generation of the action potential. The gating mechanism of K$_{mathrm{v}}$ channels is characterized experimentally by single-channel recordings and ensemble properties of the channel currents. In this work, we propose a bubble model coupled with a Poisson-Nernst-Planck (PNP) system to capture the key characteristics, particularly the delay in the opening of channels. The coupled PNP system is solved numerically by a finite-difference method and the solution is compared with an analytical approximation. We hypothesize that the stochastic behaviour of the gating phenomenon is due to randomness of the bubble and channel sizes. The predicted ensemble average of the currents under various applied voltage across the channels is consistent with experimental observations, and the Cole-Moore delay is captured by varying the holding potential.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139584980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Correction to: Initial-boundary value problem for a fractional heat equation on an interval","authors":"","doi":"10.1093/imamat/hxad036","DOIUrl":"https://doi.org/10.1093/imamat/hxad036","url":null,"abstract":"","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138606597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Estimating hyperparameters has been a long-standing problem in machine learning. We consider the case where the task at hand is modeled as the solution to an optimization problem. Here the exact gradient with respect to the hyperparameters cannot be feasibly computed and approximate strategies are required. We introduce a unified framework for computing hypergradients that generalizes existing methods based on the implicit function theorem and automatic differentiation/backpropagation, showing that these two seemingly disparate approaches are actually tightly connected. Our framework is extremely flexible, allowing its subproblems to be solved with any suitable method, to any degree of accuracy. We derive a priori and computable a posteriori error bounds for all our methods, and numerically show that our a posteriori bounds are usually more accurate. Our numerical results also show that, surprisingly, for efficient bilevel optimization, the choice of hypergradient algorithm is at least as important as the choice of lower-level solver.
{"title":"Analyzing inexact hypergradients for bilevel learning","authors":"Matthias J Ehrhardt, Lindon Roberts","doi":"10.1093/imamat/hxad035","DOIUrl":"https://doi.org/10.1093/imamat/hxad035","url":null,"abstract":"Estimating hyperparameters has been a long-standing problem in machine learning. We consider the case where the task at hand is modeled as the solution to an optimization problem. Here the exact gradient with respect to the hyperparameters cannot be feasibly computed and approximate strategies are required. We introduce a unified framework for computing hypergradients that generalizes existing methods based on the implicit function theorem and automatic differentiation/backpropagation, showing that these two seemingly disparate approaches are actually tightly connected. Our framework is extremely flexible, allowing its subproblems to be solved with any suitable method, to any degree of accuracy. We derive a priori and computable a posteriori error bounds for all our methods, and numerically show that our a posteriori bounds are usually more accurate. Our numerical results also show that, surprisingly, for efficient bilevel optimization, the choice of hypergradient algorithm is at least as important as the choice of lower-level solver.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138538042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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