Hadjer Nita, Fairouz Afroun, M. Cherfaoui, D. Aïssani
Abstract In this work, we considered the parametric estimation of the characteristics of the M / M / 1 / N {M/M/1/N} waiting model with Bernoulli feedback. Through a Monte-Carlo simulation study, we have illustrated the effect of the estimation of the starting parameters of the considered waiting system on the statistical properties of its performance measures estimates, when these latter are obtained using the plug-in method. In addition, several types of convergence (bias, variance, MSE, in law) of these performance measure estimators have also been showed by simulation.
{"title":"Statistical analysis of the estimates of some stationary performances of the unreliable M/M/1/N queue with Bernoulli feedback","authors":"Hadjer Nita, Fairouz Afroun, M. Cherfaoui, D. Aïssani","doi":"10.1515/mcma-2023-2004","DOIUrl":"https://doi.org/10.1515/mcma-2023-2004","url":null,"abstract":"Abstract In this work, we considered the parametric estimation of the characteristics of the M / M / 1 / N {M/M/1/N} waiting model with Bernoulli feedback. Through a Monte-Carlo simulation study, we have illustrated the effect of the estimation of the starting parameters of the considered waiting system on the statistical properties of its performance measures estimates, when these latter are obtained using the plug-in method. In addition, several types of convergence (bias, variance, MSE, in law) of these performance measure estimators have also been showed by simulation.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"0 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41551785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The partitioning algorithm is an iterative procedure that computes explicitly the steady-state probability of a finite Markov chain 𝑋. In this paper, we propose to adapt this algorithm to the case where the state space E := C ∪ D E:=Ccup D is composed of a continuous part 𝐶 and a finite part 𝐷 such that C ∩ D = ∅ Ccap D=emptyset . In this case, the steady-state probability 𝜋 of 𝑋 is a convex combination of two steady-state probabilities π C pi_{C} and π D pi_{D} of two Markov chains on 𝐶 and 𝐷 respectively. The obtained algorithm allows to compute explicitly π D pi_{D} . If π C pi_{C} cannot be computed explicitly, our algorithm approximates it by numerical resolution of successive integral equations. Some numerical examples are studied to show the usefulness and proper functioning of our proposal.
分划算法是显式计算有限马尔可夫链稳态概率的迭代过程𝑋。在本文中，我们提出将该算法应用于状态空间E:=C∪D E:=C cup D由连续部分和有限部分𝐷组成，使得C∩D=∅C cap D= emptyset。在这种情况下，稳态概率𝑋分别是两条马尔可夫链的两个稳态概率π C pi ＿C{和π }D pi ＿D{的凸组合。得到的算法允许显式计算π D }pi ＿D{。如果π C }pi ＿C{不能显式计算，我们的算法通过连续积分方程的数值解析近似它。通过数值算例分析，说明了该方法的有效性和良好的功能。}
{"title":"Computation of the steady-state probability of Markov chain evolving on a mixed state space","authors":"Az-eddine Zakrad, A. Nasroallah","doi":"10.1515/mcma-2023-2003","DOIUrl":"https://doi.org/10.1515/mcma-2023-2003","url":null,"abstract":"Abstract The partitioning algorithm is an iterative procedure that computes explicitly the steady-state probability of a finite Markov chain 𝑋. In this paper, we propose to adapt this algorithm to the case where the state space E := C ∪ D E:=Ccup D is composed of a continuous part 𝐶 and a finite part 𝐷 such that C ∩ D = ∅ Ccap D=emptyset . In this case, the steady-state probability 𝜋 of 𝑋 is a convex combination of two steady-state probabilities π C pi_{C} and π D pi_{D} of two Markov chains on 𝐶 and 𝐷 respectively. The obtained algorithm allows to compute explicitly π D pi_{D} . If π C pi_{C} cannot be computed explicitly, our algorithm approximates it by numerical resolution of successive integral equations. Some numerical examples are studied to show the usefulness and proper functioning of our proposal.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43124785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Guilhem Balvet, J. Minier, C. Henry, Y. Roustan, M. Ferrand
Abstract The purpose of this paper is to propose a time-step-robust cell-to-cell integration of particle trajectories in 3-D unstructured meshes in particle/mesh Lagrangian stochastic methods. The main idea is to dynamically update the mean fields used in the time integration by splitting, for each particle, the time step into sub-steps such that each of these sub-steps corresponds to particle cell residence times. This reduces the spatial discretization error. Given the stochastic nature of the models, a key aspect is to derive estimations of the residence times that do not anticipate the future of the Wiener process. To that effect, the new algorithm relies on a virtual particle, attached to each stochastic one, whose mean conditional behavior provides free-of-statistical-bias predictions of residence times. After consistency checks, this new algorithm is validated on two representative test cases: particle dispersion in a statistically uniform flow and particle dynamics in a non-uniform flow.
{"title":"A time-step-robust algorithm to compute particle trajectories in 3-D unstructured meshes for Lagrangian stochastic methods","authors":"Guilhem Balvet, J. Minier, C. Henry, Y. Roustan, M. Ferrand","doi":"10.1515/mcma-2023-2002","DOIUrl":"https://doi.org/10.1515/mcma-2023-2002","url":null,"abstract":"Abstract The purpose of this paper is to propose a time-step-robust cell-to-cell integration of particle trajectories in 3-D unstructured meshes in particle/mesh Lagrangian stochastic methods. The main idea is to dynamically update the mean fields used in the time integration by splitting, for each particle, the time step into sub-steps such that each of these sub-steps corresponds to particle cell residence times. This reduces the spatial discretization error. Given the stochastic nature of the models, a key aspect is to derive estimations of the residence times that do not anticipate the future of the Wiener process. To that effect, the new algorithm relies on a virtual particle, attached to each stochastic one, whose mean conditional behavior provides free-of-statistical-bias predictions of residence times. After consistency checks, this new algorithm is validated on two representative test cases: particle dispersion in a statistically uniform flow and particle dynamics in a non-uniform flow.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"29 1","pages":"95 - 126"},"PeriodicalIF":0.9,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48076657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-01DOI: 10.1515/mcma-2023-frontmatter1
{"title":"Frontmatter","authors":"","doi":"10.1515/mcma-2023-frontmatter1","DOIUrl":"https://doi.org/10.1515/mcma-2023-frontmatter1","url":null,"abstract":"","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"212 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136270828","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
P. J. Valades-Pelayo, M. Ramirez-Cabrera, A. Balbuena-Ortega
Abstract This manuscript presents a short route to justify the widely used Monte Carlo Radiative Transfer (MCRT) algorithm straight from the Radiative Transfer Equation (RTE). In this regard, this paper starts deriving a probability measure obtained from the integral formulation of the RTE under a unidirectional point source in an infinite domain. This derivation only requires the analytical integration of the first two terms of a perturbation expansion. Although derivations have been devised to clarify the relationship between the MCRT and the RTE, they tend to be rather long and elaborate. Considering how simple it is to justify the MCRT from a loose probabilistic interpretation of the photon’s physical propagation process, the decay in popularity of former approaches relating MCRT to the RTE is entirely understandable. Unfortunately, all of this has given the false impression that MCRT and the RTE are not that closely related, to the point that recent works have explicitly stated that no direct link exists between them. This work presents a simpler route demonstrating how the MCRT algorithm emerges to statistically sample the RTE explicitly through Markov chains, further clarifying the method’s foundations. Although compact, the derivation proposed in this work does not skip any fundamental step, preserving mathematical rigor while giving specific expressions and functions. Thus, this derivation can help devise efficient ways to statistically sample the RTE for different scenarios or when coupling the MCRT method with other methods traditionally grounded in the RTE, such as the Spherical Harmonics and Discrete Ordinates methods.
{"title":"Linking the Monte Carlo radiative transfer algorithm to the radiative transfer equation","authors":"P. J. Valades-Pelayo, M. Ramirez-Cabrera, A. Balbuena-Ortega","doi":"10.1515/mcma-2023-2001","DOIUrl":"https://doi.org/10.1515/mcma-2023-2001","url":null,"abstract":"Abstract This manuscript presents a short route to justify the widely used Monte Carlo Radiative Transfer (MCRT) algorithm straight from the Radiative Transfer Equation (RTE). In this regard, this paper starts deriving a probability measure obtained from the integral formulation of the RTE under a unidirectional point source in an infinite domain. This derivation only requires the analytical integration of the first two terms of a perturbation expansion. Although derivations have been devised to clarify the relationship between the MCRT and the RTE, they tend to be rather long and elaborate. Considering how simple it is to justify the MCRT from a loose probabilistic interpretation of the photon’s physical propagation process, the decay in popularity of former approaches relating MCRT to the RTE is entirely understandable. Unfortunately, all of this has given the false impression that MCRT and the RTE are not that closely related, to the point that recent works have explicitly stated that no direct link exists between them. This work presents a simpler route demonstrating how the MCRT algorithm emerges to statistically sample the RTE explicitly through Markov chains, further clarifying the method’s foundations. Although compact, the derivation proposed in this work does not skip any fundamental step, preserving mathematical rigor while giving specific expressions and functions. Thus, this derivation can help devise efficient ways to statistically sample the RTE for different scenarios or when coupling the MCRT method with other methods traditionally grounded in the RTE, such as the Spherical Harmonics and Discrete Ordinates methods.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"29 1","pages":"173 - 180"},"PeriodicalIF":0.9,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43267975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we propose and investigate two new kernel regression estimators based on a bias reduction transformation technique. We study the properties of these estimators and compare them with Nadaraya–Watson’s regression estimator and Slaoui’s (2016) regression estimator. It turns out that, with an adequate choice of the parameters of the two proposed estimators, the rate of convergence of two estimators will be faster than the two classical estimators, and the asymptotic MISE (mean integrated squared error) will be smaller than the two classical estimators. We corroborate these theoretical results through simulations and a real Malaria dataset.
{"title":"Methodology for nonparametric bias reduction in kernel regression estimation","authors":"Y. Slaoui","doi":"10.1515/mcma-2022-2130","DOIUrl":"https://doi.org/10.1515/mcma-2022-2130","url":null,"abstract":"Abstract In this paper, we propose and investigate two new kernel regression estimators based on a bias reduction transformation technique. We study the properties of these estimators and compare them with Nadaraya–Watson’s regression estimator and Slaoui’s (2016) regression estimator. It turns out that, with an adequate choice of the parameters of the two proposed estimators, the rate of convergence of two estimators will be faster than the two classical estimators, and the asymptotic MISE (mean integrated squared error) will be smaller than the two classical estimators. We corroborate these theoretical results through simulations and a real Malaria dataset.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"29 1","pages":"55 - 77"},"PeriodicalIF":0.9,"publicationDate":"2023-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49604358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we address a long-standing open problem in stochastic simulation: construction of a random walk on spheres (RWS) algorithm for solving a system of elasticity equations, known as the Lamé equation. Many attempts to generalize the classic probabilistic representations like the Kac formula for parabolic and scalar elliptic equations failed. A different approach based on a branching random walk on spheres (BRWS) introduced in our paper of 1995 [K. K. Sabelfeld and D. Talay, Integral formulation of the boundary value problems and the method of random walk on spheres, Monte Carlo Methods Appl. 1 1995, 1, 1–34] made little progress in solving this problem. In the present study, we further improve the BRWS algorithm by a special implementation of a branching anisotropic random walk on spheres process.
{"title":"Development and implementation of branching random walk on spheres algorithms for solving the 2D elastostatics Lamé equation","authors":"I. Shalimova, K. Sabelfeld","doi":"10.1515/mcma-2022-2131","DOIUrl":"https://doi.org/10.1515/mcma-2022-2131","url":null,"abstract":"Abstract In this paper, we address a long-standing open problem in stochastic simulation: construction of a random walk on spheres (RWS) algorithm for solving a system of elasticity equations, known as the Lamé equation. Many attempts to generalize the classic probabilistic representations like the Kac formula for parabolic and scalar elliptic equations failed. A different approach based on a branching random walk on spheres (BRWS) introduced in our paper of 1995 [K. K. Sabelfeld and D. Talay, Integral formulation of the boundary value problems and the method of random walk on spheres, Monte Carlo Methods Appl. 1 1995, 1, 1–34] made little progress in solving this problem. In the present study, we further improve the BRWS algorithm by a special implementation of a branching anisotropic random walk on spheres process.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"29 1","pages":"79 - 93"},"PeriodicalIF":0.9,"publicationDate":"2023-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46049317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We address the least-squares estimation of the drift coefficient parameter θ = ( λ , A , B , ω p ) theta=(lambda,A,B,omega_{p}) of a time-inhomogeneous Ornstein–Uhlenbeck process that is observed at high frequency, in which the discretized step size ℎ satisfies h → 0 hto 0 . In this paper, under the conditions n h → ∞ nhtoinfty and n h 2 → 0 nh^{2}to 0 , we prove the consistency and the asymptotic normality of the estimators. We obtain the convergence of the parameters at rate n h sqrt{nh} , except for ω p omega_{p} at n 3 h 3 sqrt{n^{3}h^{3}} . To verify our theoretical findings, we do a simulation study. We then illustrate the use of the proposed model in fitting the energy use of light fixtures in one Belgium household and the stock exchange.
摘要我们讨论了在高频下观测到的时间非均匀Ornstein–Uhlenbeck过程的漂移系数参数θ=（λ，A，B，ωp）θ=（lambda，A，B，omega_{p}）的最小二乘估计，其中离散化的步长ℎ 满足h→ 0小时到0。在本文中，在条件n h→ ∞ nhtoinfty和n h 2→ 0nh^｛2｝到0，我们证明了估计量的一致性和渐近正态性。我们得到了在速率n h sqrt｛nh｝下参数的收敛性，除了ω^{3}h^{3} }。为了验证我们的理论发现，我们进行了一项模拟研究。然后，我们说明了所提出的模型在拟合比利时一个家庭和证券交易所的灯具能源使用方面的用途。
{"title":"Parameter least-squares estimation for time-inhomogeneous Ornstein–Uhlenbeck process","authors":"G. Pramesti","doi":"10.1515/mcma-2022-2127","DOIUrl":"https://doi.org/10.1515/mcma-2022-2127","url":null,"abstract":"Abstract We address the least-squares estimation of the drift coefficient parameter θ = ( λ , A , B , ω p ) theta=(lambda,A,B,omega_{p}) of a time-inhomogeneous Ornstein–Uhlenbeck process that is observed at high frequency, in which the discretized step size ℎ satisfies h → 0 hto 0 . In this paper, under the conditions n h → ∞ nhtoinfty and n h 2 → 0 nh^{2}to 0 , we prove the consistency and the asymptotic normality of the estimators. We obtain the convergence of the parameters at rate n h sqrt{nh} , except for ω p omega_{p} at n 3 h 3 sqrt{n^{3}h^{3}} . To verify our theoretical findings, we do a simulation study. We then illustrate the use of the proposed model in fitting the energy use of light fixtures in one Belgium household and the stock exchange.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"29 1","pages":"1 - 32"},"PeriodicalIF":0.9,"publicationDate":"2022-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44053654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The global random walk on grid method (GRWG) is developed for solving two-dimensional nonlinear systems of equations, the Navier–Stokes and Burgers equations. This study extends the GRWG which we have earlier developed for solving the nonlinear drift-diffusion-Poisson equation of semiconductors (Physica A 556 (2020), Article ID 124800). The Burgers equation is solved by a direct iteration of a system of linear drift-diffusion equations, while the Navier–Stokes equation is solved in the stream function-vorticity formulation.
摘要提出了求解二维非线性方程组Navier–Stokes和Burgers方程的全局随机网格行走方法。本研究扩展了我们早期开发的用于求解半导体非线性漂移扩散泊松方程的GRWG（Physica A 556（2020），文章ID 124800）。Burgers方程是通过线性漂移扩散方程组的直接迭代求解的，而Navier–Stokes方程是通过流函数涡度公式求解的。
{"title":"Global random walk on grid algorithm for solving Navier–Stokes and Burgers equations","authors":"K. Sabelfeld, Oleg Bukhasheev","doi":"10.1515/mcma-2022-2126","DOIUrl":"https://doi.org/10.1515/mcma-2022-2126","url":null,"abstract":"Abstract The global random walk on grid method (GRWG) is developed for solving two-dimensional nonlinear systems of equations, the Navier–Stokes and Burgers equations. This study extends the GRWG which we have earlier developed for solving the nonlinear drift-diffusion-Poisson equation of semiconductors (Physica A 556 (2020), Article ID 124800). The Burgers equation is solved by a direct iteration of a system of linear drift-diffusion equations, while the Navier–Stokes equation is solved in the stream function-vorticity formulation.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"28 1","pages":"293 - 305"},"PeriodicalIF":0.9,"publicationDate":"2022-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43562871","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we address the problem of flow simulation at high Péclet numbers by the random walk on spheres (RWS) method. Conventional deterministic methods here face difficulties related to high solution gradients near the boundary in the region known as the boundary layer. In the finite-difference methods, this leads to introduction of very fine meshes which in turn causes problems of stability and high dimensions. The RWS algorithm is mesh free, but the high Péclet number flows should probably also affect the efficiency of simulations. However, it turns out that the RWS algorithm can be well adapted to this case. We present an analysis of the RWS algorithm for different examples of flows with high Péclet number. Simulations are carried out for different boundary conditions and for two-layered material with different diffusion coefficients of exciton’s mobility.
{"title":"Simulation of drift-diffusion process at high Péclet numbers by the random walk on spheres method","authors":"K. Sabelfeld, Ivan Aksyuk","doi":"10.1515/mcma-2022-2128","DOIUrl":"https://doi.org/10.1515/mcma-2022-2128","url":null,"abstract":"Abstract In this paper, we address the problem of flow simulation at high Péclet numbers by the random walk on spheres (RWS) method. Conventional deterministic methods here face difficulties related to high solution gradients near the boundary in the region known as the boundary layer. In the finite-difference methods, this leads to introduction of very fine meshes which in turn causes problems of stability and high dimensions. The RWS algorithm is mesh free, but the high Péclet number flows should probably also affect the efficiency of simulations. However, it turns out that the RWS algorithm can be well adapted to this case. We present an analysis of the RWS algorithm for different examples of flows with high Péclet number. Simulations are carried out for different boundary conditions and for two-layered material with different diffusion coefficients of exciton’s mobility.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"28 1","pages":"349 - 367"},"PeriodicalIF":0.9,"publicationDate":"2022-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42307771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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