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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
Abstract We apply the Monte Carlo method to solving the Dirichlet problem of linear parabolic equations with fractional Laplacian. This method exploits the idea of weak approximation of related stochastic differential equations driven by the symmetric stable Lévy process with jumps. We utilize the jump-adapted scheme to approximate Lévy process which gives exact exit time to the boundary. When the solution has low regularity, we establish a numerical scheme by removing the small jumps of the Lévy process and then show the convergence order. When the solution has higher regularity, we build up a higher-order numerical scheme by replacing small jumps with a simple process and then display the higher convergence order. Finally, numerical experiments including ten- and one hundred-dimensional cases are presented, which confirm the theoretical estimates and show the numerical efficiency of the proposed schemes for high-dimensional parabolic equations.
{"title":"Monte Carlo method for parabolic equations involving fractional Laplacian","authors":"Caiyu Jiao, Changpin Li","doi":"10.1515/mcma-2022-2129","DOIUrl":"https://doi.org/10.1515/mcma-2022-2129","url":null,"abstract":"Abstract We apply the Monte Carlo method to solving the Dirichlet problem of linear parabolic equations with fractional Laplacian. This method exploits the idea of weak approximation of related stochastic differential equations driven by the symmetric stable Lévy process with jumps. We utilize the jump-adapted scheme to approximate Lévy process which gives exact exit time to the boundary. When the solution has low regularity, we establish a numerical scheme by removing the small jumps of the Lévy process and then show the convergence order. When the solution has higher regularity, we build up a higher-order numerical scheme by replacing small jumps with a simple process and then display the higher convergence order. Finally, numerical experiments including ten- and one hundred-dimensional cases are presented, which confirm the theoretical estimates and show the numerical efficiency of the proposed schemes for high-dimensional parabolic equations.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44779092","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 many Monte Carlo applications, one can substitute the use of pseudorandom numbers with quasirandom numbers and achieve improved convergence. This is because quasirandom numbers are more uniform that pseudorandom numbers. The most common measure of that uniformity is the star discrepancy. Moreover, the main error bound in quasi-Monte Carlo methods, called the Koksma–Hlawka inequality, has the star discrepancy in the formulation. A difficulty with this bound is that computing the star discrepancy is very costly. The star discrepancy can be computed by evaluating a function called the local discrepancy at a number of points. The supremum of these local discrepancy values is the star discrepancy. If we have a point set in [ 0 , 1 ] s {[0,1]^{s}} with N members, we need to compute the local discrepancy at N s {N^{s}} points. In fact, computing star discrepancy is NP-hard. In this paper, we will consider an approximate algorithm for a lower bound on the star discrepancy based on using a random walk through some of the N s {N^{s}} points. This approximation is much less expensive that computing the star discrepancy, but still accurate enough to provide information on convergence. Our numerical results show that the random walk algorithm has the same convergence rate as the Monte Carlo method, which is O ( N - 1 2 {O(N^{-frac{1}{2}}} ).
摘要在许多蒙特卡罗应用中，可以用拟随机数代替伪随机数的使用，从而达到提高收敛性的目的。这是因为准随机数比伪随机数更均匀。这种均匀性最常见的测量方法是恒星差异。此外，拟蒙特卡罗方法的主要误差界，称为Koksma-Hlawka不等式，在公式中具有星形差异。这个界限的一个困难是，计算恒星差异的成本非常高。星形差异可以通过在若干点上计算一个称为局部差异的函数来计算。这些局部差值的最大值是星形差值。如果我们有一个在[0,1]s {[0,1]^{s}}中有N个成员的点集，我们需要计算N s {N^{s}}点上的局部差异。事实上，计算恒星差异是np困难的。在本文中，我们将考虑一种基于随机遍历一些N s {N^{s}}点的星差下界的近似算法。这种近似方法比计算恒星差异要便宜得多，但仍然足够精确，可以提供关于收敛的信息。我们的数值结果表明，随机漫步算法具有与蒙特卡罗方法相同的收敛速度，即O(N - 1 2 {O(N^{-frac{1}{2}}})。
{"title":"A random walk algorithm to estimate a lower bound of the star discrepancy","authors":"Maryam Alsolami, M. Mascagni","doi":"10.1515/mcma-2022-2125","DOIUrl":"https://doi.org/10.1515/mcma-2022-2125","url":null,"abstract":"Abstract In many Monte Carlo applications, one can substitute the use of pseudorandom numbers with quasirandom numbers and achieve improved convergence. This is because quasirandom numbers are more uniform that pseudorandom numbers. The most common measure of that uniformity is the star discrepancy. Moreover, the main error bound in quasi-Monte Carlo methods, called the Koksma–Hlawka inequality, has the star discrepancy in the formulation. A difficulty with this bound is that computing the star discrepancy is very costly. The star discrepancy can be computed by evaluating a function called the local discrepancy at a number of points. The supremum of these local discrepancy values is the star discrepancy. If we have a point set in [ 0 , 1 ] s {[0,1]^{s}} with N members, we need to compute the local discrepancy at N s {N^{s}} points. In fact, computing star discrepancy is NP-hard. In this paper, we will consider an approximate algorithm for a lower bound on the star discrepancy based on using a random walk through some of the N s {N^{s}} points. This approximation is much less expensive that computing the star discrepancy, but still accurate enough to provide information on convergence. Our numerical results show that the random walk algorithm has the same convergence rate as the Monte Carlo method, which is O ( N - 1 2 {O(N^{-frac{1}{2}}} ).","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41818243","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 I consider black body radiation. The wall of the black body exchanges photons with the radiation field in equilibrium, therefore with a common temperature in Planck’s radiation law. The underlying process of radiation consists of creation and annihilation of photons. I want to present an alternate model of motions, where the process of radiation consists of small steps in positive and negative direction, not zero in mean. The detection of radiation consists of storing and restoring of packages of energy. I get an analogue of Planck’s radiation law, where the common temperature emerges from the underlying common model of small steps. The object of the law is not the radiation, but a storage of packages of energy, which belongs to the wall of the black body.
{"title":"Superposition of forward and backward motion","authors":"Manfred Harringer","doi":"10.1515/mcma-2022-2124","DOIUrl":"https://doi.org/10.1515/mcma-2022-2124","url":null,"abstract":"Abstract I consider black body radiation. The wall of the black body exchanges photons with the radiation field in equilibrium, therefore with a common temperature in Planck’s radiation law. The underlying process of radiation consists of creation and annihilation of photons. I want to present an alternate model of motions, where the process of radiation consists of small steps in positive and negative direction, not zero in mean. The detection of radiation consists of storing and restoring of packages of energy. I get an analogue of Planck’s radiation law, where the common temperature emerges from the underlying common model of small steps. The object of the law is not the radiation, but a storage of packages of energy, which belongs to the wall of the black body.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45461671","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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