The process of dynamic evolution in dispersed systems due to simultaneous particle coagulation and deposition is described mathematically by general dynamic equation (GDE). Monte Carlo (MC) method is an important appr...The process of dynamic evolution in dispersed systems due to simultaneous particle coagulation and deposition is described mathematically by general dynamic equation (GDE). Monte Carlo (MC) method is an important approach of numerical solu- tions of GDE. However, constant-volume MC method exhibits the contradictory of low computation cost and high computation precision owing to the fluctuation of the number of simulation particles; constant-number MC method can hardly be applied to engineering application and general scientific quantitative analysis due to the continual contraction or expansion of computation domain. In addition, the two MC methods depend closely on the “subsystem” hypothesis, which constraints their expansibility and the scope of application. A new multi-Monte Carlo (MMC) method is promoted to take account of GDE for simulta- neous particle coagulation and deposition. MMC method introduces the concept of “weighted fictitious particle” and is based on the “time-driven” technique. Furthermore MMC method maintains synchronously the computational domain and the total number of fictitious particles, which results in the latent expansibility of simulation for boundary con- dition, the space evolution of particle size distribution and even particle dynamics. The simulation results of MMC method for two special cases in which analytical solutions exist agree with analytical solutions well, which proves that MMC method has high and stable computational precision and low computation cost because of the constant and limited number of fictitious particles. Lastly the source of numerical error and the relative error of MMC method are analyzed, respectively.展开更多
可再生能源的间歇性出力以及负荷的波动给综合能源系统(integrated energy systems,IES)引入了大量不确定性因素。提出了一种基于K-means聚类技术改进的多线性蒙特卡洛概率能流计算方法。首先,引入输入随机变量整体灵敏度系数概念,并以...可再生能源的间歇性出力以及负荷的波动给综合能源系统(integrated energy systems,IES)引入了大量不确定性因素。提出了一种基于K-means聚类技术改进的多线性蒙特卡洛概率能流计算方法。首先,引入输入随机变量整体灵敏度系数概念,并以权重系数的形式修正输入随机变量样本,改进K-means聚类效果,确保各聚类簇均具有较小的波动范围。然后,采用多线性化求解思路进行概率能流计算,即对聚类中心进行确定性能流计算,而各聚类簇中输入随机变量样本利用同一簇聚类中心处得到的状态变量和雅可比矩阵进行线性化能流求解,从而减少了迭代过程,提高计算效率。以IEEE57节点电力系统和14节点天然气网络构成的IES为算例,验证了所提方法比传统蒙特卡洛法具有更高的计算效率,相比现有多线性蒙特卡洛算法具有更高的准确性和计算效率。展开更多
In this article, we propose a space-time Multi-Index Monte Carlo （MIMC） estimator for a one-dimensional parabolic stochastic partial differential equation （SPDE） of Zakai type. We compare the complexity with the M...In this article, we propose a space-time Multi-Index Monte Carlo （MIMC） estimator for a one-dimensional parabolic stochastic partial differential equation （SPDE） of Zakai type. We compare the complexity with the Multilevel Monte Carlo （MLMC） method of Giles and Reisinger （2012）, and find, by means of Fourier analysis, that the MIMC method： （i） has suboptimal complexity of 0（ε^-21 ｜ogε｜） for a root mean square error （RMSE） z if the same spatial discretisation as in the MLMC method is used; （ii） has a better complexity of 0（ε^-21 ｜ogε｜） if a carefully adapted discretisation is used; （iii） has to be adapted for non-smooth functionals. Numerical tests confirm these findings empirically.展开更多
基金This work was supported by the National Key Basic Research and Development Program (Grant No. 2002CB211602)the National Natural Science Foundation of China (Grant No. 90410017).
文摘The process of dynamic evolution in dispersed systems due to simultaneous particle coagulation and deposition is described mathematically by general dynamic equation (GDE). Monte Carlo (MC) method is an important approach of numerical solu- tions of GDE. However, constant-volume MC method exhibits the contradictory of low computation cost and high computation precision owing to the fluctuation of the number of simulation particles; constant-number MC method can hardly be applied to engineering application and general scientific quantitative analysis due to the continual contraction or expansion of computation domain. In addition, the two MC methods depend closely on the “subsystem” hypothesis, which constraints their expansibility and the scope of application. A new multi-Monte Carlo (MMC) method is promoted to take account of GDE for simulta- neous particle coagulation and deposition. MMC method introduces the concept of “weighted fictitious particle” and is based on the “time-driven” technique. Furthermore MMC method maintains synchronously the computational domain and the total number of fictitious particles, which results in the latent expansibility of simulation for boundary con- dition, the space evolution of particle size distribution and even particle dynamics. The simulation results of MMC method for two special cases in which analytical solutions exist agree with analytical solutions well, which proves that MMC method has high and stable computational precision and low computation cost because of the constant and limited number of fictitious particles. Lastly the source of numerical error and the relative error of MMC method are analyzed, respectively.
文摘可再生能源的间歇性出力以及负荷的波动给综合能源系统(integrated energy systems,IES)引入了大量不确定性因素。提出了一种基于K-means聚类技术改进的多线性蒙特卡洛概率能流计算方法。首先,引入输入随机变量整体灵敏度系数概念,并以权重系数的形式修正输入随机变量样本,改进K-means聚类效果,确保各聚类簇均具有较小的波动范围。然后,采用多线性化求解思路进行概率能流计算,即对聚类中心进行确定性能流计算,而各聚类簇中输入随机变量样本利用同一簇聚类中心处得到的状态变量和雅可比矩阵进行线性化能流求解,从而减少了迭代过程,提高计算效率。以IEEE57节点电力系统和14节点天然气网络构成的IES为算例,验证了所提方法比传统蒙特卡洛法具有更高的计算效率,相比现有多线性蒙特卡洛算法具有更高的准确性和计算效率。
文摘In this article, we propose a space-time Multi-Index Monte Carlo （MIMC） estimator for a one-dimensional parabolic stochastic partial differential equation （SPDE） of Zakai type. We compare the complexity with the Multilevel Monte Carlo （MLMC） method of Giles and Reisinger （2012）, and find, by means of Fourier analysis, that the MIMC method： （i） has suboptimal complexity of 0（ε^-21 ｜ogε｜） for a root mean square error （RMSE） z if the same spatial discretisation as in the MLMC method is used; （ii） has a better complexity of 0（ε^-21 ｜ogε｜） if a carefully adapted discretisation is used; （iii） has to be adapted for non-smooth functionals. Numerical tests confirm these findings empirically.
