Considering the stochastic spatial variation of geotechnical parameters over the slope, a Stochastic Finite Element Method (SFEM) is established based on the combination of the Shear Strength Reduction (SSR) concept a...Considering the stochastic spatial variation of geotechnical parameters over the slope, a Stochastic Finite Element Method (SFEM) is established based on the combination of the Shear Strength Reduction (SSR) concept and quasi-Monte Carlo simulation. The shear strength reduction FEM is superior to the slice method based on the limit equilibrium theory in many ways, so it will be more powerful to assess the reliability of global slope stability when combined with probability theory. To illustrate the performance of the proposed method, it is applied to an example of simple slope. The results of simulation show that the proposed method is effective to perform the reliability analysis of global slope stability without presupposing a potential slip surface.展开更多
Deep learning has achieved great success in solving partial differential equations(PDEs),where the loss is often defined as an integral.The accuracy and efficiency of these algorithms depend greatly on the quadrature ...Deep learning has achieved great success in solving partial differential equations(PDEs),where the loss is often defined as an integral.The accuracy and efficiency of these algorithms depend greatly on the quadrature method.We propose to apply quasi-Monte Carlo(QMC)methods to the Deep Ritz Method(DRM)for solving the Neumann problems for the Poisson equation and the static Schr¨odinger equation.For error estimation,we decompose the error of using the deep learning algorithm to solve PDEs into the generalization error,the approximation error and the training error.We establish the upper bounds and prove that QMC-based DRM achieves an asymptotically smaller error bound than DRM.Numerical experiments show that the proposed method converges faster in all cases and the variances of the gradient estimators of randomized QMC-based DRM are much smaller than those of DRM,which illustrates the superiority of QMC in deep learning over MC.展开更多
As one of the major methods for the simulation of option pricing,Monte Carlo method assumes random fluctuations in the distribution of asset prices.Under certain uncertainties process,different evolution paths could b...As one of the major methods for the simulation of option pricing,Monte Carlo method assumes random fluctuations in the distribution of asset prices.Under certain uncertainties process,different evolution paths could be simulated so as to finally yield the expectation value of the asset price,which requires a lot of simulations to ensure the accuracy based on huge and expensive calculations.In order to solve the above computational problem,quantum Monte Carlo(QMC)has been established and applied in the relevant systems such as European call options.In this work,both MC and QM methods are adopted to simulate European call options.Based on the preparation of quantum states in QMC algorithm and the construction of quantum circuits by simulating a quantum hardware environment on a traditional computer,the amplitude estimation(AE)algorithm is found to play a secondary role in accelerating the pricing of European options.More importantly,the payoff function and the time required for the simulation in QMC method show some improvements than those in MC method.展开更多
针对障碍期权的定价问题,给出了一种高效的蒙特卡罗(Monte Carlo,MC)模拟方法——基于布朗桥构造路径的随机化拟蒙特卡罗(Brownian bridge path randomization quasi Monte Carlo,BBPR-QMC)方法.首先,用Faure序列代替MC方法中的随机序列...针对障碍期权的定价问题,给出了一种高效的蒙特卡罗(Monte Carlo,MC)模拟方法——基于布朗桥构造路径的随机化拟蒙特卡罗(Brownian bridge path randomization quasi Monte Carlo,BBPR-QMC)方法.首先,用Faure序列代替MC方法中的随机序列,得到了Faure序列的拟蒙特卡罗(quasi Monte Carlo,QMC)模拟方法;其次,应用Moro算法得到了随机化拟蒙特卡罗(randomization quasi Monte Carlo,R-QMC)模拟方法;最后,将QMC方法和R-QMC方法结合,利用布朗桥技术来降低有效维,得到障碍期权定价的BBPR-QMC方法.数值试验表明,与MC方法和R-QMC方法相比较,BBPR-QMC方法模拟的价格与真实价格更接近、收敛速度更快.数值试验证实,BBPR-QMC方法是一种高效求解障碍期权定价的数值方法.展开更多
The development of the option price theory provides business enterprise a beneficial tool to carry through property risk management, but a variety of option price theories are established on certain environments, and ...The development of the option price theory provides business enterprise a beneficial tool to carry through property risk management, but a variety of option price theories are established on certain environments, and they can not deal with crisis in uncertain environments precisely and quickly, especially when multi-factors change at the same time. Thus, price the option in uncertain environment has been becoming an important direction of research. In this paper, wc take the stock option for example~ using Quasi-Monte Carlo method to price the American-style option, and then provide an example to explain. The powerful assistant decision-making ability of the computer simulation is clearly expressed when we study and analyze the Quasi-Monte Carlo method＇s characteristics.展开更多
In the present note the convergence problem of the sequential number-theoretic method for optimization proposed by Fang and Wang is studied, the convergence criteria and the estimation of errors concerning this algori...In the present note the convergence problem of the sequential number-theoretic method for optimization proposed by Fang and Wang is studied, the convergence criteria and the estimation of errors concerning this algorithm are given.展开更多
文摘Considering the stochastic spatial variation of geotechnical parameters over the slope, a Stochastic Finite Element Method (SFEM) is established based on the combination of the Shear Strength Reduction (SSR) concept and quasi-Monte Carlo simulation. The shear strength reduction FEM is superior to the slice method based on the limit equilibrium theory in many ways, so it will be more powerful to assess the reliability of global slope stability when combined with probability theory. To illustrate the performance of the proposed method, it is applied to an example of simple slope. The results of simulation show that the proposed method is effective to perform the reliability analysis of global slope stability without presupposing a potential slip surface.
基金supported by the National Natural Science Foundation of China(Grant No.72071119).
文摘Deep learning has achieved great success in solving partial differential equations(PDEs),where the loss is often defined as an integral.The accuracy and efficiency of these algorithms depend greatly on the quadrature method.We propose to apply quasi-Monte Carlo(QMC)methods to the Deep Ritz Method(DRM)for solving the Neumann problems for the Poisson equation and the static Schr¨odinger equation.For error estimation,we decompose the error of using the deep learning algorithm to solve PDEs into the generalization error,the approximation error and the training error.We establish the upper bounds and prove that QMC-based DRM achieves an asymptotically smaller error bound than DRM.Numerical experiments show that the proposed method converges faster in all cases and the variances of the gradient estimators of randomized QMC-based DRM are much smaller than those of DRM,which illustrates the superiority of QMC in deep learning over MC.
基金This work was financially supported by the National Natural Science Foundation of China Granted No.11764028。
文摘As one of the major methods for the simulation of option pricing,Monte Carlo method assumes random fluctuations in the distribution of asset prices.Under certain uncertainties process,different evolution paths could be simulated so as to finally yield the expectation value of the asset price,which requires a lot of simulations to ensure the accuracy based on huge and expensive calculations.In order to solve the above computational problem,quantum Monte Carlo(QMC)has been established and applied in the relevant systems such as European call options.In this work,both MC and QM methods are adopted to simulate European call options.Based on the preparation of quantum states in QMC algorithm and the construction of quantum circuits by simulating a quantum hardware environment on a traditional computer,the amplitude estimation(AE)algorithm is found to play a secondary role in accelerating the pricing of European options.More importantly,the payoff function and the time required for the simulation in QMC method show some improvements than those in MC method.
文摘针对障碍期权的定价问题,给出了一种高效的蒙特卡罗(Monte Carlo,MC)模拟方法——基于布朗桥构造路径的随机化拟蒙特卡罗(Brownian bridge path randomization quasi Monte Carlo,BBPR-QMC)方法.首先,用Faure序列代替MC方法中的随机序列,得到了Faure序列的拟蒙特卡罗(quasi Monte Carlo,QMC)模拟方法;其次,应用Moro算法得到了随机化拟蒙特卡罗(randomization quasi Monte Carlo,R-QMC)模拟方法;最后,将QMC方法和R-QMC方法结合,利用布朗桥技术来降低有效维,得到障碍期权定价的BBPR-QMC方法.数值试验表明,与MC方法和R-QMC方法相比较,BBPR-QMC方法模拟的价格与真实价格更接近、收敛速度更快.数值试验证实,BBPR-QMC方法是一种高效求解障碍期权定价的数值方法.
文摘The development of the option price theory provides business enterprise a beneficial tool to carry through property risk management, but a variety of option price theories are established on certain environments, and they can not deal with crisis in uncertain environments precisely and quickly, especially when multi-factors change at the same time. Thus, price the option in uncertain environment has been becoming an important direction of research. In this paper, wc take the stock option for example~ using Quasi-Monte Carlo method to price the American-style option, and then provide an example to explain. The powerful assistant decision-making ability of the computer simulation is clearly expressed when we study and analyze the Quasi-Monte Carlo method＇s characteristics.
基金the National Natural Science Foundation of China (No.19871083)
文摘In the present note the convergence problem of the sequential number-theoretic method for optimization proposed by Fang and Wang is studied, the convergence criteria and the estimation of errors concerning this algorithm are given.
