包含分数阶拉普拉斯式的抛物方程的蒙特卡罗方法

IF 0.8 Q3 STATISTICS & PROBABILITY Monte Carlo Methods and Applications Pub Date : 2022-10-27 DOI:10.1515/mcma-2022-2129
Caiyu Jiao, Changpin Li
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引用次数: 0

摘要

摘要应用蒙特卡罗方法求解了具有分数阶拉普拉斯式的线性抛物型方程的Dirichlet问题。该方法利用了由具有跳跃的对称稳定lsamvy过程驱动的相关随机微分方程的弱逼近思想。我们利用跳跃适应方案来近似lsamvy过程,给出了精确的边界退出时间。当解的正则性较低时,通过去掉lsamvy过程的小跳变,建立了数值格式,并给出了收敛阶。当解具有较高的正则性时,我们用一个简单的过程代替小的跳跃,建立一个高阶的数值格式,然后显示更高的收敛阶。最后,给出了十维和一百维情况下的数值实验,验证了理论估计,并证明了所提格式对高维抛物方程的数值效率。
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Monte Carlo method for parabolic equations involving fractional Laplacian
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.
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来源期刊
Monte Carlo Methods and Applications
Monte Carlo Methods and Applications STATISTICS & PROBABILITY-
CiteScore
1.20
自引率
22.20%
发文量
31
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