A Multilevel Monte Carlo Ensemble Scheme for Solving Random Parabolic PDEs
A Multilevel Monte Carlo Ensemble Scheme for Solving Random Parabolic PDEs
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发表时间：
2018-02
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通讯作者：
Yan Luo;Zhu Wang
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作者：
Yan Luo;Zhu Wang
最近在[26]中引入了一种一阶Monte Carlo系综方法来求解随机系数抛物方程，这是基于系综的Monte Carlo采样算法和基于系综的一阶时间步进方案的自然合成。通过引入扩散函数的系综平均，该算法得到了一组实现的具有多个右手边的单个离散系统，该系统可以比线性系统序列更有效地求解。在本文中，我们沿着同样的方向发展了一种新的多层Monte Carlo系综方法来求解随机抛物型偏微分方程。与文献[26]中的方法相比，该方法在时间上具有更高的精度，并且由于采用了多层蒙特卡罗方法，进一步降低了计算量。严格的数值分析表明，该方法达到了最佳的收敛速度。几个数值实验来说明理论结果。
A first-order, Monte Carlo ensemble method has been recently introduced for solving parabolic equations with random coefficients in [26], which is a natural synthesis of the ensemble-based, Monte Carlo sampling algorithm and the ensemble-based, first-order time stepping scheme. With the introduction of an ensemble average of the diffusion function, this algorithm leads to a single discrete system with multiple right-hand sides for a group of realizations, which could be solved more efficiently than a sequence of linear systems. In this paper, we pursue in the same direction and develop a new multilevel Monte Carlo ensemble method for solving random parabolic partial differential equations. Comparing with the approach in [26], this method possesses a high-order accuracy in time and further reduces the computational cost by using the multilevel Monte Carlo method. Rigorous numerical analysis shows the method achieves the optimal rate of convergence. Several numerical experiments are presented to illustrate the theoretical results.