论文标题
动力学SPDE的数值解决方案通过随机Magnus膨胀
Numerical solution of kinetic SPDEs via stochastic Magnus expansion
论文作者
论文摘要
在本文中，我们展示了如何使用ITô-Stochastic Magnus扩展来有效地求解具有两个空间变量的随机偏微分方程（SPDE）。为此，我们首先仅通过利用有限差异方法来将SPDE离散在空间中，并矢量化所得的方程利用其稀疏性。作为基准，我们将其应用于具有恒定系数的随机Langevin方程的情况，其中有明确的解决方案，并将Magnus方案与Euler-Maruyama方案进行比较。我们将看到，通过使用单个GPU并在可变系数情况下对其进行验证，Magnus的扩展在精度和尤其是计算时间方面都是优越的。值得注意的是，根据准确性目标和空间分辨率，我们将看到与欧拉 - 玛鲁山方案相比，订单范围为20到200的速度加速。
In this paper, we show how the Itô-stochastic Magnus expansion can be used to efficiently solve stochastic partial differential equations (SPDE) with two space variables numerically. To this end, we will first discretize the SPDE in space only by utilizing finite difference methods and vectorize the resulting equation exploiting its sparsity. As a benchmark, we will apply it to the case of the stochastic Langevin equation with constant coefficients, where an explicit solution is available, and compare the Magnus scheme with the Euler-Maruyama scheme. We will see that the Magnus expansion is superior in terms of both accuracy and especially computational time by using a single GPU and verify it in a variable coefficient case. Notably, we will see speed-ups of order ranging form 20 to 200 compared to the Euler-Maruyama scheme, depending on the accuracy target and the spatial resolution.
