We will introduce Euler-Maruyama approximations given by an orthogonal system in $L^{2}[0,1]$ for high dimensional SDEs, which could be finite dimensional approximations of SPDEs. In general, the higher the dimension is, the more one needs to generate uniform random numbers at every time step in numerical simulation. The schemes proposed in this paper, in contrast, can deal with this problem by generating very few uniform random numbers at every time step. The schemes save time in the simulation of very high dimensional SDEs. In particular, we conclude that an Euler-Maruyama approximation based on the Walsh system is efficient in high dimensions.
翻译:我们将引入由正方形系统以$L ⁇ 2}[0,1]美元提供的高维SDE近似值。 高维SDE可能是SPDE的有限维近近似值。 一般来说, 维度越高, 就越需要在数字模拟的每个阶段生成统一的随机数字。 相反, 本文中提议的计划可以通过在每一个步骤生成非常少的统一随机数字来解决这个问题。 计划节省了非常高维SDE模拟的时间。 特别是, 我们得出结论, 基于 Walsh 系统的Euler- Maruyama近近似值在高维度上是有效的。