The paper proposes a new automatic/algorithmic differentiation for the solutions to partial differential equations of parabolic type. In particular, we provide a higher order discretization scheme which is a natural extension of the standard automatic differentiation. A Brownian polynomial approach is introduced to avoid the Lévy area simulation. The Lie brackets of vector fields associated with stochastic differential equation play an important role in the proposed scheme. The case that the test function is non-smooth but has Gateaux derivative is considered. Numerical examples are shown to confirm the effectiveness

中文翻译：

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抛物型偏微分方程解的自动微分加速：高阶离散化

针对抛物线型偏微分方程的解，本文提出了一种新的自动/算法微分方法。特别是，我们提供了高阶离散化方案，这是标准自动微分的自然扩展。引入布朗多项式方法来避免Lévy区域模拟。与随机微分方程相关的矢量场的李括号在所提出的方案中起着重要的作用。考虑测试函数不平滑但具有Gateaux导数的情况。数值例子证实了有效性