论文标题
用布朗运动解决椭圆方程：减少偏差和时间差异学习
Solving Elliptic Equations with Brownian Motion: Bias Reduction and Temporal Difference Learning
论文作者
论文摘要
Feynman-Kac公式提供了一种方法，可以从连续时间马尔可夫过程的期望方面了解椭圆形偏微分方程的解决方案。这种连接允许基于这些马尔可夫过程的样本来创建用于解决方案的数值方案，在某些情况下，这些方法比传统数值方法具有优势。但是，幼稚的数值实现遭受统计偏差和采样误差的影响。我们提出了分散Feynman-KAC公式中出现的随机过程的方法，该过程减少了数值方案的偏差。我们还建议使用时间差学习以比传统的Monte Carlo方法更有效的方式从随机样本中组装信息。
The Feynman-Kac formula provides a way to understand solutions to elliptic partial differential equations in terms of expectations of continuous time Markov processes. This connection allows for the creation of numerical schemes for solutions based on samples of these Markov processes which have advantages over traditional numerical methods in some cases. However, naïve numerical implementations suffer from statistical bias and sampling error. We present methods to discretize the stochastic process appearing in the Feynman-Kac formula that reduce the bias of the numerical scheme. We also propose using temporal difference learning to assemble information from random samples in a way that is more efficient than the traditional Monte Carlo method.
