Stochastic Partial Differential Equations and Their Numerical Solution

随机偏微分方程（SPDE）模拟了各种时空现象，如湍流，相场动力学，表面生长，神经元活动，人口动态和利率波动。 与偏微分方程（PDE）相比，SPDE是有利的，因为它们允许建模者显式地将时空随机性和捕获随机波动的影响，固有的系统与许多自由度。 然而，他们是棘手的数值求解比偏微分方程，需要新的专门技术。 实际上，由于SPDE解非常粗糙，数值PDE方法的直接适应经常导致不能准确表示实际解的行为的近似。 为了解决这个问题，该项目开发了一种新的数值方法，称为Spectrwm [spek-trum]（表示谱随机游走方法），专门针对SPDE。 REU的学生参与了这个项目的工作。Spectrwm是一个关于随机微分方程的马尔可夫跳跃过程近似族。 因此，这些近似可以被视为布朗运动的随机游走方法的无限维类似物。 除了它们的数学兴趣之外，这些方法以自然的方式解决了模拟SPDE时遇到的几个问题，包括长时间模拟，准确地表示乘性噪声的影响，以及它们的路径相关期望值的近似。 通过适当地构造它们，Spectrwm忠实于它们的解的域。 研究者和他的合作者的目标是使用概率技术来定量地理解Spectrwm方法在具有反射边界条件的适定SPDE中的有限和长时间行为，并为更广泛的SPDE问题开发Spectrwm方法的加速变体，包括经典不适定或底层过程的算子不太清楚的问题。 作为一个主要的副产品，该项目有助于深入了解SPDE本身：与SPDE相关的Kolmogorov方程的性质，边界对SPDE解的影响，SPDE解的亚稳定性，以及经典不适定SPDE的重正化。 REU学生也参与了该项目的工作。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Space-time phenomena as varied as turbulence, phase-field dynamics, surface growth, neuronal activity, population dynamics, and interest rate fluctuations are modelled by stochastic partial differential equations (SPDE). In comparison with partial differential equations (PDE), SPDEs are advantageous because they allow modelers to explicitly incorporate space-time randomness and capture the effect of random fluctuations inherent to systems with many degrees of freedom. However, they are trickier to numerically solve than PDEs and require new specialized techniques. Indeed, since SPDE solutions are very rough, straightforward adaptations of numerical PDE methods often lead to approximations that do not accurately represent the behaviors of the actual solutions. To address this issue, the project develops a new numerical method, called Spectrwm [spek-trum] (signifying spectral random walk method), that is tailored specifically to SPDEs. REU students are included in the work of the project.Spectrwm is a family of Markov jump process approximations for SPDEs. As such, these approximations can be viewed as infinite-dimensional analogues of random walk methods for Brownian motion. Aside from their mathematical interest, these methods address in a natural way several issues encountered when simulating SPDEs, including long-time simulation, accurately representing the effect of multiplicative noise, and approximation of their path-dependent expected values. by suitably constructing them, Spectrwm is faithful to the domain of their solution. The investigator and his collaborators aim to use probabilistic techniques to quantitatively understand the finite- and long-time behavior of Spectrwm methods for well-posed SPDEs with reflecting boundary conditions, and to develop accelerated variants of Spectrwm methods for a wider range of SPDE problems, including ones that are classically ill-posed or where the operators of the underlying processes are not well understood. As a major byproduct, the project lends insight into SPDEs themselves: the nature of the Kolmogorov equations associated to SPDEs, the influence of boundaries on SPDE solutions, metastability in SPDE solutions, and renormalizations for classically ill-posed SPDEs. REU students are included in the work of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.