Stochastic Partial Differential Equations and Their Numerical Solution

各种时空现象，如湍流、相场动力学、表面生长、神经元活动、种群动力学和利率波动，都由随机偏微分方程组(SPDE)模拟。与偏微分方程组(PDE)相比，SPDEs具有明显的优势，因为它们允许建模人员明确地考虑时空随机性，并捕捉具有多个自由度的系统固有的随机波动的影响。然而，它们比偏微分方程组更难数值求解，并且需要新的专门技术。事实上，由于SPDE解是非常粗糙的，直接适应数值PDE方法通常会导致不能准确地代表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[speck-trum](表示光谱随机游动方法)，它是专门为SPDEs量身定做的。REU的学生被包括在这个项目的工作中。Spectrwm是一族关于SPDEs的马尔可夫跳跃过程近似。因此，这些近似可以看作是布朗运动的随机游动方法的无限维类似。除了数学上的兴趣之外，这些方法还以自然的方式解决了在模拟SPDE时遇到的几个问题，包括长期模拟、准确地表示乘性噪声的影响以及它们与路径相关的期望值的近似。通过适当地构造它们，Spectrwm忠于它们的解决方案的域。这位研究人员和他的合作者的目标是使用概率技术来定量地了解具有反射边界条件的适定SPDE的Spectrwm方法的有限和长期行为，并为更广泛的SPDE问题开发Spectrwm方法的加速变体，包括那些经典的不适定的问题或潜在过程的算子不被很好地理解的问题。作为一个主要的副产品，该项目提供了对SPDEs本身的洞察：与SPDEs相关的Kolmogorov方程的性质，边界对SPDEs解的影响，SPDEs解的亚稳定性，以及经典不适定SPDEs的重整化。该奖项反映了NSF的法定使命，通过使用基金会的学术价值和更广泛的影响审查标准进行评估，被认为是值得支持的。

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.