Stochastic Partial Differential Equations and Their Numerical Solution

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等都是由随机偏微分方程（SPDE）建模的。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合并时空随机性，并捕获具有多个自由度的系统固有的随机波动的影响。然而，它们比偏微分方程更难于数值求解，并且需要新的专门技术。实际上，由于SPDE解非常粗糙，直接采用数值PDE方法通常会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（光谱随机游走法），专为spde量身定制。REU的学生也参与了该项目的工作。Spectrwm是一类用于spde的马尔可夫跳跃过程逼近。因此，这些近似可以看作是布朗运动的随机游走方法的无限维类似物。除了它们的数学意义之外，这些方法以一种自然的方式解决了在模拟spde时遇到的几个问题，包括长时间模拟，准确表示乘法噪声的影响，以及它们的路径依赖期望值的近似值。通过适当地构建它们，Spectrwm忠实于其解决方案的领域。研究者和他的合作者的目标是使用概率技术定量地理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，包括那些经典不适定的问题或底层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SPDE本身的见解：与SPDE相关的Kolmogorov方程的性质，边界对SPDE解的影响，SPDE解的亚稳性，以及经典病态SPDE的重整化。REU的学生也参与了该项目的工作。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.