RUI: Analysis and Numerical Solutions for Stochastic Stokes Equations

Pi和他的同事研究不可压缩粘性流体的数值模拟，遵循Stokes方程。该研究将研究附加的随机强迫项、随机粘性和随机边值的影响。将使用和比较构造解族的方法。PI感兴趣的是输入中的不确定性(强迫项)和输出中的不确定性(用于估计期望值的解值)之间的关系。主要包括三个部分：1.随机输入参数随机Stokes方程的灵敏度导数增强的蒙特卡罗方法；2.白噪声强迫项随机Stokes方程的有限元逼近；3.具有随机系数和边界条件的多项式混沌空间的随机谱有限元方法。计算机计算可能看起来精准得令人难以置信。但科学家们发现，在所有物理系统中都存在着相当大的不确定性；不仅是在测量某件东西的时候，甚至在试图描述系统如何变化的时候也是如此。任何计算机计算都不可能考虑到正在研究的系统的测量和动力学中的每一个微小变化。然而，我们知道，至少在某些情况下，少量的不确定性可能会导致计算结果中的重大误差，甚至是灾难性的误差。拟议的研究是努力了解、量化和控制不确定性影响的一个例子，在这种情况下，对于涉及流体流动的标准物理计算。这项研究的一个重要好处是让一群以少数族裔为主的学生参与进来，他们将得到开始科学生涯所需的支持、指导和培训。

The PI and colleagues study the numerical simulation of an incompressible The PI and colleagues study the numerical simulation of an incompressible viscous fluid, obeying the Stokes equations. The research will study the effect of an additional stochastic forcing term, random viscosity and random boundary values . Methods of constructing families of solutions will be used and compared. The PI is interested in the relationship between uncertainty in the input (forcing term) to the uncertainty in the output (solution values used to estimate expected values.) The research has three major components: 1. Monte Carlo methods enhanced with sensitivity derivatives, for the stochastic Stokes equations with random input parameters; 2. finite element approximation of stochastic Stokes equations with white noise forcing terms; 3. the stochastic spectral finite element method with polynomial chaos spaces, with random coefficients and boundary conditions. Computer calculations can seem deceptively precise. But scientists have discovered that there is a significant amount of uncertainty in all physical systems; not just when something is measured, but even when an attempt is made to describe how the system changes. No computer calculation can possibly consider every slight variation in the measurements and dynamics of a system under study. And yet it is known that, at least in some cases, small amounts of uncertainty can lead to significant, and even disastrous, errors in the computed results. The proposed research is one example of the effort to understand, quantify and control the effect of uncertainties, in this case, for a standard physical computation involving the flow of a fluid. An important bonus of this research is the involvement of a group of largely minority students, who will receive the support, guidance and training necessary to begin scientific careers.