RUI: Analysis and Numerical Solutions for Stochastic Stokes Equations

根据Stokes方程，PI及其同事研究了一种不可压缩粘性流体的数值模拟。该研究将研究附加的随机强迫项、随机粘度和随机边界值的影响。构造解族的方法将被使用和比较。PI感兴趣的是输入（强制项）的不确定性与输出（用于估计期望值的解值）的不确定性之间的关系。本研究有三个主要组成部分：1。具有随机输入参数的随机Stokes方程的灵敏度导数增强蒙特卡罗方法2. 带白噪声强迫项的随机Stokes方程的有限元逼近；3. 具有多项式混沌空间、随机系数和边界条件的随机谱有限元方法。计算机计算可能看起来很精确。但科学家们发现，在所有的物理系统中都存在大量的不确定性；不仅仅是在测量某些东西的时候，甚至在试图描述系统如何变化的时候。任何计算机计算都不可能考虑所研究系统的测量和动力学中的每一个细微变化。然而，众所周知，至少在某些情况下，少量的不确定性会导致计算结果出现重大甚至灾难性的错误。拟议的研究是努力理解、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有一群主要是少数民族学生的参与，他们将得到开始科学事业所需的支持、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.