Numerical solutions of time-dependent stochastic partial differential equations

本课题是研究含时随机偏微分方程组的数值解。特别是在建立快速、实用的数值算法的同时，也将为这些算法在正确的误差分析基础上提供坚实的理论基础。该项目打算集中于随机抛物型偏微分方程作为我们的模型问题。然而，预计算法和分析将扩展到许多其他类型的依赖于时间的SPDEs。将作出协调和全面的努力，以开发高效、准确和健壮的计算方法，从一开始就考虑不确定性影响。这项研究有三个主要部分：1.研究高维抛物型SPDEs的有限元逼近；2.研究具有乘性色噪声或随机场的强迫项的高维抛物型SPDEs的有限元逼近；2.研究快速配置法，以数值计算具有随机边界输入数据的抛物型SPDEs的统计矩；3.利用灵敏度导数构造改进的蒙特卡罗方法，用于具有随机参数(如扩散系数)的SPDEs。科学家们已经发现，在所有物理系统中都存在着相当大的不确定性；不仅当测量某物时，甚至当试图描述系统如何变化时也是如此。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。这项研究旨在通过数值计算来理解、量化和控制不确定性的影响。下面的数学方程描述了基本的物理现象，如热传递、扩散过程和流体流动动力学。这项研究的一个重要好处是让一群本科生和研究生参与进来，其中一些人来自代表性不足的群体。预计他们参与这一项目将使他们接触到科学研究，促使他们继续接受培训，并将科学事业视为未来。

This project is to study numerical solutions for time-dependent stochastic partial differential equations (SPDEs). In particular the investigator will construct fast, practical numerical algorithms.At the same time, A solid theoretical basis for these algorithms based on proper error analysis will be provided. The project intends to concentrate on stochastic parabolic partial differential equations as our model problem. However, it is expected that the algorithms and analysis will be extended to many other types of timedependent SPDEs. A concerted and comprehensive effort will be madeto develop efficient, accurate, and robust computational methodologies, which from the very beginning incorporate uncertainty effects. The research has three major components: 1. Study of finite element approximations for high dimensional parabolic SPDEs with a forcing term involving either multiplicative colored noise or a random field; 2. Investigation of fast collocation methods to numerically evaluate statistical moments of parabolic SPDEs with random boundary input data; 3. Construction of enhanced Monte Carlo methods, using sensitivity derivatives, for SPDEs with random parameters such as diffusion coefficients.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 intends to make an effort to understand, quantify and control the effect of uncertainties through numerical computations. The underling mathematical equations describe basic physical phenomena such as heat transfer, diffusion processes and fluid flow dynamics. An important bonus of this research is the involvement of a group of undergraduate and graduate students, some of them from under represented groups. It is expected that their participation in this project will expose them to scientific research, induce them to pursue further training, and to consider a scientific career as a future.