Numerical solutions of time-dependent stochastic partial differential equations

本课题研究时变随机偏微分方程（SPDEs）的数值解。特别是研究者将构建快速，实用的数值算法。同时，通过适当的误差分析，为这些算法提供坚实的理论基础。本课题拟集中研究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时间相关spde。将作出协调一致和全面的努力，以发展有效、准确和健壮的计算方法，从一开始就纳入不确定性的影响。本研究有三个主要组成部分：1。高维抛物型SPDEs的有限元逼近研究，其中强迫项包含乘性有色噪声或随机场2. 随机边界输入抛物型SPDEs统计矩数值计算的快速配置方法研究3. 使用灵敏度导数的增强蒙特卡罗方法，用于具有随机参数（如扩散系数）的spde。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。本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，如传热、扩散过程和流体流动动力学。这项研究的一个重要好处是参与了一组本科生和研究生，其中一些来自代表性不足的群体。预计他们参与这个项目将使他们接触到科学研究，促使他们继续接受进一步的培训，并考虑将来从事科学事业。

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.