Algorithms for Discrete and Stochastic Partial Differential Equations

本项目关注于在计算建模中出现的问题的有效数值算法的开发和分析，重点放在两个主要主题：从随机有限元方法中产生的方程系统的算法，以及流体动力学模型中产生的代数系统的算法。第一个问题解决了这样一个事实，即物理现象的模型经常包含参数或方程系数，其精确性质尚未得到很好的理解。例子包括数量（例如，地下水中的污染物）流动或扩散的介质的渗透性特性，以及边界条件（例如，沿着海底）。在随机有限元法中，问题的随机方面以一种类似于引入新空间维度的方式处理。这种方法似乎具有比蒙特卡罗方法更有效的潜力，提供有效的算法可用于离散化后生成的代数系统。我们的目标是研究这种方法产生的算法问题。对于第二个项目，我们将开发和研究解决不可压缩流动模型中出现的方程组的有效算法，主要是由稳定解的线性稳定性分析导出的特征值问题的方法，以及离散转换扩散方程的多网格算法。这些都是流体动力学中出现的基本问题，它们的有效解决对于开发有效的计算模型至关重要。该项目的总体目标是提高数学建模在理解科学和工程现象方面的效用和有效性。对于许多不同的物理过程，包括血液流动、环境污染物的扩散、航空航天飞行器的性能以及大气和海洋现象，都有有用的模型。通过纯粹的实验技术来理解这些过程是非常昂贵或不可能的，而使用建模和算法解决方案通过提供流量和压力等量的近似值来介绍对物理的基本理解。然而，只有使用可靠和快速的求解算法，才能获得准确的解。此外，通常情况下，模型的某些方面，如输送介质的地质性质或沿边界流动的速度，是不确定的。我们这项工作的目标是为数学模型开发快速解决算法，并确保解决策略能够处理不确定性，并以低计算成本产生关于解决方案的可靠统计信息。

This project concerns the development and analysis of efficientnumerical algorithms for problems arising in computational modeling,with emphasis on two main topics: algorithms for systems of equationsarising from the stochastic finite element method, and algorithms foralgebraic systems arising in models of fluid dynamics. The first ofthese addresses the fact that models of physical phenomena oftencontain parameters or equation coefficients whose precise propertiesare not well understood. Examples include permeability properties ofmedia in which quantities (e.g., pollutants in groundwater) areflowing or diffusing, and boundary conditions (e.g., along the oceanbottom). In the stochastic finite element method, the random aspectsof problems are handled in a manner analogous to the introduction ofnew spatial dimensions. This methodology appears to have thepotential to be more efficient than Monte-Carlo methods, providedefficient algorithms are available for the algebraic systems that aregenerated after discretization. Our aim is to study the algorithmicissues that arise from this approach. For the second project, we willdevelop and study efficient algorithms for solving systems ofequations arising in models of incompressible flow, principally,methods for eigenvalue problems derived from linear stability analysisof steady solutions, and multigrid algorithms for the discreteconvection-diffusion equation. These are fundamental problems arisingthroughout fluid dynamics, and their efficient solution is criticalfor development of effective computational models.The general aim of this project is to enhance the utility andeffectiveness of mathematical modeling for understanding scientificand engineering phenomena. There are useful models for many disparatephysical processes, including blood flows, dispersal of environmentalpollutants, performance of aerospace vehicles, and atmospheric andoceanographic phenomena. Understanding such processes through purelyexperimental techniques is prohibitively expensive or impossible,whereas the use of modeling and together with algorithmic solutionintroduces a basic understanding of the physics by providingapproximations to quantities such as flow rates and pressures.Accurate solutions are only available, however, if reliable and fastsolution algorithms can be used. Moreover, it is often the case thatcertain aspects of models, such as the geologic properties oftransporting media or the velocities of flows along boundaries, arenot known with certainty. Our goal for this work is to develop fastsolution algorithms for mathematical models and to ensure that thesolution strategies are able to handle uncertainty and to producereliable statistical information about solutions at low computationalcost.