Algorithms for Discrete and Stochastic Partial Differential Equations

本项目涉及对计算建模中出现的问题的高效数值算法的开发和分析，重点是两个主要主题：随机有限元方法产生的方程系统的算法和流体动力学模型中产生的代数系统的算法。第一个问题是物理现象的模型常常包含参数或方程系数，这些参数或方程系数的精确性质还没有被很好地理解。例如，流动或扩散的数量(例如，地下水中的污染物)的介质的渗透性，以及边界条件(例如，沿海底)。在随机有限元方法中，处理问题的随机方面的方式类似于引入新的空间维度。这种方法似乎有可能比蒙特卡罗方法更有效，前提是有有效的算法可用于离散化后重新生成的代数系统。我们的目的是研究这种方法产生的算法问题。对于第二个项目，我们将开发和研究求解不可压缩流动模型中的方程组的有效算法，主要是由定常解的线性稳定性分析导出的特征值问题的方法，以及离散对流-扩散方程的多重网格算法。这些都是流体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。本项目的总体目标是提高数学模型的实用性和有效性，以了解科学和工程现象。有许多不同的物理过程的有用模型，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。通过纯粹的实验技术来理解这样的过程是昂贵得令人望而却步的，甚至是不可能的，而使用建模和算法解决方案通过提供对流量和压力等量的近似来引入对物理学的基本理解。然而，只有在可以使用可靠和快速的求解算法的情况下，精确的解决方案才是可用的。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的流动速度，并不是确定的。我们这项工作的目标是为数学模型开发快速求解算法，并确保求解策略能够处理不确定性，并以较低的计算成本产生关于解的可靠统计信息。

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.