Algorithms for Discrete and Stochastic Partial Differential Equations

这个项目关注的发展和分析的efficientnumerical算法的问题所产生的计算建模，重点是两个主要议题：算法系统的方程所产生的随机有限元法，算法代数系统所产生的模型中的流体动力学。 第一个解决的事实是，物理现象的模型往往包含参数或方程系数，其精确的属性没有很好地理解。 例子包括介质的渗透性，地下水中的污染物）流动或扩散，以及边界条件（例如，沿着海底）。 在随机有限元法中，问题的随机方面是以类似于引入新的空间维度的方式处理的。 这种方法似乎有潜力比蒙特-卡罗方法更有效，提供有效的算法可用于离散后生成的代数系统。 我们的目标是研究这种方法所产生的算法问题。 在第二个项目中，我们将开发和研究求解不可压缩流模型中方程组的有效算法，主要是从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。 这些都是贯穿于流体力学的基本问题，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建模的实用性和有效性，以理解科学和工程现象。 对于许多复杂的物理过程，包括血液流动、环境污染物的扩散、航空航天器的性能和大气的地形现象，都有有用的模型。 通过纯粹的实验技术来理解这样的过程是极其昂贵的，甚至是不可能的，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，引入了对物理的基本理解。然而，只有在使用可靠和快速的求解算法的情况下，才能得到精确的解。 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沿着流动的速度，是不确定的。 我们这项工作的目标是开发快速解决算法的数学模型，并确保解决策略能够处理不确定性，并产生可靠的统计信息的解决方案在低计算成本。

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.