Multilevel Methods for Numerical Modeling with Applications in Hydrogeology

该项目的重点是开发新的数学工具，以提高我们对先进计算方法的理解和应用。该研究探索了新颖的以及建立了实用的算法，然后在各种水文地质情况下进行验证和验证。计算数学研究的一个主要目标，一般来说，是提高我们对数值算法的理解，使它们更有效和准确。然而，改进方法然后应用它们的任务通常由不同的研究人员小组完成，并且在开发和应用之间有很长的转换时间。 通常，理论发现和实践者开发的启发式算法遵循不同的轨迹。事实上，许多有价值的数值技术已被提出和使用的从业者没有太多的理论依据或一组狭窄的问题。基于新的数学见解的并行改进通常与实际问题无关。本研究通过将高级和抽象的数学理论与实践重新联系起来，解决了两个学科轨迹之间的差距。 该项目有可能影响广泛的应用，包括例如饱和渗流的模拟。该项目关注的是可扩展到任何阶的自适应、保守和单调离散化的开发和分析，用于求解非线性偏微分方程，例如用于模拟饱和渗流的理查兹方程和孔隙弹性中的毕奥模型。通常，这样的离散化方法导致大规模的病态线性系统。这类系统通常是非对称和不确定的，其有效解对于整个数值模拟的性能至关重要，因为它消耗了大部分的计算资源。 该项目的一部分包括开发一类高效和自适应的多级求解器，能够生成灵活的空间层次结构。这样的层次结构在过滤和表示稀疏数据集时也是有用的，相对于数据的结构和类型是鲁棒的：平滑的、振荡的或两者的组合。在水文地质的目标应用中，将研究有效近似高程、地下水位、降水量和其他相关水文地质空间数据的技术。

The project focuses on the development of new mathematical tools which improve our understanding and application of advanced computational methods. The research explores novel as well as established practical algorithms which are then validated and verified in a variety of hydrogeological scenarios. One major goal of computational mathematics research, in general, is to improve our understanding of numerical algorithms and to make them more efficient and accurate. However, the tasks of improving methods and then applying them are often completed by distinct groups of researchers and have long transfer times between development and application. Often, the theoretical findings and the heuristic algorithms developed by practitioners follow different trajectories. Indeed, many valuable numerical techniques have been proposed and used by practitioners without much theoretical justification or for a narrow set of problems. Concurrent improvements based on new mathematical insights are often unrelated to practical problems. This research addresses the disparity between the two disciplinary trajectories by reconnecting advanced and abstract mathematical theories with practice. The project has the potential to impact a wide range of applications, including for example the simulation of variably saturated flow.This project is concerned with the development and analysis of adaptive, conservative and monotone discretizations that are extendable to any order for the solution of nonlinear partial differential equations, such as Richards' equation used to simulate variably saturated flow and Biot's model in poroelasticity. Typically, such discretization methods result in large-scale, ill-conditioned linear systems. The efficient solution of such systems, generally non-symmetric and indefinite, is crucial for the performance of overall numerical simulation as it consumes the larger part of the computing resources. Part of this project includes the development of a class of efficient and adaptive multilevel solvers capable of generating flexible hierarchies of spaces. Such hierarchies are useful also in filtering and representing sparse data sets, robust with respect to the structure and the type of data: smooth, oscillatory or combinations of the two. In the targeted applications in hydrogeology, the research will be on techniques which efficiently approximate elevation, groundwater head, precipitation, and other relevant hydrogeological spatial data.