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。这项研究探索了新颖的和已建立的实用算法，然后在各种水文地质场景中进行了验证和验证。总的来说，计算数学研究的一个主要目标是提高我们对数值算法的理解，使其更加高效和准确。然而，改进方法并应用它们的任务往往是由不同的研究人员小组完成的，并且从开发到应用需要很长的转移时间。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经被实践者提出并使用，而没有太多的理论理由或针对一系列狭隘的问题。基于新的数学见解的并行改进通常与实际问题无关。这项研究通过将先进和抽象的数学理论与实践重新联系起来，解决了这两个学科轨迹之间的差距。本项目致力于发展和分析可扩展到任意阶次的自适应、守恒和单调离散格式，用于求解非线性偏微分方程组，如用于模拟变饱和渗流的Richards方程和多孔弹性力学中的Biot模型。通常，这种离散化方法会导致大规模的、病态的线性系统。这类系统的有效解通常是非对称的和不确定的，这对整个数值模拟的性能至关重要，因为它消耗了很大一部分计算资源。该项目的一部分包括开发一类高效和自适应的多级解算器，能够生成灵活的空间层次结构。这种层次结构在过滤和表示稀疏数据集时也很有用，这些稀疏数据集相对于数据的结构和类型是健壮的：平滑、振荡或两者的组合。在水文地质学的有针对性的应用中，将研究有效地逼近高程、地下水位、降水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.