Analysis of Numerical Methods for Partial Differential Equations with Random Data

物理过程，如多孔岩石中的流体流动或受面内应力影响的金属板的变形，使用偏微分方程（PDEs）进行数学建模。在传统的确定性建模中，pde的输入数据（如材料参数、边界数据和源项等）在计算域中的每个点都是已知的。然后寻求与输入的特定实现相对应的解的高精度数值近似值。例如，一个典型的地下水流动模型需要将感兴趣区域的几何形状、流体流经的多孔介质的渗透系数、流域边界的压头测量以及流中任何源或汇的位置作为输入。如果我们假设我们知道这些量是什么，那么我们可以继续找到单一的解（流体速度和相应的压力）如果它存在，对应于这些特定的输入。然而，人类的资源有限，特别是渗透率系数不可能准确地记录在流域的每个点上。事实上，在大多数工程问题中，建模者不知道控制偏微分方程的一个或多个输入，可能在一些孤立的位置除外。如果我们对PDE输入中的不完美知识（“认知不确定性”）诚实，那么我们需要以不同的方式处理我们的建模，并在概率框架中提出PDE问题。我们需要能够容纳不确定输入的数值近似方案，然后允许我们量化输出变量中产生的不确定性。例如，我们需要估计对公共安全至关重要的不良事件的概率，例如化学品被运输到地下水中的特定位置，或者受应力的金属板断裂。简而言之，必须将不确定性纳入物理过程的数学模型，以便进行风险评估。我们可以很容易地将偏微分方程中的未知输入表示为随机量；然后我们要解决所谓的随机偏微分方程。对于材料参数，如渗透率系数或弹性体的弹性模量，通常情况下，它们的值在两个不同的空间位置是相关的，因此讨论相关的随机数据（而不是白噪声）是合适的。在过去，由于计算资源的限制或非常原始的近似方案的阻碍，避免了求解具有相关随机数据的偏微分方程。简单地平均对应于输入的特定实现的多个解决方案可能会导致大量的计算时间浪费。近年来，人们提出了更复杂的数值方法来逼近具有相关随机数据的偏微分方程的解。不幸的是，这项工作仅限于标量、椭圆偏微分方程，而由此产生的线性方程组的有效线性代数问题在很大程度上被忽视了。特别是所谓的随机伽辽金方法，具有吸引人的近似性质，但由于缺乏鲁棒解算器而被忽视。更简单的方案，需要更少的用户知识，但最终更多的计算时间来实现，已经普及。本项目的目的是研究由具有两个输出变量的偏微分方程系统（例如随机多孔介质中的地下水流动）建模的更复杂工程问题中的不确定性量化的近似方案。我们将扩展和测试引入随机数据的标量偏微分方程的近似方案的效率，重点关注求解所得线性方程组的有效线性代数技术的发展。

Physical processes such as fluid flow in porous rocks or the deformation of a metal plate subject to an in-plane stress are modelled mathematically using partial differential equations (PDEs). In traditional deterministic modelling, input data for PDEs (e.g. material parameters, boundary data and source terms etc.) are assumed to be known at every point in the computational domain. Highly accurate numerical approximations of solutions that correspond to particular realizations of the inputs are then sought. For example, a typical groundwater flow model requires, as inputs, the geometry of the domain of interest, the permeability coefficients of the porous medium through which the fluid is moving, pressure head measurements at the boundary of the flow domain and the locations of any sources or sinks in the flow. If we assume we know what these quantities are, everywhere, then we can proceed to find the single solution (fluid velocity and corresponding pressure), if it exists, that corresponds to those particular inputs. However, we as human beings have limited resources and the permeability coefficients, in particular, can never be recorded exactly at every point in the flow domain. Indeed, in most engineering problems, one or more inputs to the governing PDEs will not be known to the modeller, save perhaps at a few isolated locations. If we are honest about our imperfect knowledge ('epistemic uncertainty') in inputs to PDEs then we need to approach our modelling in a different way and pose PDE problems in a probabilistic framework. We need numerical approximation schemes that can accommodate uncertain inputs and then allow us to quantify the resulting uncertainty in the output variables. For instance, we need to estimate probabilities of undesirable events that are critical to public safety such as a chemical being transported to a particular location in groundwater, or the fracture of a stressed metal plate. In short, it is imperative to incorporate uncertainty into mathematical models of physical processes so that risk assessments can be performed.We can easily represent the unknown inputs in PDEs as random quantities; we are then faced with solving so-called stochastic PDEs. For material parameters such as permeability coefficients or the modulus of elasticity of an elastic body, it is usually the case that their values, at two distinct spatial locations, are associated and so it is appropriate to talk about correlated random data (rather than white noise). In the past, solving PDEs with correlated random data has been avoided due to limitations in computing resources or else hampered by very primitive approximation schemes. Simply averaging multiple solutions that correspond to particular realisations of the inputs can result in a huge amount of wasted computation time. Recently, more sophisticated numerical methods for approximating solutions to PDEs with correlated random data have been proposed. Unfortunately, this work has been restricted to scalar, elliptic PDEs and the question of efficient linear algebra for the resulting linear systems of equations has been largely overlooked. So-called stochastic Galerkin methods, in particular, have attractive approximation properties but have been somewhat ignored due to a lack of robust solvers. More simplistic schemes which require less user know-how but ultimately more computing time to implement, have been popularised. The aim of this project is to investigate approximation schemes for quantifying uncertainty in more complex engineering problems modelled by systems of PDEs with two output variables (e.g. groundwater flow in a random porous medium). We will extend and test the efficiency of approximation schemes introduced for scalar PDEs with random data, paying significant attention to the development of efficient linear algebra techniques for solving the resulting linear systems of equations.