Analysis of Numerical Methods for Partial Differential Equations with Random Data

物理过程，如流体在多孔岩石中的流动或金属板面内应力的变形，都是使用偏微分方程组(PDE)进行数学建模的。在传统的确定性建模中，偏微分方程的输入数据(如材料参数、边界数据和源项等)。假设在计算域中的每一点都是已知的。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。例如，典型的地下水流动模型需要输入感兴趣区域的几何形状、流体通过的多孔介质的渗透系数、流动区域边界处的压头测量以及流动中任何源或汇的位置。如果我们假设我们在任何地方都知道这些量是什么，那么我们就可以继续找到对应于这些特定输入的单一解(流体速度和相应的压力)，如果它存在的话。然而，作为人类，我们拥有有限的资源，特别是渗透率系数，永远不可能准确地记录在流动领域的每一个点。事实上，在大多数工程问题中，对管理PDE的一个或多个输入将不为建模者所知，可能只有几个孤立的位置。如果我们诚实地承认我们在PDE输入中的不完善知识(“认知不确定性”)，那么我们需要以一种不同的方式来处理我们的建模，并在概率框架中提出PDE问题。我们需要能够适应不确定输入的数值近似方案，然后允许我们量化由此产生的输出变量中的不确定性。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，例如化学物质被运送到地下水中的特定位置，或受力的金属板断裂。简而言之，将不确定性融入到物理过程的数学模型中，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，然后我们就面临着求解所谓的随机偏微分方程。对于材料参数，如渗透系数或弹性体的弹性模数，通常情况下，它们在两个不同的空间位置上的值是相关的，因此适当地谈论相关的随机数据(而不是白噪声)。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，求解具有相关随机数据的偏微分方程组是被避免的。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计算时间的浪费。最近，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组的解。遗憾的是，这项工作仅限于标量的、椭圆型的偏微分方程组，而所得到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Galerkin方法，具有吸引人的逼近性质，但由于缺乏健壮的求解器，在某种程度上被忽略了。更简单的方案已经流行起来，它们需要较少的用户知识，但最终需要更多的计算时间来实现。这个项目的目的是研究在更复杂的工程问题中量化不确定性的近似方案，这些问题由具有两个输出变量的偏微分方程组(例如，随机多孔介质中的地下水流动)建模。我们将扩展和测试为具有随机数据的标量偏微分方程组引入的近似方案的效率，重点关注用于求解由此产生的线性方程组的高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.