Unifying Probabilistic Computation for PDEs and Linear Systems

这项研究计划专注于加速计算机模型，特别是当它们被合并到逆问题中时。计算机模型通过求解模拟现实的方程来发挥作用，例如线性系统或偏微分方程组(PDE)。这些模型可以用来模拟不同的过程，从人类心脏的电导率到恒星的演化。然而，如果没有对其参数的估计，计算机模型就毫无用处，而这些估计通常是从被称为逆问题的问题中的数据中获得的。求解反问题通常需要对不同参数值重复从一个模型进行模拟。为了解决这个问题，我们将与图宾根大学合作开发新的概率数值方法(PNMS)。PNMS是一种数值方法，它返回描述离散化误差引起的不确定性的概率分布。用户可以使用更粗的离散化来更快地逼近偏微分方程组或线性系统的解，并因此获得更广泛的不确定性量化，反映出求解器对解的信心较低。重要的是，这种不确定性可以严格地传播到反问题的解中，因此参数估计反映了求解器中的精度水平。这允许用户减少求解逆问题的计算费用，同时保留统计上严格的参数估计。更详细地说，我们将重点解决概率线性求解器和偏微分方程求解器的两个核心问题。在前一种情况下，我们将开发比现有例程更快、更准确的解算器，方法是承认并纠正这些解算器的基本方法中的一个矛盾(即，所使用的过程不承认数据依赖于解决方案的事实)。对于偏微分方程组的解算器，我们将专注于求解非线性偏微分方程组，这是更具挑战性的问题，并提供了对物理现象的更真实的描述，但到目前为止还没有得到严格的概率处理。为了展示这些解算器的影响，我们将与鲁汶大学的合作伙伴将它们应用于天体物理学中一个具有挑战性的逆问题：基于恒星风的测量结果对恒星进行3D反投影。这一影响可能是深远的。计算机模型在应用科学和工业中被广泛使用，例如在制造业和工程学、生物学和医疗保健中，加速它们可能是实现更广泛使用的垫脚石。与此同时，在能源成本上升和芯片短缺的背景下，降低已经部署的机型的成本可能会带来重大的经济效益。

This research programme focuses on accelerating computer models, particularly when they are incorporated into inverse problems. Computer models function by solving equations that model reality, such as linear systems or partial differential equations (PDEs). These can be used to model diverse processes, from electrical conductivity in the human heart to stellar evolution. However computer models are useless without estimates of their parameters, and these estimates are often obtained from data in a problem referred to as the inverse problem. Solving an inverse problem usually requires simulating from a model repeatedly for different values of the parameters. As a result it is highly computationally expensive, and is often a limiting factor in the complexity of models that can be used.To address this we will develop novel probabilistic numerical methods (PNMs) in close partnership with the University of Tuebingen. PNMs are numerical methods that return a probability distribution describing uncertainty due to discretisation error. A user can use a coarser discretisation to approximate the solution to the PDE or linear system faster, and obtain "wider" uncertainty quantification as a result, reflecting that the solver is less confident in the solution. Importantly, this uncertainty can be propagated rigorously into the solution of an inverse problem, so that parameter estimates reflect the level of accuracy in the solver. This allows the user to reduce the computational expense of solving the inverse problem while retaining statistically rigorous parameter estimates. In more detail, we will focus on solving two problems at the heart of probabilistic linear solvers and PDE solvers. In the former case, we will develop solvers that are faster and more accurate than existing routines, by acknowledging and correcting for a contradiction in the fundamental methodology of those solvers (namely, that the procedure employed does not acknowledge the fact that the data depend on the solution). For PDE solvers, we will focus on solving nonlinear PDEs, which are more challenging to solve and provide a more realistic description of physical phenomena, but have thus far eluded a rigorous probabilistic treatment. To demonstrate the impact of these solvers we will apply them to a challenging inverse problem in astrophysics with partners at KU Leuven: 3D deprojection of stars based on measurements of their stellar wind.The impact of this could be far reaching. Computer models are used widely in the applied sciences and industry, for example in manufacturing and engineering, biology and healthcare, and accelerating them could be a stepping stone to enabling more widespread use. At the same time, in the context of rising energy costs and chip shortages, reducing the cost of models that have already been deployed could provide major economic benefits.