Unifying Probabilistic Computation for PDEs and Linear Systems

统一偏微分方程和线性系统的概率计算

• 批准号: EP/Y001028/1
• 负责人: Jonathan Cockayne
• 金额: $ 18.2万
• 依托单位: University of Southampton
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY001028%2F1
• 关键词: Unifying; Probabilistic; Computation; PDEs; Linear

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.

该研究计划的重点是加速计算机模型，特别是当它们被纳入逆问题。计算机模型通过求解模拟现实的方程来运行，例如线性系统或偏微分方程（PDE）。这些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，计算机模型是无用的，没有估计其参数，这些估计往往是从数据中获得的问题称为反问题。求解逆问题通常需要针对不同的参数值重复地从模型进行模拟。因此，它是非常昂贵的计算，往往是一个限制因素，在复杂的模型，可以使用。为了解决这个问题，我们将开发新的概率数值方法（PNM）与图宾根大学密切合作。PNM是返回描述由于离散化误差引起的不确定性的概率分布的数值方法。用户可以使用较粗糙的离散化来更快地近似PDE或线性系统的解，并因此获得“更宽”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决方案中，因此参数估计反映了求解器的精度水平。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，我们将专注于解决概率线性求解器和PDE求解器的核心问题。在前一种情况下，我们将开发比现有例程更快、更准确的求解器，方法是承认并纠正这些求解器的基本方法中的矛盾（即，所采用的程序没有承认数据依赖于解决方案的事实）。对于偏微分方程求解器，我们将专注于求解非线性偏微分方程，这是更具有挑战性的解决，并提供一个更现实的描述物理现象，但迄今为止，逃避严格的概率处理。为了展示这些求解器的影响，我们将与鲁汶大学的合作伙伴一起将它们应用于天体物理学中一个具有挑战性的逆问题：基于恒星风测量的恒星3D反投影。计算机模型广泛应用于应用科学和工业，例如制造和工程，生物学和医疗保健，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，降低已部署机型的成本可以带来重大的经济效益。