Modern Linear Algebra for PDE-Constrained Optimisation Models for Huge-Scale Data Analysis

什么才能准确地描述诸如流体流动机制或工业产品制造的化学反应等真实世界的过程？什么样的数学形式能使实践者保证流体的特定物理行为或运动，或使特定物质的产量最大化？问题的答案在于偏微分方程约束优化这一重要的科学领域。偏微分方程是一种叫做偏微分方程的数学工具。它们使我们能够建模和预测各种现实世界物理系统的行为。从优化的观点来看，一组特别重要的这样的问题是可以以某种期望的方式控制动力学的那些问题，例如通过向流体流动发生的区域施加力，或者以一定的速率插入化学反应物。通过以这种方式影响系统，我们能够生成现实世界过程的优化结果。因此，研究和理解偏微分方程约束优化问题是非常必要的，它所提供的可能性是巨大的，影响着应用数学、工程和实验科学领域的突破性研究。这些问题的关键现实世界应用出现在流体动力学，化学和生物机制，天气预报，图像处理，包括医学成像，金融市场和期权定价，以及许多其他领域。虽然大量的理论工作已经进行了这样的问题，它只是在过去的十年左右，重点放在解决他们的准确性和鲁棒性的计算机上，通过处理矩阵方程组的结果。支持这一建议的大部分研究涉及构建由“预条件子”加速的强大迭代方法，这些方法是通过以精确的方式近似相关矩阵来构建的，这样预条件子比求解矩阵系统本身便宜得多。应用我们的方法，然后可以打开大门，以前遥不可及的科学挑战，只存储和工作的矩阵是微小的系统相比，整体solved.Recently，偏微分方程约束优化问题已发现关键的适用性，从数据分析的问题。这是由于当今可用的巨大计算能力，这意味着存在存储和处理来自商业记录，在线新闻网站或健康数据库的大规模数据集的潜力。反过来，这导致了许多应用程序的数据驱动的过程被成功地建模由偏微分方程约束的优化问题。解决这些数据科学应用中的问题的算法必须能够跟上现实世界中数据爆炸的速度。在这个项目中，我们将研究不确定数据、图像处理问题、生物信息学应用和深度学习过程下的偏微分方程约束优化问题。对于每个问题，我们将设计最先进的数学模型来描述这个过程，然后我们将构建有效的迭代求解器和预处理器来处理得到的矩阵系统。我们的新算法将在理论和数值上得到验证，然后我们将发布一个开源代码库，以最大限度地提高其适用性和对现代优化和数据科学问题的影响。

What accurately describes such real-world processes as fluid flow mechanisms, or chemical reactions for the manufacture of industrial products? What mathematical formalism enables practitioners to guarantee a specific physical behaviour or motion of a fluid, or to maximise the yield of a particular substance? The answer lies in the important scientific field of PDE-constrained optimisation.PDEs are mathematical tools called partial differential equations. They enable us to model and predict the behaviour of a wide range of real-world physical systems. From the optimisation point-of-view, a particularly important set of such problems are those in which the dynamics may be controlled in some desirable way, for instance by applying forces to a domain in which fluid flow takes place, or inserting chemical reactants at certain rates. By influencing a system in this way, we are able to generate an optimised outcome of a real-world process. It is hence essential to study and understand PDE-constrained optimisation problems.The possibilities offered by such problems are immense, influencing groundbreaking research in applied mathematics, engineering, and the experimental sciences. Crucial real-world applications for such problems arise in fluid dynamics, chemical and biological mechanisms, weather forecasting, image processing including medical imaging, financial markets and option pricing, and many others. Although a great deal of theoretical work has been undertaken for such problems, it has only been in the past decade or so that a focus has been placed on solving them accurately and robustly on a computer, by tackling the matrix systems of equations which result. Much of the research underpinning this proposal involves constructing powerful iterative methods accelerated by 'preconditioners', which are built by approximating the relevant matrix in an accurate way, such that the preconditioner is much cheaper to apply than solving the matrix system itself. Applying our methodology can then open the door to scientific challenges which were previously out of reach, by only storing and working with matrices that are tiny compared to the systems being solved overall.Recently, PDE-constrained optimisation problems have found crucial applicability to problems from data analysis. This is due to the vast computing power that is available today, meaning that there exists the potential to store and work with huge-scale datasets arising from commercial records, online news sites, or health databases, for example. In turn, this has led to a number of applications of data-driven processes being successfully modelled by optimisation problems constrained by PDEs. It is essential that algorithms for solving problems from these applications of data science can keep pace with the explosion of data which arises from real-world processes. Our novel numerical methods for solving the resulting huge-scale matrix systems aim to do exactly this.In this project, we will examine PDE-constrained optimisation problems under the presence of uncertain data, image processing problems, bioinformatics applications, and deep learning processes. For each problem, we will devise state-of-the-art mathematical models to describe the process, for which we will then construct potent iterative solvers and preconditioners to tackle the resulting matrix systems. Our new algorithms will be validated theoretically and numerically, whereupon we will then release an open source code library to maximise their applicability and impact on modern optimisation and data science problems.