Fast Solvers for Real-World PDE-Constrained Optimization

运筹学中大量重要且具有挑战性的应用都受到最优化问题的制约。这类问题中最重要的一类是偏微分方程(PDE)约束优化问题，这类问题对实际过程有很大的适用性，其中优化问题的求解以偏微分方程作为约束。举个例子，这样的公式广泛出现在图像处理应用中：这产生了与来自世界各地的科学和技术挑战的关键联系，例如确定大脑等复杂人体器官的健康，探索地下地质结构，以及使谷歌汽车能够通过评估交通状况而在没有人工驾驶的情况下工作。偏微分方程约束优化问题提供的可能性是巨大的，因此它们最近引起了数学研究人员以及更广泛的应用科学家的极大兴趣。这些公式还可以用来描述广泛领域的过程，如流体动力学、化学和生物机制、其他图像处理问题，如医学成像、天气预报、金融市场和期权定价问题、电磁逆问题和许多其他重要的应用。因此，对这些问题的研究是一个前沿的研究领域，可以在运筹学和优化领域取得巨大的进步。在这些问题上已经有了很多的理论工作，但数值求解这些优化问题的策略的构建是一个相对较新的发展。在这个项目中，我希望为所涉及的矩阵系统(这些系统包含问题产生的所有方程)建立快速有效的求解器。求解器与强大的“预条件器”的开发相结合(其想法是在某种意义上准确地逼近相应的矩阵，但以一种在计算机上应用成本较低的方式)。实现这一点是非常不平凡的挑战，原因有很多，具体而言，在任何时刻存储整个矩阵通常是不可行的，很难建立以有效方式捕捉矩阵的性质并且应用也很便宜的近似，经常需要建立可并行的求解器(意味着计算可以在许多不同的计算机上同时执行)，并且经常需要执行重新计算许多不同的矩阵的昂贵的过程。该项目的目的是建立强大的求解器，其抵消上述问题，用于具有重要现实和工业价值的偏微分方程约束优化问题。我将考虑四个具体的应用：医学成像应用中的最优控制问题，图像处理问题的PDE约束优化公式，流体流动的最优控制模型，以及化学和生物过程中出现的控制问题。我将考虑具有最大实际潜力的问题陈述，并为这些问题制定可行、快速和有效的解决策略。

A huge number of important and challenging applications in operational research are governed by optimization problems. One crucial class of these problems, which has significant applicability to real-world processes, is that of partial differential equation (PDE)-constrained optimization, where an optimization problem is solved with PDEs acting as constraints. To provide one illustration, such formulations arise widely in image processing applications: this produces a crucial link to scientific and technological challenges from far-and-wide, for example determining the health of complex human organs such as the brain, exploring underground geological structures, and enabling Google cars to function without a human driver by assessing traffic situations. The possibilities offered by PDE-constrained optimization problems are immense, and consequently they have recently attracted tremendous interest from researchers in mathematics, as well as applied scientists more widely. These formulations may also be used to describe processes in fields as wide-ranging as fluid dynamics, chemical and biological mechanisms, other image processing problems such as medical imaging, weather forecasting, problems in financial markets and option pricing, electromagnetic inverse problems, and many other applications of importance. The study of these problems is therefore a cutting-edge research area, and one which can forge a huge advance in the fields of operational research and optimization.There has been much theoretical work undertaken on these problems, however the construction of strategies for solving these optimization problems numerically is a relatively recent development. In this project I wish to build fast and effective solvers for the matrix systems involved (these systems contain all of the equations which arise from the problem). The solvers are coupled with the development of a powerful 'preconditioner' (the idea of which is to approximate the corresponding matrix accurately in some sense, but in a way that is cheap to apply on a computer). Carrying this out is a highly non-trivial challenge for many reasons, specifically that it is often infeasible to store the matrix in its entirety at any one time, it is very difficult to build an approximation that captures the properties of the matrix in an effective way and is also cheap to apply, it is frequently necessary to build solvers which are parallelizable (meaning that computations may be carried out on many different computers at one time), and one is often required to carry out the expensive process of re-computing many different matrices.The aim of this project is to build powerful solvers, which counteract the above issues, for PDE-constrained optimization problems of significant real-world and industrial value. I will consider four specific applications: optimal control problems arising from medical imaging applications, PDE-constrained optimization formulations of image processing problems, models for the optimal control of fluid flow, and control problems arising in chemical and biological processes. I will consider problem statements that have the maximum practical potential, and generate viable, fast and effective solution strategies for these problems.