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优化问题的影响。这些问题中的一个关键类别，这对现实世界的过程具有重要的适用性，是偏微分方程（PDE）约束优化，其中优化问题是用PDE作为约束来解决的。举个例子，这种公式在图像处理应用中广泛出现：这与来自四面八方的科学和技术挑战产生了关键联系，例如确定大脑等复杂人体器官的健康状况，探索地下地质结构，以及通过评估交通状况使谷歌汽车能够在没有人类驾驶员的情况下运行。偏微分方程约束优化问题提供的可能性是巨大的，因此，他们最近吸引了数学研究人员的巨大兴趣，以及更广泛的应用科学家。这些公式也可用于描述流体动力学、化学和生物机制等领域的过程，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、电磁逆问题以及许多其他重要应用。因此，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。对这些问题进行了大量的理论工作，但是，解决这些优化问题的策略的建设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“预处理器”的开发相结合（其思想是在某种意义上准确地近似相应的矩阵，但在计算机上应用的方式很便宜）。由于许多原因，执行这一点是一个非常重要的挑战，特别是在任何时候都不可能完整地存储矩阵，很难建立一个以有效方式捕获矩阵属性的近似，并且应用起来也很便宜，通常需要构建可并行化求解器，（这意味着计算可以在许多不同的计算机上进行一次），并且经常需要执行重新计算许多不同矩阵的昂贵过程。该项目的目的是构建强大的求解器，其抵消了上述问题，用于具有重要现实世界和工业价值的PDE约束优化问题。我将考虑四个具体的应用：最优控制问题所产生的医学成像应用，偏微分方程约束的优化配方的图像处理问题，模型的流体流动的最优控制，以及控制问题所产生的化学和生物过程。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，快速和有效的解决方案。

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.