Numerical Analysis of Partial Differential Equation Constrained Optimization Problems

自然界中的每一个过程都试图在现有资源的情况下优化某些数量。有限的可用资源是约束，因此，约束优化问题在科学和工程中是普遍存在的。由于约束的存在，这类优化问题本质上是非线性的。这使得开发和分析用于模拟相关现象的算法和计算技术具有极大的挑战性和实际意义。需要一种系统的方法来分配资源，而不违反限制，并保持有效地将成本降至最低。该项目将导致开发先进、高效的算法来设计微米和纳米级的芯片实验室设备，以及具有所需配置和用户指定特性的新材料。作为形状优化工具的结果，将开发用于图像处理和结构设计的稳健算法。大量的工作将致力于开发心脏电反应和脑组织中水分子扩散的新模型。许多描述约束的偏微分方程是几何的、非线性的、多尺度的、具有未知区域的，即自由边界问题(FBPs)。这使得解决这些问题的数值技术的开发和分析具有挑战性。本项目主要研究具有偏微分方程约束的约束优化问题--即所谓的偏微分方程约束优化问题。事实上，它的目标是开发和研究数值技术，以解决经典思想和方法无法解决的这类新问题。它们是：以Stokes FBP(包括表面张力)为约束的优化，在代价泛函中出现海森的问题，形状优化和图像分割的多层次技术，非局部(分数)算子的优化，以及退化抛物方程作为约束。其应用范围从微米到纳米，其中表面效应占主体效应的主导地位。这将对设计下一代芯片上实验室设备产生影响。通过操纵曲率，人们可以将粒子推向所需的组件或特定的构型。这将导致具有用户指定属性的高度复杂的材料。为解决形状优化问题而开发的多层次技术将提供一个通用工具来研究反问题(例如，声散射)、图像处理、结构设计和流体动力学(例如，在飞机设计中)。

Every process in nature tries to optimize certain quantities under the available resources. The limited available resources act as constraints and, for this reason, constrained optimization problems are ubiquitous in science and engineering. Such optimization problems are inherently nonlinear due to the presence of constraints. This makes the development and analysis of algorithms and computational techniques to simulate the relevant phenomena extremely challenging and of practical relevance. A systematic approach is needed to allocate resources, without violating constraints, and remain effective in minimizing cost. This project will lead to development of advanced, efficient algorithms to design micro- and nano-scale lab-on-chip devices, as well as novel materials with desired configuration and user specified properties. As an outcome of shape optimization tools, robust algorithms for image processing and structural design will be developed. A significant amount of effort will be devoted to develop new models for cardiac electrical response and diffusion of water molecules in brain tissue. Many of the partial differential equations (PDEs) that describe the constraints are geometric, nonlinear, and multiscale, with an unknown domain, i.e. free boundary problems (FBPs). This makes the development and analysis of numerical techniques for the solution of these problems challenging. This project focuses on constrained optimization problems with PDE constraints -- the so-called PDE constrained optimization problem. In fact, it aims to develop and study numerical techniques for a new collection of such problems for which classical ideas and methods do not work. They are: optimization with Stokes FBP (including surface tension) as constraint, problems where the Hessian appears in the cost functional, multilevel techniques for shape optimization and image segmentation problems, optimization with nonlocal (fractional) operators, and degenerate parabolic equations as constraints. The applications range from micro- to nano-scales, where surface effects dominate bulk effects. This will have impact in designing next generation lab-on-chip devices. By manipulating curvature, one can drive particles towards a desired assembly or a specific configuration. This will lead to highly sophisticated materials with user specified properties. The multilevel techniques developed to solve shape optimization problems will provide a general tool to study inverse problems (acoustic scattering, for example), image processing, structural design, and fluid dynamics (for example, in aircraft design).