Numerical Analysis of Partial Differential Equation Constrained Optimization Problems

自然界中的每一个过程都试图在可用资源下优化一定的数量。有限的可用资源作为约束，因此，约束优化问题在科学和工程中普遍存在。由于约束条件的存在，这种优化问题本质上是非线性的。这使得模拟相关现象的算法和计算技术的开发和分析极具挑战性和实际意义。需要一种系统的方法来分配资源，而不违反约束，并保持有效的最小化成本。该项目将开发先进、高效的算法来设计微纳米级芯片实验室设备，以及具有所需配置和用户指定属性的新材料。作为形状优化工具的结果，图像处理和结构设计的鲁棒算法将被开发出来。大量的努力将致力于开发新的模型心脏电反应和水分子在脑组织中的扩散。许多描述约束的偏微分方程（PDEs）是几何的、非线性的、多尺度的，具有未知域，即自由边界问题（fbp）。这使得解决这些问题的数值技术的发展和分析具有挑战性。本项目主要研究带PDE约束的约束优化问题——即所谓的PDE约束优化问题。事实上，它的目的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新问题。它们是：以Stokes FBP（包括表面张力）为约束的优化，在代价函数中出现Hessian的问题，用于形状优化和图像分割问题的多层技术，非局部（分数）算子的优化，以及退化抛物方程作为约束。应用范围从微观到纳米尺度，其中表面效应主导体效应。这将对设计下一代芯片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合或特定的配置。这将导致高度复杂的材料与用户指定的属性。为解决形状优化问题而开发的多层次技术将为研究反问题（例如声散射）、图像处理、结构设计和流体动力学（例如飞机设计）提供通用工具。

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).