Nonlinear Optimization Algorithms for Large-Scale and Nonsmooth Applications

研究者、他的同事和他的学生研究大规模pde约束和非光滑优化算法的开发、分析和实现。在这两种框架下工作的新颖之处在于，在每种情况下，研究者和他的团队都在寻找强有力的方法，在这些方法中，最先进的非线性规划方法可以得到增强和扩展，从而在它们以前效率低下或不适用的应用领域保持有效。在大规模pde约束问题的背景下，例如优化设计，参数估计和图像配准，这是通过消除对矩阵分解的需要和允许大规模线性系统解中的不精确来实现的，同时仍然保证收敛到解点。在非光滑应用的背景下，例如压缩感知和鲁棒稳定性和控制，这是通过通过梯度采样过程增强领先的算法框架来实现的，允许放松问题函数处处可微的假设。这些领域的工作将不同领域的算法和计算技术结合在一起，并提供了数值方法和收敛理论。这个项目更广泛的影响是，它将铅笔和纸的工程思想推进到可以在高性能计算软件中实现的程度，并且能够解决复杂系统设计和分析中的挑战性问题。例如，在医疗保健领域，特别是在癌症治疗和治疗领域，对这些优化工具的需求很高。通过为医生和医疗技术人员提供新颖的计算工具，他们将能够以一种考虑到人体内部复杂性（如血液流动）的方式，对热疗进行最佳管理。他们还将能够有效和适应性地设计放射治疗计划，以尽量减少对目标区域附近健康（通常是关键）组织的损害。令人惊讶的是，这些相同的计算工具也可以用于医学图像配准，帮助医疗专业人员检测不同时间和不同扫描（例如PET， CT， MRI）之间的不规则性。所有这些领域的目标都是为用户提供复杂的软件，可以实时回答诸如“管理这种辐射的最佳方法是什么？”和“这张图像中是否有任何变化或引起警报？”

The investigator, his colleagues, and his students study the development, analysis, and implementation of algorithms for large-scale PDE-constrained and nonsmooth optimization. The novelty of the work in both of these frameworks is that in each case the investigator and his group are finding powerful ways in which the most advanced methods for nonlinear programming can be enhanced and broadened to remain effective for application areas in which they have previously been inefficient or inapplicable. In the context of large-scale PDE-constrained problems, such as those in optimal design, parameter estimation, and image registration, this is being achieved by removing the need for the factorization of matrices and allowing for inexactness in the solution of large-scale linear systems, while still guaranteeing convergence to a solution point. In the context of nonsmooth applications, such as those in compressed sensing and robust stability and control, this is being achieved by enhancing leading algorithmic frameworks through a process of gradient sampling, allowing for a loosening of the assumption that the problem functions are differentiable everywhere. These works in these fields tie together algorithms and computational techniques from diverse areas, and both numerical methods and convergence theory are being provided.The broader impact of this project is that it advances pencil-and-paper engineering ideas to the point where they can be implemented in high-performance computing software and are able to solve challenging problems in the design and analysis of complex systems. For example, there is a high demand for optimization tools such as these in healthcare, particularly in the area of cancer treatment and therapy. By providing doctors and medical technicians with novel computational tools, they will be able to optimally administer hyperthermia treatment in a manner that takes into account the inner complexities of the human body, such as blood flow. They will also be able to effectively and adaptively design plans for radiation therapy that minimize damage to healthy -- and often critical -- tissue near the target area(s). Amazingly enough, these same computational tools can also be employed in medical image registration, aiding medical professionals in the detection of irregularities over time and between different (e.g., PET, CT, MRI) scans. The goal in all of these areas is to provide the user with sophisticated software that can answer, in real-time, difficult questions such as "What is the optimal way of administering this radiation?" and "Is there anything in this image that has changed or is cause for alarm?"