Fast Algorithms for Nonlinear Optimal Control of Geodesic Flows of Diffeomorphisms

最优控制问题在包括医学、地球科学、制造业、国家安全或经济学在内的众多计算科学应用中起着至关重要的作用。最优控制问题是一种从数据中推断知识的系统工具，使科学发现和决策成为可能。它们通常被描述为以动态系统(模拟问题)为约束的数据拟合问题。这个模拟问题描述了在给定输入变量(例如，脑瘤和肿瘤生长速率)的情况下，自然或工程系统的可能行为。在实践中，这些值通常是未知的，不能直接测量。人们需要通过优化性能目标从观察数据(例如，一系列患者图像)中推断它们。这一过程构成了控制问题，未知变量是仿真问题的控制变量。解决最优控制问题带来了重大的数学挑战。该项目将考虑可能有多达数十亿个未知数的控制问题。对于决策，人们需要为控制问题的解配备置信度区间。这是使用统计框架实现的，这增加了大量的计算成本。此外，不同的控制变量实现可以产生与观测数据同样匹配的模拟输出，从而导致众所周知的不适定问题。为了减少这种模糊性，可以基于正则化模型引入关于合理解的先验知识。然而，选择适当的正则化模型仍然是一个巨大的挑战。该项目旨在提供为现代计算架构量身定做的快速、可扩展和健壮的软件，以解决巨大的计算成本。该项目将侧重于从数据中学习适当的正规化模型。应用领域是用于对对象进行分类的统计形状分析，特别是基于器官的解剖形状变异性对患者(疾病与健康)的分类。这项研究完成后，将为与运输有关的最优控制问题和更一般的反问题产生一个通用的数学和算法框架，以及适用于(生物医学)成像科学、大气科学、计算机视觉、遥感、数据科学和深度学习等一系列问题的软件基础设施。该项目将为两名研究生提供培训，并为本科生提供暑期研究项目。研究将开发有效的、可扩展的计算方法，用于微分同胚测地线流的非线性最优控制。其新奇之处在于设计了硬件加速的计算内核和高效的数值方案，这些方案利用问题结构并严格遵循数学原理来研究形状可变性。这是通过设计用于统计形状分析的贝叶斯框架来实现的。研究对象的不同实现的形状可变性的量化是通过将一个对象映射到另一个对象的微分同胚的测地线流的透镜来完成的。特别是，人们通过连接两个形状的测地线路径的长度来量化两个形状之间的接近程度。从统计学的角度来看，人们可以通过识别特定被调查对象的平均几何形状(“统计模板”)来研究数据库中的形状可变性，然后研究单个数据集如何偏离该平均值。该项目将专注于自适应、分层的数值方案，如果需要，可以进行高精度计算，如果可能，可以进行低精度近似。解算器将具有快速计算内核、最大化单节点和单GPU吞吐量的特点，同时在(异类)高性能计算平台上保持可伸缩性。工作包包括预条件、用于评估正向和伴随算子的快速分层计算内核、在不同架构上的混合精度实现，以及利用问题结构来加速求解并允许在从高维概率分布中采样时获得高接受率的计算方法。特别是，该项目将提供(I)解决非线性初值控制问题的方法，(Ii)不确定性量化，(Iii)从大型成像数据库中进行统计模板估计，以及(Iv)从数据中学习正则化运算符。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Optimal control problems play a critical role in numerous computational sciences applications, including those in medicine, geosciences, manufacturing, national security, or economics. Optimal control problems are a systematic tool to infer knowledge from data, enabling scientific discovery and decision making. They are typically formulated as data-fitting problems with dynamical systems (the simulation problem) as constraints. This simulation problem describes the possible behavior of a natural or engineered system under investigation for given values of input variables (for example, brain tumor and tumor growth rates). In practice, the values are typically not known and cannot be measured directly. One needs to infer them from observational data (for example, a series of patient images) by optimizing a performance goal. This process constitutes the control problem; the unknown variables are the controls of the simulation problem. Solving optimal control problems poses significant mathematical challenges. The project will consider control problems that can have up to billions of unknowns. For decision making, one needs to equip the solutions of the control problem with confidence intervals. This is achieved using a statistical framework, which adds massive computational costs. Moreover, distinct control variable realizations can yield simulation outputs that match the observational data equally well, leading to what is known as ill-posed problems. To alleviate this ambiguity, prior knowledge about plausible solutions can be introduced based on regularization models. However, choosing adequate regularization models remains a significant challenge. This project aims to provide fast, scalable, and robust software tailored to modern computing architectures to address the massive computational costs. The project will focus on learning appropriate regularization models from data. The area of application is statistical shape analysis for classifying objects, and, in particular, the classification of patients (diseased versus healthy) based on the anatomical shape variability of organs. Upon completion, the research will produce a generic mathematical and algorithmic framework for transport-related optimal control problems and more generally inverse problems, along with software infrastructure that applies to a range of problems in (biomedical) imaging sciences, atmospheric sciences, computer vision, remote sensing, data science, and deep learning. The project will provide training for two graduate students and summer research projects for undergraduates.The research will develop effective, scalable computational methods for nonlinear optimal control of geodesic flows of diffeomorphisms. The novelty is the design of hardware-accelerated computational kernels and efficient numerical schemes that exploit problem structure and rigorously follow mathematical principles for studying shape variability. This is achieved through the design of a Bayesian framework for statistical shape analysis. The quantification of shape variability of distinct realizations of an object under investigation is done through the lens of geodesic flows of diffeomorphisms that map one object to another. In particular, one quantifies the proximity between two shapes by the length of the geodesic path that connects them. From a statistical point of view, one can study shape variability in a database by identifying an average geometry (the "statistical template") of a particular object under investigation, and then studies how individual datasets deviate from this average. The project will focus on adaptive, hierarchical numerical schemes, enabling high-accuracy computations if desired, and low-accuracy approximations when possible. The solvers will feature fast computational kernels, maximizing single-node, and single-GPU throughput while maintaining scalability on (heterogeneous) high-performance computing platforms. Work packages include preconditioning, fast hierarchical computational kernels for evaluating forward and adjoint operators, mixed-precision implementations on heterogeneous architectures, and computational methods that exploit problem structure to speed up the solution and allow for high acceptance rates when sampling from high-dimensional probability distributions. In particular, the project will provide methodology for (i) the solution of nonlinear initial value control problems, (ii) uncertainty quantification, (iii) statistical template estimation from large imaging databases, and (iv) learning regularization operators from data.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.