CAREER: Scalable Algorithms for Nonlinear, Large-Scale Inverse Problems Governed by Dynamical Systems

由动力系统控制的大规模反问题在许多科学学科中具有极其重要的意义。例子包括地球科学，医学，气候科学，制造业，国家安全和经济学。反演是一种不可或缺的工具，可以以一致和可预测的方式从数据中推断知识，从而实现科学发现，决策以及最终可靠的模型和数据预测。然而，反演的实际应用仍然有限，除非不确定性可以量化，因为它们通过模型和算法传播，不确定性量化增加了显着的数学复杂性和巨大的计算成本已经具有挑战性的问题。该项目将开发一个通用的数学框架，用于大规模的统计逆问题，以及软件基础设施，其算法可在现代和未来的计算架构上扩展。它将数学方法和理论与数据密集型应用程序和良好的算法实践相结合，以推进计算和数据科学的前沿，最终目标是促进数据驱动的科学发现和基于模型的预测，从而促进一般科学。除了研究活动，还制定了一项教育和传播计划，将这项工作的结果传达给STEM学生和研究人员，以及广大的计算科学家和应用专家。该项目将在近年来美国行业需求特别高的领域培训学生，例如优化，统计推断，数据支持的科学，性能评估和工作负载表征。教育活动包括为研究生和本科生提供实践研究经验，明确鼓励少数民族和代表性不足的群体参与。公共领域的软件模块将提供给广泛的STEM受众和从业人员。本研究的应用领域包括医学、影像学、地球科学等，主要研究不确定性条件下的最优化、统计推断、大规模反问题求解等基础数学和计算问题，以促进科学发现和数据探索的进步。总体目标是设计快速计算内核和可扩展的黑盒算法，严格遵循数学和物理原理，具有良好的理论基础，并可证明收敛到独立于问题维度的最优解。这包括开发自适应、分层数值方案和混合精度算法，在需要时实现高精度计算，并在可能时实现低精度近似，以高数据吞吐量应用为目标。该项目探索（i）基础数学方面和基于运输的变分推理的快速（可扩展）算法的部署，（ii）非线性逆问题的问题通知正则化方案的设计，以及（iii）随机算法和学习的集成，用于构建低阶代理模型，用于优化，推理，采样和预处理。有效的数值技术和计算内核的梯度和曲率信息的快速评估和他们的近似设计的方法是至关重要的。 性能将被评估为抛物线（扩散为主）和双曲线（对流为主）的动力系统的不同复杂性与计算医学，气候科学和地球科学中的应用。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Large-scale inverse problems governed by dynamical systems are of paramount importance in numerous scientific disciplines. Examples include geoscience, medicine, climate science, manufacturing, national security, and economics. Inversion is an indispensable tool to infer knowledge from data in a consistent and predictable way, enabling scientific discovery, decision-making, and ultimately dependable model- and data-informed predictions. However, the practical use of inversion remains limited unless uncertainties can be quantified as they propagate through models and algorithms, Uncertainty quantification adds significant mathematical complications and massive computational costs to an already challenging problem. This project will develop a generic mathematical framework for large-scale, statistical inverse problems alongside software infrastructure with algorithms that scale on modern and future computing architectures. It blends mathematical methods and theory with data-intensive applications and good algorithmic practices to advance the frontiers of computational and data-enabled sciences, with the ultimate aspiration to promote data-driven scientific discovery and model-based prediction and by that, science in general. Alongside research activities, an educational and dissemination program is developed to communicate the results under this work to STEM students and researchers, and a broad audience of computational scientists and application specialists. The project will train students in areas that have seen exceptionally high industry demand in the US in recent years, such as optimization, statistical inference, data-enabled science, performance evaluation, and workload characterization. Educational activities include hands-on research experiences for graduate and undergraduate students, explicitly encouraging participation by minorities and underrepresented groups. Public domain software modules will be made available to a broad STEM audience and practitioners. Applications of this work include medicine, imaging, and geosciences.Fundamental mathematical and computational aspects of optimization under uncertainty, statistical inference, and the solution of large-scale inverse problems will be investigated in this project to promote the progress of scientific discovery and data exploration. The overarching aim is the design of fast computational kernels and scalable, black-box algorithms that rigorously follow mathematical and physical principles, have a sound theoretical basis, and provably converge to an optimal solution independent of the problem dimension. This includes the development of adaptive, hierarchical numerical schemes and mixed-precision algorithms, enabling high-accuracy computations if desired, and low-accuracy approximations when possible, targeting high data-throughput applications. The project explores (i) foundational mathematical aspects and the deployment of fast (scalable) algorithms for transport-based variational inference, (ii) the design of problem-informed regularization schemes for nonlinear inverse problems, and (iii) the integration of randomized algorithms and learning for the construction of low-order surrogate models for optimization, inference, sampling, and preconditioning. Effective numerical techniques and computational kernels for a fast evaluation of gradient and curvature information and their approximation are of paramount importance for the designed methodology. The performance will be assessed for parabolic (diffusion-dominated) and hyperbolic (advection-dominated) dynamical systems of varying complexity with applications in computational medicine, climate science, and geoscience.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.