Collaborative Research: Computational Methods for Optimal Transport via Fluid Flows

流体中的输运和混合是工程和自然科学中一个基本感兴趣的话题，其广泛应用范围从工业和化学混合到防止污染物在地球物理流动中的扩散。本项目的重点是控制流体流动中某些感兴趣的量的最优输送和混合的计算方法。在复杂动力系统领域，什么样的流体流动能使混合速度最大化、减慢混合速度，甚至将大量的流体引向理想的目标分布，这一问题引起了科学家和工程师的极大关注。该项目的目标是将这些问题置于一个灵活的计算框架内，并开发基于最优控制工具、数据压缩策略和方法的解决策略，以降低数学模型的复杂性。该项目还将帮助不同学科的研究生在最优输送和混合、流动控制和解决这些问题的计算方法方面进行合作研究。该项目致力于开发和分析流体流动中混合的最优控制的数值方法。更准确地说，输运方程被用来描述不可压缩的Stokes流和Navier-Stokes流平流的无耗散标量场。研究的目的是通过主动控制流速来实现最优混合，并构造有效的数值格式来解决这一问题。将研究各种控制设计来控制流体流动。通过在目标泛函中引入非光滑惩罚项，提高最优边界控制的稀疏性。这实质上导致了抛物型和双曲型耦合系统或半耗散系统的高度挑战性的非线性非光滑控制问题。该项目将为这些困难的最优控制问题建立一个新颖而严格的数学框架和新的准确而高效的计算技术。采用流动与输送耦合的相容离散化方法对受控系统进行离散化，实现最优控制设计。对于高度复杂的最优化系统，将以系统的方式构造和分析数值格式。新的增量数据压缩技术将被用来避免在迭代求解器中存储极大的解数据集，并且将开发专门为最优混合问题设计的新的模型降阶技术以提高效率。最优控制和数值近似的综合将使人们能够研究在许多其他复杂和真实世界的流动动力学中出现的类似现象。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。

Transport and mixing in fluids is a topic of fundamental interest in engineering and natural sciences, with broad applications ranging from industrial and chemical mixing on small and large scales, to preventing the spreading of pollutants in geophysical flows. This project focuses on computational methods for control of optimal transport and mixing of some quantity of interest in fluid flows. The question of what fluid flow maximizes mixing rate, slows it down, or even steers a quantity of interest toward a desired target distribution draws great attention from a broad range of scientists and engineers in the area of complex dynamical systems. The goal of this project is to place these problems within a flexible computational framework, and to develop a solution strategy based on optimal control tools, data compression strategies, and methods to reduce the complexity of the mathematical models. This project will also help the training and development of graduate students across different disciplines to conduct collaborative research in optimal transport and mixing, flow control, and computational methods for solving these problems.The project is concerned with the development and analysis of numerical methods for optimal control for mixing in fluid flows. More precisely, the transport equation is used to describe the non-dissipative scalar field advected by the incompressible Stokes and Navier-Stokes flows. The research aims at achieving optimal mixing via an active control of the flow velocity and constructing efficient numerical schemes for solving this problem. Various control designs will be investigated to steer the fluid flows. Sparsity of the optimal boundary control will be promoted via a non-smooth penalty term in the objective functional. This essentially leads to a highly challenging nonlinear non-smooth control problem for a coupled parabolic and hyperbolic system, or a semi-dissipative system. The project will establish a novel and rigorous mathematical framework and also new accurate and efficient computational techniques for these difficult optimal control problems. Compatible discretization methods for coupled flow and transport will be employed to discretize the controlled system and implement the optimal control designs numerically. Numerical schemes for the highly complicated optimality system will be constructed and analyzed in a systematic fashion. New incremental data compression techniques will be utilized to avoid storing extremely large solution data sets in the iterative solvers, and new model order reduction techniques specifically designed for the optimal mixing problem will be developed to increase efficiency. The synthesis of optimal control and numerical approximation will enable the study of similar phenomena arising in many other complex and real-world flow dynamics.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.