Collaborative Research: Reduced Order Model Approaches for Time Dependent Nonlinear PDE Constrained Optimization

该项目开发、分析和实现了基于投影的降阶模型(ROMs)，用于解决与非线性发展偏微分方程组(PDE)相关的优化问题。这些只读存储器确定一个子空间，该子空间包含非线性演化偏微分方程组的基本动力学(用于优化)，并将这些偏微分方程组投影到子空间上。当子空间较小时，可以用一个小的常微分方程组代替原优化问题中的非线性偏微分方程组，从而有效地求解近似优化问题。有效地生成只读存储器以及能够监测只读存储器质量的误差估计是具有挑战性的。该项目扩展和整合了面向目标的自适应网格细化、适当的正交分解(POD)和优化中的模型管理方法的思想，以克服这些挑战。具体地说，使用来自优化的模型管理思想来确定需要在哪些优化参数处求解非线性演化PDE以生成用于ROM的快照。此外，对于偏微分方程组的数值解和快照的生成，将结合使用面向目标的基于双重加权的自适应时空有限元近似的偏微分方程组和离散Galerkin-POD。具体地，将获得用于控制时间误差和空间误差的时间局部和空间局部双重加权残差，其还提供对拍摄快照的适当时间步长的预测。我们的目标是推导出用于只读存储器误差的后验误差估计器，该估计器给出关于需要包括的缩减基函数的数量的信息。这种新的方法将产生一种自适应离散Galerkin-POD(ADGPOD)算法，用于高效可靠地求解偏微分方程组约束优化的基于只读存储器的数值解。此外，所得到的ROMS还将在几个应用中得到演示，包括流动控制/设计问题和用于快速分离纳米粒子、蛋白质和其他大分子的非对称流场流动分离过程的最优控制。工程和生命科学应用中的过程和系统的优化设计通常需要对非线性偏微分方程组(PDE)进行最优控制/优化。这类问题的数值解通常相当于需要大量存储和计算时间的大型非线性代数系统的解。另一方面，设计工程师感兴趣的是在几分钟内在他们的PC上运行最佳设计。这只能通过大幅降低问题的维度来实现，即，通过为底层的PDE系统开发一个简化的模型来捕获昂贵的高保真模拟的基本动态。虽然降阶模型已被证明适用于广泛的应用，但从理论的角度来看，它们还没有被很好地理解，特别是对于非线性问题。该项目将为非线性问题降维模型的研究提供更好的理论基础，为高效地生成可靠的降维模型开发新的算法工具，并将在重要的科学和工程应用中展示算法。

This project develops, analyses and implements projection based reduced order models (ROMs) for optimization problems associated with nonlinear evolution partial differential equations (PDEs). These ROMs determine a subspace that contains the essential (for the optimization) dynamics of the nonlinear evolution PDEs and project these PDEs onto the subspace. If the subspace is small, the original nonlinear PDEs in the optimization problem can be replaced by a small system of ordinary differential equations and the resulting approximate optimization problem can be solved efficiently. The efficient generation of ROMs together with error estimates that can monitor the quality of the ROMs is challenging. This project expands and integrates ideas from goal oriented adaptive mesh refinement, proper orthogonal decomposition (POD), and model management approaches in optimization to overcome these challenges. Specifically, model management ideas from optimization are used determine at which optimization parameters the nonlinear evolution PDE needs to be solved to generate snapshots for the ROM. Furthermore, for the numerical solution of the PDE and generation of snapshots a combination of goal-oriented dual weighted based adaptive space-time finite element approximations of the PDE and discrete Galerkin-POD will be used. In particular, local-in-time and local-in-space dual weighted residuals for the control of the error in time and the error in space will be obtained that also provide a prediction of appropriate time steps at which snapshots are taken. The goal is the derivation of an a posteriori error estimator for the ROM error that gives us information about the number of reduced basis functions that need to be included. This novel approach will result in an Adaptive Discrete Galerkin-POD (ADGPOD) algorithm for an efficient and reliable ROM-based numerical solution of PDE constrained optimization. In addition the resulting ROMs will be demonstrated on several applications, including flow control/design problems and the optimal control of Asymmetrical-Flow Field-Flow-Fractionation processes for the fast separation of nanoparticles, proteins, and other macromolecules.The optimal design of processes and systems in engineering and life science applications often requires the optimal control/optimization of systems of nonlinear partial differential equations (PDE). The numerical solution of such problems typically amounts to the solution of large nonlinear algebraic systems requiring extensive storage and computational time. On the other hand, the design engineers are interested to run optimal designs on their PCs within a couple of minutes. This can be achieved only by a dramatic reduction of the dimension of the problem, i.e., by developing a reduced model for the underlying PDE system that captures the essential dynamics of the expensive high fidelity simulation. Although reduced order models have been shown to work well for a wide spectrum of applications, they not yet well understood from a theoretical point of view, especially for nonlinear problems. This project will provide a better theoretical foundation of reduced order models for nonlinear problems, it will develop novel algorithmic tools for the efficient generation of reliable reduced order models, and it will demonstrate the algorithms on important science and engineering applications.