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

协作研究：用于瞬态非线性 PDE 约束优化的降阶模型方法

• 批准号: 1115345
• 负责人: Matthias Heinkenschloss
• 金额: $ 15万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1115345&HistoricalAwards=false
• 关键词: Collaborative; Research; Reduced; Order; Model

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.

该项目开发、分析和实现基于投影的降阶模型 (ROM)，用于解决与非线性演化偏微分方程 (PDE) 相关的优化问题。这些 ROM 确定一个子空间，其中包含非线性演化 PDE 的基本（用于优化）动力学，并将这些 PDE 投影到该子空间上。如果子空间很小，优化问题中的原始非线性偏微分方程可以用一个小的常微分方程组代替，并且可以有效地求解所得的近似优化问题。高效生成 ROM 并进行误差估计以监控 ROM 的质量具有挑战性。该项目扩展并集成了面向目标的自适应网格细化、适当正交分解 (POD) 和模型管理方法的优化思想，以克服这些挑战。具体来说，来自优化的模型管理思想用于确定需要求解非线性演化 PDE 的优化参数以生成 ROM 的快照。此外，对于 PDE 的数值求解和快照的生成，将使用 PDE 的面向目标的双加权自适应时空有限元近似和离散 Galerkin-POD 的组合。特别地，将获得用于控制时间误差和空间误差的局部时间和局部空间双加权残差，其还提供对拍摄快照的适当时间步长的预测。目标是推导 ROM 误差的后验误差估计器，为我们提供有关需要包含的简化基函数数量的信息。这种新颖的方法将产生自适应离散伽辽金-POD (ADGPOD) 算法，用于高效可靠的基于 ROM 的 PDE 约束优化数值解。此外，所得的 ROM 将在多种应用中进行演示，包括流量控制/设计问题以及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分级过程的优化控制。过程的优化设计工程和生命科学应用中的系统通常需要对非线性偏微分方程 (PDE) 系统进行最优控制/优化。此类问题的数值求解通常相当于大型非线性代数系统的求解，需要大量的存储和计算时间。另一方面，设计工程师有兴趣在几分钟内在他们的 PC 上运行最佳设计。这只能通过大幅缩小问题的维度来实现，即通过为底层 PDE 系统开发简化模型来捕获昂贵的高保真度模拟的基本动态。尽管降阶模型已被证明适用于广泛的应用，但从理论角度来看，它们尚未得到很好的理解，特别是对于非线性问题。该项目将为非线性问题的降阶模型提供更好的理论基础，将开发新的算法工具来有效生成可靠的降阶模型，并将在重要的科学和工程应用中演示算法。