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CDS&E: Space-Time Parallel Algorithms for Solving PDE-Constrained Optimization Problems

CDS

基本信息

批准号：
1709727
负责人：
Adrian Sandu
金额：
$ 50万
依托单位：
Virginia Polytechnic Institute and State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2021-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1709727&HistoricalAwards=false
关键词：
CDS&amp Space Time Parallel Algorithms

项目摘要

Many fields of science and engineering, from atmospheric science to aeronautics, and from material science to cosmology, rely on complex models built from "first principles" in order to study the phenomena of interest; the fundamental physical laws are often described by time-dependent partial differential equations (PDEs). These models are implemented in complex software that performs vast amount of computations and processes large data sets in order to simulate the physical reality. PDE-based models typically run long times on parallel computers, using large numbers of cores. A central problem in these fields of science and engineering is that of optimizing the system of interest according to specific design criteria. For example, in aeronautics, one wants not only to simulate the flight of an airplane, but also to design the best aircraft using shape optimization. In numerical weather prediction, one needs not only simulate the evolution of the atmosphere, but also to optimally utilize the information coming from satellite, aircraft, and ground based measurements in order to keep the forecasts accurate. All these applications seek to optimize systems governed by PDEs. This is an extremely challenging quest, since solving a PDE-constrained optimization problem is one-two orders of magnitude costlier than the underlying forward PDE simulation. There is considerable need for novel highly-parallel solution methodologies. This project develops the algorithmic infrastructure to support large-scale optimization of systems governed by time-dependent PDEs. New ideas will be used to unravel and exploit the inherent parallelism. First, we seek to parallelize the computations in both space and time. The space is divided in subdomains, the time in subintervals, and sub-models on each time subinterval and on each spatial subdomain are run concurrently on different sets of processors. Next, to further increase computational effectiveness, we will build surrogate models, i.e., inexpensive approximate models that capture the main dynamical characteristics of the full PDE-based models. Parallel construction of new surrogates is proposed using local-in-space-and-time information. The main idea is to perform optimization using the inexpensive surrogate models, transferring the improved design to the full PDE-model, re-computing a surrogate for the new configuration, and iterating. Enormous computational savings can be realized this way. Lastly, the new algorithms will be laid on solid theoretical foundations, and will be applied to speed up the incorporation of measurement data in a numerical weather prediction model. The tools developed in this project will enable leap developments in many fields in science and engineering where time-dependent PDE-constrained optimization problems are central. Important examples include aircraft shape optimization, seismic imaging, medical imaging, optimal control of fabrication processes, and inverse problems. The project will directly train one doctoral student and one postdoctoral researcher, will involve undergraduates in research, will develop graduate level educational materials, and will attract students from under-represented groups in parallel computing and large-scale simulations of the physical world.This project develops the algorithmic infrastructure to support large-scale optimization of systems governed by time-dependent partial differential equations (PDEs). PDE optimization problems are central to many fields in science and engineering. They are considerably more complex, and costlier to solve, than the underlying PDE simulations. There is considerable need for highly-parallel solution methodologies. In order to address this, the project proposes a space-time parallel formalism, and new reduced order modeling techniques, that have the potential to speed up the PDE-constrained optimization solution process by several orders of magnitude. (1) Intellectual merit: This work develops a space-time parallel formalism for the solution of large scale PDE-constrained optimization problems. The space is divided in subdomains, the time in subintervals, and the forward and adjoint models are run in parallel on each time subinterval and on each spatial subdomain. Solution continuity equations are imposed more stringently as the optimization process advances. This work formulates reduced-order PDE-constrained optimization problems using local-in-space-and-time reduced order models. Such models can represent the system dynamics much better than traditional global approaches. Moreover, both the off-line construction of local reduced order models and the on-line reduced order simulations can be carried out concurrently on each time subinterval and on each spatial subdomain, resulting in considerable speed-ups. A trust region framework is employed for provably convergent reduced order optimization algorithms. The new methodologies are demonstrated on large atmospheric data assimilation applications. (2) Broader impact: The tools developed in this project will enable leap developments in many fields in science and engineering where time-dependent PDE-constrained optimization problems are central. Important examples include aircraft shape optimization, seismic imaging, medical imaging, optimal control of fabrication processes, and inverse problems. One doctoral student and one postdoctoral researcher are directly trained in PDE-constrained optimization, reduced order modeling, high performance computing, and science applications. Graduate level educational materials are developed.This project is supported by the Office of Advanced Cyberinfrastructure in the Directorate for Computer & Information Science & Engineering and the Division of Mathematical Sciences in the Directorate of Mathematical and Physical Sciences.

许多科学和工程领域，从大气科学到航空学，从材料科学到宇宙学，都依赖于从“第一原理”建立的复杂模型来研究感兴趣的现象;基本物理定律通常由时间相关的偏微分方程（PDE）描述。这些模型在复杂的软件中实现，该软件执行大量计算并处理大型数据集，以模拟物理现实。基于PDE的模型通常在并行计算机上运行很长时间，使用大量的内核。在这些科学和工程领域中的一个中心问题是根据特定的设计标准优化感兴趣的系统。例如，在航空学中，人们不仅要模拟飞机的飞行，还要使用形状优化来设计最好的飞机。在数值天气预报中，人们不仅需要模拟大气的演变，而且还需要最佳地利用来自卫星、飞机和地面测量的信息，以保持预报的准确性。所有这些应用程序都试图优化由PDE管理的系统。这是一个极具挑战性的探索，因为解决PDE约束优化问题的成本比基础的正向PDE模拟高出一到两个数量级。有相当大的需求，为新的高度并行的解决方案的方法。该项目开发了算法基础设施，以支持由时间依赖的偏微分方程管理的系统的大规模优化。新的想法将被用来解开和利用固有的并行性。首先，我们寻求在空间和时间上并行计算。空间被划分为子域，时间被划分为子区间，并且每个时间子区间和每个空间子域上的子模型在不同的处理器集合上并发地运行。接下来，为了进一步提高计算效率，我们将构建代理模型，即，廉价的近似模型，捕捉的主要动态特性的完整的偏微分方程为基础的模型。提出了利用时空局部信息并行构造新代理的方法。其主要思想是使用廉价的代理模型进行优化，将改进的设计转换为完整的PDE模型，重新计算新配置的代理，并迭代。通过这种方式可以实现巨大的计算节省。最后，新算法将建立在坚实的理论基础上，并将用于加速将测量数据纳入数值天气预报模式。在这个项目中开发的工具将使飞跃的发展，在许多领域的科学和工程中，时间依赖的偏微分方程约束优化问题是中央。重要的例子包括飞机形状优化，地震成像，医学成像，制造过程的最佳控制，以及逆问题。该项目将直接培养一名博士生和一名博士后研究员，将让本科生参与研究，将开发研究生水平的教育材料，并将吸引来自并行计算和大规模模拟物理世界的代表性不足的群体的学生。该项目开发算法基础设施，以支持由时间依赖偏微分方程（PDE）控制的系统的大规模优化。偏微分方程优化问题是科学和工程中许多领域的核心问题。他们是相当复杂的，更昂贵的解决，比基本的偏微分方程模拟。高度并行的解决方案方法有相当大的需求。为了解决这个问题，该项目提出了一个时空并行形式主义，和新的降阶建模技术，有可能加快几个数量级的PDE约束优化求解过程。(1)智力价值：这项工作开发了一个空间-时间并行形式主义的大规模偏微分方程约束优化问题的解决方案。空间被划分为子域，时间被划分为子区间，并且前向和伴随模型在每个时间子区间和每个空间子域上并行运行。随着优化过程的推进，解连续性方程被施加得更严格。这项工作制定降阶PDE约束优化问题，使用本地的空间和时间降阶模型。这样的模型可以更好地代表系统的动态比传统的全球方法。此外，无论是离线建设的本地降阶模型和在线降阶仿真可以同时进行每个时间子区间和每个空间子域，从而在相当大的速度。信赖域框架被用于可证明收敛的降阶优化算法。新方法在大型大气数据同化应用中得到了验证。(2)更广泛的影响：在这个项目中开发的工具将使飞跃的发展，在许多领域的科学和工程中，时间依赖的偏微分方程约束优化问题是中央。重要的例子包括飞机形状优化，地震成像，医学成像，制造过程的最佳控制，以及逆问题。一名博士生和一名博士后研究员直接接受PDE约束优化，降阶建模，高性能计算和科学应用的培训。该项目得到了计算机信息科学工程局高级网络基础设施办公室和数学和物理科学局数学科学处的支持。