Automated extension of fixed point PDE solvers for optimal design with bounded retardation

定点 PDE 求解器的自动扩展，用于具有有限延迟的优化设计

• 批准号: 25207711
• 负责人: Professor Dr. Nicolas R. Gauger
• 金额: --
• 依托单位: Institut für Informatik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2006
• 资助国家: 德国
• 起止时间: 2005-12-31 至 2014-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/25207711?language=en
• 关键词: Automated; extension; fixed; point; PDE

This proposal concerns the development of mathematical methods, algorithmic techniques and software tools for the transition from simulation to optimization. We focus in particular on applications in aerodynamics to optimize wing shapes and in climate studies to identify parameters. The methodology is applicable to all areas of scientific computing, where large scale governing equations involving discretized PDEs are treated by custom made fixed point solvers. To exploit the domain specific experience and expertise invested in these simulation tools we propose to extend them in a semi-automated fashion. First they are augmented with an adjoint solvers to obtain (reduced) derivatives and then this sensitivity information is immediately used to determine optimization corrections. In other words, rather than applying an outer optimization loop we prefer the one-shot strategy of pursuing optimality simultaneously with the goals of primal and adjoint feasibility.To complete an optimization run in a small multiple of the number of iterations required for a simulation run it must be ensured that the contractivity factor of the combined one-shot scheme is only a certain fraction closer to 1 than that of the user supplied fixed point solver. The realization of this principle of bounded retardation of the convergence rate requires a significant amount of mathematical analysis, algorithmic experimentation, and software development. More specifically we propose to analyze and optimize the spectral properties of the combined iteration in a Hubert space setting, to constructively identify the required first and second derivatives, and to verify the efficacy of the scheme on full scale models in aerodynamics and climatology. The analysis and algorithm development will be carried out at the two Berlin Universities and the application specific work at the HU Berlin and the TU Berlin for aerodynamics and climatology, respectively.

该提案涉及数学方法、算法技术和软件工具的开发，以实现从模拟到优化的过渡。我们特别关注空气动力学中的应用以优化机翼形状以及气候研究中的应用以确定参数。该方法适用于科学计算的所有领域，其中涉及离散偏微分方程的大规模控制方程由定制的定点求解器处理。为了利用在这些模拟工具中投入的特定领域的经验和专业知识，我们建议以半自动化的方式扩展它们。首先，用伴随求解器来增强它们以获得（减少的）导数，然后立即使用此敏感性信息来确定优化校正。换句话说，我们更喜欢同时追求最优性和原始可行性和伴随可行性目标的一次性策略，而不是应用外部优化循环。为了以模拟运行所需迭代次数的小倍数完成优化运行，必须确保组合一次性方案的收缩因子仅比用户提供的定点求解器的收缩因子更接近 1。收敛速度有限延迟原理的实现需要大量的数学分析、算法实验和软件开发。更具体地说，我们建议在休伯特空间设置中分析和优化组合迭代的光谱特性，建设性地识别所需的一阶和二阶导数，并验证该方案在空气动力学和气候学全尺寸模型上的有效性。分析和算法开发将在柏林​​的两所大学进行，具体的应用工作将分别在柏林大学和柏林工业大学进行，分别用于空气动力学和气候学。