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

这项建议涉及从模拟向优化过渡的数学方法、算法技术和软件工具的开发。我们特别关注在空气动力学方面的应用，以优化机翼形状，并在气候研究中确定参数。这种方法适用于科学计算的所有领域，其中包含离散的偏微分方程组的大规模控制方程由定制的不动点求解器来处理。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，我们建议以半自动的方式扩展它们。首先，用伴随求解器对它们进行扩充，以获得(减少)导数，然后立即使用该灵敏度信息来确定优化修正。换言之，与外部优化循环相比，我们更喜欢同时追求最优性和伴随可行性目标的一次性策略。为了在模拟运行所需迭代次数的较小倍数内完成优化运行，必须确保组合一次性方案的对比度因子仅比用户提供的定点求解器的对比度因子更接近1。要实现收敛速度的有界延迟这一原理，需要进行大量的数学分析、算法实验和软件开发。更具体地说，我们建议在Hubert空间环境中分析和优化组合迭代的频谱特性，建设性地确定所需的一阶和二阶导数，并在空气动力学和气候学的全尺寸模型上验证该方案的有效性。分析和算法开发将分别在两所柏林大学和胡柏林大学和柏林理工大学进行空气动力学和气候学的具体应用工作。

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.