论文标题
自适应信任区域的不合格双重方法减少了PDE受限优化的基础近似
A non-conforming dual approach for adaptive Trust-Region Reduced Basis approximation of PDE-constrained optimization
论文作者
论文摘要
在此贡献中，我们提出并严格分析具有PDE约束和双边参数约束的参数优化的自适应信任区域方法的新变体。该方法采用了在外部优化循环期间构建的替代基础替代模型，并用作信任区域方法的模型函数。通过投影的BFGS方法解决每个信任区域子问题。此外，我们提出了一种不合格的双（NCD）方法，以改善最佳系统的标准RB近似。严格改善的后验误差界限被得出，并用于证明所得NCD校正的自适应信任区域减少基础算法的收敛。数值实验表明，这种方法能够大大减少对大规模或多尺度PDE受约束优化问题的计算需求。
In this contribution we propose and rigorously analyze new variants of adaptive Trust-Region methods for parameter optimization with PDE constraints and bilateral parameter constraints. The approach employs successively enriched Reduced Basis surrogate models that are constructed during the outer optimization loop and used as model function for the Trust-Region method. Each Trust-Region sub-problem is solved with the projected BFGS method. Moreover, we propose a non-conforming dual (NCD) approach to improve the standard RB approximation of the optimality system. Rigorous improved a posteriori error bounds are derived and used to prove convergence of the resulting NCD-corrected adaptive Trust-Region Reduced Basis algorithm. Numerical experiments demonstrate that this approach enables to reduce the computational demand for large scale or multi-scale PDE constrained optimization problems significantly.
