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Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables

机译:具有局部优化变量的PDE约束优化问题数值解的域分解和模型约简

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We introduce a technique for the dimension reduction of a class of PDE constrained optimization problems governed by linear time dependent advection diffusion equations for which the optimization variables are related to spatially localized quantities. Our approach uses domain decomposition applied to the optimality system to isolate the subsystem that explicitly depends on the optimization variables from the remaining linear optimality subsystem. We apply balanced truncation model reduction to the linear optimality subsystem. The resulting coupled reduced optimality system can be interpreted as the optimality system of a reduced optimization problem. We derive estimates for the error between the solution of the original optimization problem and the solution of the reduced problem. The approach is demonstrated numerically on an optimal control problem and on a shape optimization problem.
机译:我们介绍了一种用于减小一类PDE约束优化问题的维数的技术,该问题受线性时间相关的对流扩散方程控制,其优化变量与空间局部量有关。我们的方法是将域分解应用于最优系统,以将明确依赖于优化变量的子系统与其余线性最优子系统隔离开来。我们将平衡截断模型归约应用于线性最优子系统。所得的耦合的减少的最优系统可以解释为减少的优化问题的最优系统。我们得出原始优化问题的解决方案与简化问题的解决方案之间的误差估计。在最佳控制问题和形状优化问题上通过数值论证了该方法。

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