标题： 用于生成半线性最优控制问题POD模型快照的简化牛顿方法 显示英文标题

标题： A simplified Newton method to generate snapshots for POD models of semilinear optimal control problems

摘要： 在PDE约束优化中，本征正交分解（POD）提供了一个（可能昂贵的）PDE离散化的代理模型，在该模型上执行优化迭代。 由于POD模型通常仅在局部提供良好的近似质量，因此在优化过程中必须对其进行更新。 然而，更新POD模型通常是昂贵的，因此在模型预测控制（MPC）背景下往往不可行。 因此，可能接受质量一般的降阶模型。 我们采用求解半线性演化方程的简化牛顿法的观点，推导出一个算法，该算法可以作为离线阶段来生成POD模型。 在这一背景下，通过脉冲响应快照构建POD模型的方法可以被视为第一个牛顿步骤。 特别是，基于脉冲响应快照的POD模型通过添加第二个简化牛顿步骤进行扩展。 此过程通过在优化或MPC循环中引入适量的额外计算成本，显著提高了POD模型的近似质量。 我们通过一个满足我们假设的示例来说明我们的发现。 显示英文摘要

摘要： In PDE-constrained optimization, proper orthogonal decomposition (POD) provides a surrogate model of a (potentially expensive) PDE discretization, on which optimization iterations are executed. Because POD models usually provide good approximation quality only locally, they have to be updated during optimization. Updating the POD model is usually expensive, however, and therefore often impossible in a model-predictive control (MPC) context. Thus, reduced models of mediocre quality might be accepted. We take the view of a simplified Newton method for solving semilinear evolution equations to derive an algorithm that can serve as an offline phase to produce a POD model. Approaches that build the POD model with impulse response snapshots can be regarded as the first Newton step in this context. In particular, POD models that are based on impulse response snapshots are extended by adding a second simplified Newton step. This procedure improves the approximation quality of the POD model significantly by introducing a moderate amount of extra computational costs during optimization or the MPC loop. We illustrate our findings with an example satisfying our assumptions.