Partial differential equation (PDE)–constrained optimization problems with control or state constraints are challenging from an analytical and numerical perspective. The combination of these constraints with a sparsity‐promoting L¹ term within the objective function requires sophisticated optimization methods. We propose the use of an interior‐point scheme applied to a smoothed reformulation of the discretized problem and illustrate that such a scheme exhibits robust performance with respect to parameter changes. To increase the potency of this method, we introduce fast and efficient preconditioners that enable us to solve problems from a number of PDE applications in low iteration numbers and CPU times, even when the parameters involved are altered dramatically.

中文翻译：

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涉及稀疏项的PDE约束优化问题的内点方法和预处理

从分析和数值的角度来看，偏微分方程（PDE）约束的具有控制或状态约束的优化问题都具有挑战性。这些约束与目标函数内的稀疏度提升L ¹项的组合需要复杂的优化方法。我们建议将内点方案应用于离散化问题的平滑重新表述，并说明这种方案在参数变化方面表现出强大的性能。为了提高此方法的效力，我们引入了快速有效的预处理器，即使涉及的参数发生了巨大变化，这些预处理器也使我们能够以较低的迭代次数和CPU时间来解决许多PDE应用程序中的问题。