Sparse Solutions in Optimal Control of PDEs with Uncertain Parameters: The Linear Case
Sparse Solutions in Optimal Control of PDEs with Uncertain Parameters: The Linear Case
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DOI：
10.1137/18m1181419
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发表时间：
2018-04
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通讯作者：
Chen Li-;G. Stadler
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文献类型：
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作者：
Chen Li-;G. Stadler
研究了具有不确定系数的偏微分方程组控制的最优控制问题的稀疏解。我们提出了两种公式，一种是解是优化平均目标的确定性控制，另一种是针对具有相同稀疏结构的随机控制的公式。在这两种公式中，控制不消失的区域可被解释为放置控制装置的最佳位置。本文主要研究具有线性输入不确定参数的线性偏微分方程组。在这些假设下，确定性的表述归结为一个已知结构的问题，因此我们主要关注随机控制表述。这里，共享稀疏性是通过将逐点平方控制的平均值的$L^1$-范数合并到目标中来实现的。我们使用仅在物理空间上定义的范数重加权函数来重新表述问题，从而有助于避免使用样本或求积来逼近随机空间。我们证明了应用于范数重加权公式的不动点算法导致了研究得很好的迭代重加权最小二乘(IRLS)算法的变体，并提出了一种新的预条件牛顿-共轭梯度法来加速IRLS算法。我们将我们的算法与低阶算子近似相结合，并给出了截断误差的估计。我们仔细检查了所得到的算法的计算复杂性。利用Laplace方程和Helmholtz方程所描述的控制问题，数值研究了最优控制的稀疏结构和求解算法的性能。在这些实验中，牛顿变种明显优于IRLS方法。
We study sparse solutions of optimal control problems governed by PDEs with uncertain coefficients. We propose two formulations, one where the solution is a deterministic control optimizing the mean objective, and a formulation aiming at stochastic controls that share the same sparsity structure. In both formulations, regions where the controls do not vanish can be interpreted as optimal locations for placing control devices. In this paper, we focus on linear PDEs with linearly entering uncertain parameters. Under these assumptions, the deterministic formulation reduces to a problem with known structure, and thus we mainly focus on the stochastic control formulation. Here, shared sparsity is achieved by incorporating the $L^1$-norm of the mean of the pointwise squared controls in the objective. We reformulate the problem using a norm reweighting function that is defined over physical space only and thus helps to avoid approximation of the random space using samples or quadrature. We show that a fixed point algorithm applied to the norm reweighting formulation leads to a variant of the well-studied iterative reweighted least squares (IRLS) algorithm, and we propose a novel preconditioned Newton-conjugate gradient method to speed up the IRLS algorithm. We combine our algorithms with low-rank operator approximations, for which we provide estimates of the truncation error. We carefully examine the computational complexity of the resulting algorithms. The sparsity structure of the optimal controls and the performance of the solution algorithms are studied numerically using control problems governed by the Laplace and Helmholtz equations. In these experiments the Newton variant clearly outperforms the IRLS method.