We develop both first and second order numerical optimization methods to solve non-smooth optimization problems featuring a shared sparsity penalty, constrained by differential equations with uncertainty. To alleviate the curse of dimensionality we use tensor product approximations. To handle the non-smoothness of the objective function we introduce a smoothed version of the shared sparsity objective. We consider both a benchmark elliptic PDE constraint, and a more realistic topology optimization problem. We demonstrate that the error converges linearly in iterations and the smoothing parameter, and faster than algebraically in the number of degrees of freedom, consisting of the number of quadrature points in one variable and tensor ranks. Moreover, in the topology optimization problem, the smoothed shared sparsity penalty actually reduces the tensor ranks compared to the unpenalised solution. This enables us to find a sparse high-resolution design under a high-dimensional uncertainty.

翻译：本文开发了一阶和二阶数值优化方法，用于求解具有共享稀疏性惩罚的非光滑优化问题，该问题受含不确定性的微分方程约束。为缓解维度灾难，我们采用张量积近似。为处理目标函数的非光滑性，我们引入了共享稀疏性目标的平滑版本。我们同时考虑了基准椭圆偏微分方程约束和更实际的拓扑优化问题。实验证明，误差在迭代次数和平滑参数上呈线性收敛，且在自由度数量（包括单变量积分点数和张量秩）上收敛速度快于代数收敛。此外，在拓扑优化问题中，平滑的共享稀疏性惩罚相较于无惩罚解实际上降低了张量秩。这使我们能够在高维不确定性下找到稀疏的高分辨率设计方案。