SIAM Journal on Optimization, Volume 31, Issue 1, Page 348-376, January 2021.
In this work, we present a novel approach for solving stochastic shape optimization problems. Our method is the extension of the classical stochastic gradient method to infinite-dimensional shape manifolds. We prove convergence of the method on Riemannian manifolds and then make the connection to shape spaces. The method is demonstrated on a model shape optimization problem from interface identification. Uncertainty arises in the form of a random partial differential equation, where underlying probability distributions of the random coefficients and inputs are assumed to be known. We verify some conditions for convergence for the model problem and demonstrate the method numerically.

中文翻译：

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形状空间优化中的随机逼近

SIAM优化杂志，第31卷，第1期，第348-376页，2021
年1月。在这项工作中，我们提出了一种解决随机形状优化问题的新颖方法。我们的方法是将经典的随机梯度法扩展到无限维的形状流形。我们证明了该方法在黎曼流形上的收敛性，然后将其连接到形状空间。通过界面识别对模型形状优化问题进行了论证。不确定性以随机偏微分方程的形式出现，其中假定随机系数和输入的潜在概率分布是已知的。我们验证了模型问题收敛的一些条件，并通过数值方法进行了验证。