We study optimal control of PDEs under uncertainty with the state variable subject to joint chance constraints. The controls are deterministic, but the states are probabilistic due to random variables in the governing equation. Joint chance constraints ensure that the random state variable meets pointwise bounds with high probability. For linear governing PDEs and elliptically distributed random parameters, we prove existence and uniqueness results for almost-everywhere state bounds. Using the spherical-radial decomposition (SRD) of the uncertain variable, we prove that when the probability is very large or small, the resulting Monte Carlo estimator for the chance constraint probability exhibits substantially reduced variance compared to the standard Monte Carlo estimator. We further illustrate how the SRD can be leveraged to efficiently compute derivatives of the probability function, and discuss different expansions of the uncertain variable in the governing equation. Numerical examples for linear and bilinear PDEs compare the performance of Monte Carlo and quasi-Monte Carlo sampling methods, examining probability estimation convergence as the number of samples increases. We also study how the accuracy of the probabilities depends on the truncation of the random variable expansion, and numerically illustrate the variance reduction of the SRD.

翻译：我们研究了在状态变量受联合机会约束条件下偏微分方程的不确定性最优控制问题。控制变量是确定性的，但状态变量由于控制方程中的随机变量而具有概率性。联合机会约束确保随机状态变量以高概率满足逐点边界条件。对于线性控制偏微分方程和椭圆分布的随机参数，我们证明了几乎处处状态边界的存在性与唯一性结果。利用不确定变量的球面-径向分解，我们证明了当概率极大或极小时，所得机会约束概率的蒙特卡洛估计器相比标准蒙特卡洛估计器展现出显著降低的方差。我们进一步阐述了如何利用球面-径向分解高效计算概率函数的导数，并讨论了控制方程中不确定变量的不同展开形式。针对线性和双线性偏微分方程的数值算例比较了蒙特卡洛与拟蒙特卡洛采样方法的性能，检验了概率估计随样本数量增加的收敛性。我们还研究了概率精度对随机变量展开截断的依赖性，并通过数值实验展示了球面-径向分解的方差缩减效果。