Sample Size Estimates for Risk-Neutral Semilinear PDE-Constrained Optimization
研究了样本平均逼近方法在随机输入半线性椭圆偏微分方程约束的风险中性优化问题中的应用，推导了非渐近样本量估计，给出了获得精确临界点所需样本数的上界，并用数值例子验证。
.The sample average approximation (SAA) approach is applied to risk-neutral optimization problems governed by semilinear elliptic partial differential equations with random inputs. After constructing a compact set that contains the SAA critical points, we derive nonasymptotic sample size estimates for SAA critical points using the covering number approach. Thereby, we derive upper bounds on the number of samples needed to obtain accurate critical points of the risk-neutral PDE-constrained optimization problem through SAA critical points. We quantify accuracy using expectation and exponential tail bounds. Numerical illustrations are presented.Keywordsstochastic optimizationPDE-constrained optimization under uncertaintysample average approximationMonte Carlo samplingsample complexityuncertainty quantificationMSC codes90C1590C3090C6049J2049J5549K4549K2035J61