多项式混沌展开改进采样策略的性能和可靠性比较
Comparison of the performance and reliability between improved sampling strategies for polynomial chaos expansion
摘要：
随着复杂模型的不确定性和灵敏度分析变得越来越重要，及时实现它们的困难突出了对更有效的数值运算的需求。非侵入式多项式混沌方法是一种高效、精确的输入输出映射方法，可以用来研究复杂的模型。就选定的抽样方案而言，有很大的潜力提高该方法的效力。我们研究国家的最先进的抽样方案分类的空间填充最优设计，如拉丁超立方体抽样和L1最优抽样，并比较其经验表现对标准随机抽样。在L1最小化的背景下使用最小角度回归算法进行分析，以拟合GPCE回归模型。由于抽样方案的随机性，我们使用统计稳定性措施比较了不同的抽样方法，并评估了构建替代模型的成功率，相对误差分别为$<0.1$ \%、$<1$ \%和$<10$ \%。通过评估为各种不同的测试用例构建的代理模型的y，代表不同的问题类，涵盖低，中，高维问题的抽样方案进行了深入的研究。最后，采样方案进行了测试的应用实例，以估计的灵敏度，用于测量生物组织的阻抗在不同的频率的探头的自阻抗。我们观察到强烈的差异，在所分析的检验函数之间的方法的收敛特性。
英文摘要：
As uncertainty and sensitivity analysis of complex models grows ever more important, the difficulty of their timely realizations highlights a need for more efficient numerical operations. Non-intrusive Polynomial Chaos methods are highly efficient and accurate methods of mapping input-output relationships to investigate complex models. There is substantial potential to increase the efficacy of the method regarding the selected sampling scheme. We examine state-of-the-art sampling schemes categorized in space-filling-optimal designs such as Latin Hypercube sampling and L1-optimal sampling and compare their empirical performance against standard random sampling. The analysis was performed in the context of L1 minimization using the least-angle regression algorithm to fit the GPCE regression models. Due to the random nature of the sampling schemes, we compared different sampling approaches using statistical stability measures and evaluated the success rates to construct a surrogate model with relative errors of $<0.1$\%, $<1$\%, and $<10$\%, respectively. The sampling schemes are thoroughly investigated by evaluating the y of surrogate models constructed for various distinct test cases, which represent different problem classes covering low, medium and high dimensional problems. Finally, the sampling schemes are tested on an application example to estimate the sensitivity of the self-impedance of a probe that is used to measure the impedance of biological tissues at different frequencies. We observed strong differences in the convergence properties of the methods between the analyzed test functions.
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