低阶多项式混沌逼近的统计精度改进
On Improved Statistical Accuracy of Low-Order Polynomial Chaos Approximations
摘要：
多项式混沌展开是一种常用的方法来建立具有任意随机变量的随机系统的代理模型。标准的技术，如伽辽金投影，随机配置，和最小二乘逼近，应用于确定多项式混沌系数，定义的代理模型。由于代理模型是从函数近似的角度开发的，因此没有理由期望这些模型的统计准确性。从代理模型估计的统计矩可能与真实矩显著不同，特别是对于低阶近似。通常需要任意高阶来恢复，例如，二阶矩。在本文中，我们提出了修改的标准技术和确定多项式混沌系数，通过解决一个约束最优化问题。我们提出了这种新的方法，代数函数和微分方程的随机参数，并证明了代理模型的新方法能够恢复的前两个时刻。
英文摘要：
Polynomial chaos expansion is a popular way to develop surrogate models for stochastic systems with arbitrary random variables. Standard techniques such as Galerkin projection, stochastic collocation, and least squares approximation, are applied to determine polynomial chaos coefficients, which define the surrogate model. Since the surrogate models are developed from a function approximation perspective, there is no reason to expect accuracy of statistics from these models. The statistical moments estimated from the surrogate model may significantly differ from the true moments, especially for lower order approximations. Often arbitrary high orders are required to recover, for example, the second moment. In this paper, we present modifications of standard techniques and determine polynomial chaos coefficients by solving a constrained optimization problem. We present this new approach for algebraic functions and differential equations with random parameters, and demonstrate that the surrogate models from the new approach are able to recover the first two moments exactly.
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