On Improved Statistical Accuracy of Low-Order Polynomial Chaos Approximations
On Improved Statistical Accuracy of Low-Order Polynomial Chaos Approximations
复制标题
关于提高低阶多项式混沌近似的统计精度
DOI：
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
R. Bhattacharya
中科院分区：
文献类型：
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作者：
Vedang M. Deshpande;R. Bhattacharya
多项式混沌展开是开发具有任意随机变量的随机系统代理模型的流行方法。应用伽辽金投影、随机配置和最小二乘近似等标准技术来确定定义替代模型的多项式混沌系数。由于代理模型是从函数逼近的角度开发的，因此没有理由期望这些模型的统计准确性。从替代模型估计的统计矩可能与真实矩显着不同，特别是对于低阶近似。通常需要任意高阶才能恢复，例如二阶矩。在本文中，我们提出了标准技术的修改，并通过解决约束优化问题来确定多项式混沌系数。我们提出了这种用于具有随机参数的代数函数和微分方程的新方法，并证明了新方法的代理模型能够准确地恢复前两个矩。
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.