一种改进的稀疏多项式混沌展开估计的预处理方法
A preconditioning approach for improved estimation of sparse polynomial chaos expansions
摘要：
压缩采样已被广泛应用于随机函数的稀疏多项式混沌（PC）逼近。压缩采样的恢复精度高度依赖于测量矩阵的非相干性。本文考虑欠定方程组的预处理问题。用非奇异矩阵预乘线性方程组得到等效方程组，但它可以潜在地改善所得预处理测量矩阵的不相干性，并导致更好的恢复精度。然而，当测量有噪声时，预处理也可能导致更差的信噪比，从而降低恢复精度。在这项工作中，我们提出了一个预处理方案，提高了测量矩阵的不相干性，并在同一时间防止不希望的恶化的信噪比。我们提供的理论动机和数值例子表明，所提出的方法在提高估计多项式混沌展开的准确性的承诺。
英文摘要：
Compressive sampling has been widely used for sparse polynomial chaos (PC) approximation of stochastic functions. The recovery accuracy of compressive sampling highly depends on the incoherence properties of the measurement matrix. In this paper, we consider preconditioning the underdetermined system of equations that is to be solved. Premultiplying a linear equation system by a non-singular matrix results in an equivalent equation system, but it can potentially improve the incoherence properties of the resulting preconditioned measurement matrix and lead to a better recovery accuracy. When measurements are noisy, however, preconditioning can also potentially result in a worse signal-to-noise ratio, thereby deteriorating recovery accuracy. In this work, we propose a preconditioning scheme that improves the incoherence properties of measurement matrix and at the same time prevents undesirable deterioration of signal-to-noise ratio. We provide theoretical motivations and numerical examples that demonstrate the promise of the proposed approach in improving the accuracy of estimated polynomial chaos expansions.
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