不确定性量化中旋转稀疏近似的一般框架
A general framework of rotational sparse approximation in uncertainty quantification
摘要：
本文提出了一个通用的框架，估计系数的广义多项式混沌（gPC）的不确定性量化通过旋转稀疏近似。特别是，我们的目标是确定一个旋转矩阵，使得旋转后的一组随机变量的gPC展开具有稀疏表示。然而，这种旋转方法改变了要求解的基本线性系统，这使得找到稀疏系数比没有旋转的情况更困难。为了解决这个问题，我们研究了压缩感知（CS）中几种流行的非凸正则化方法，这些方法在经验上优于经典的l1方法。所有这些正则化都可以通过交替方向乘法器（ADMM）来最小化。数值例子表明，所提出的旋转和非凸稀疏促进正则化的组合，没有旋转和旋转，但使用凸l1方法的优越性能。
英文摘要：
This paper proposes a general framework to estimate coefficients of generalized polynomial chaos (gPC) used in uncertainty quantification via rotational sparse approximation. In particular, we aim to identify a rotation matrix such that the gPC expansion of a set of random variables after the rotation has a sparser representation. However, this rotational approach alters the underlying linear system to be solved, which makes finding the sparse coefficients more difficult than the case without rotation. To solve this problem, we examine several popular nonconvex regularizations in compressive sensing (CS) that perform better than the classic l1 approach empirically. All these regularizations can be minimized by the alternating direction method of multipliers (ADMM). Numerical examples show superior performance of the proposed combination of rotation and nonconvex sparse promoting regularizations over the ones without rotation and with rotation but using the convex l1 approach.
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