稀疏多项式混沌展开的交叉验证和停止采样压缩感知
Compressive Sensing with Cross-Validation and Stop-Sampling for Sparse Polynomial Chaos Expansions
摘要：
压缩感知是一种用于恢复欠定线性系统稀疏解的强大技术，这在昂贵的高维物理模型的不确定性量化分析中经常遇到。我们进行数值研究，采用几个压缩传感求解器，目标无约束LASSO制定，重点是线性系统中出现的多项式混沌展开的建设。利用l1_ls，SpaRSA，CGIST，FPC_AS和ADMM的核心求解器，我们开发了通过基于交叉验证的正则化常数的自动选择来减轻过拟合的技术，以及指导停止采样决策的启发式策略。这些技术的参数设置的实用建议，并进行了讨论。整体方法适用于一系列的数值例子越来越复杂，包括大涡模拟的超音速湍流射流在横流涉及24维输入。通过经验相变图和收敛图，我们说明了稀疏恢复性能的多项式混沌，精度和不同程度的多项式基之间的计算权衡，以及进行压缩传感的现实，高维物理应用的实用性引起的结构。在本文研究的测试案例中，我们发现ADMM通过一致的较低误差和更快的计算时间证明了经验优势。
英文摘要：
Compressive sensing is a powerful technique for recovering sparse solutions of underdetermined linear systems, which is often encountered in uncertainty quantification analysis of expensive and high-dimensional physical models. We perform numerical investigations employing several compressive sensing solvers that target the unconstrained LASSO formulation, with a focus on linear systems that arise in the construction of polynomial chaos expansions. With core solvers of l1_ls, SpaRSA, CGIST, FPC_AS, and ADMM, we develop techniques to mitigate overfitting through an automated selection of regularization constant based on cross-validation, and a heuristic strategy to guide the stop-sampling decision. Practical recommendations on parameter settings for these techniques are provided and discussed. The overall method is applied to a series of numerical examples of increasing complexity, including large eddy simulations of supersonic turbulent jet-in-crossflow involving a 24-dimensional input. Through empirical phase-transition diagrams and convergence plots, we illustrate sparse recovery performance under structures induced by polynomial chaos, accuracy and computational tradeoffs between polynomial bases of different degrees, and practicability of conducting compressive sensing for a realistic, high-dimensional physical application. Across test cases studied in this paper, we find ADMM to have demonstrated empirical advantages through consistent lower errors and faster computational times.
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