标题： 随机过程分解中的稀疏表示与Karhunen-Loeve展开的AFD类型对比 显示英文标题

标题： AFD Types Sparse Representations vs. the Karhunen-Loeve Expansion for Decomposing Stochastic Processes

摘要： 本文介绍了自适应傅里叶分解（AFD）类方法，重点介绍那些可以应用于随机过程和随机场的方法，主要包括随机自适应傅里叶分解和随机预正交自适应傅里叶分解。 我们基于协方差函数建立它们的算法，并证明它们具有与Karhunen-Loève（KL）分解相同的收敛速度。 将AFD类方法与KL分解进行比较。 与后者相比，AFD类方法无需计算由协方差函数诱导的核积分算子的特征值和特征函数，从而大大降低了计算复杂度和计算机消耗。 各种字典为AFD提供了灵活性，用于解决不同类型的问题，包括确定性和随机性方程。 所进行的实验表明，尽管后者具有全局最优性，但AFD类分解在描述局部细节方面优于KL类分解，同时具有数值上的便利性和快速收敛性。 显示英文摘要

摘要： This article introduces adaptive Fourier decomposition (AFD) type methods, emphasizing on those that can be applied to stochastic processes and random fields, mainly including stochastic adaptive Fourier decomposition and stochastic pre-orthogonal adaptive Fourier decomposition. We establish their algorithms based on the covariant function and prove that they enjoy the same convergence rate as the Karhunen-Lo\`eve (KL) decomposition. The AFD type methods are compared with the KL decomposition. In contrast with the latter, the AFD type methods do not need to compute eigenvalues and eigenfunctions of the kernel-integral operator induced by the covariance function, and thus considerably reduce the computation complexity and computer consumes. Various kinds of dictionaries offer AFD flexibility to solve problems of a great variety, including different types of deterministic and stochastic equations. The conducted experiments show, besides the numerical convenience and fast convergence, that the AFD type decompositions outperform the KL type in describing local details, in spite of the proven global optimality of the latter.