标题： AFD类型稀疏表示与Karhunen-Loeve展开在分解随机过程中的比较 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）类型方法，重点介绍可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了它们的算法，并证明了它们具有与Karhunen-Loeve（KL）分解相同的收敛速度。 AFD类型方法与KL分解进行了比较。 与后者相比，AFD类型方法不需要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。 各种类型的字典为AFD提供了灵活性，以解决各种问题，包括不同类型确定性和随机方程。 进行的实验表明，除了数值上的便利性和快速收敛外，AFD类型分解在描述局部细节方面优于KL类型，尽管后者已被证明具有全局最优性。 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-Loeve (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.