论文标题
AFD类型稀疏表示与用于分解随机过程的Karhunen-Loeve扩展
AFD Types Sparse Representations vs. the Karhunen-Loeve Expansion for Decomposing Stochastic Processes
论文作者
论文摘要
本文介绍了自适应傅立叶分解（AFD）类型方法，强调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立叶分解和随机的前骨前自适应傅立叶分解。我们基于协方差函数建立了它们的算法，并证明它们与Karhunen-Loève（KL）分解相同。将AFD类型的方法与KL分解进行了比较。与后者相反，AFD类型方法无需计算由协方差函数引起的内核 - 集成运算符的特征值和特征，因此大大降低了计算复杂性和计算机消耗。各种词典提供了AFD灵活性，可以解决各种各样的问题，包括不同类型的确定性和随机方程。进行的实验表明，除了数值的便利性和快速收敛外，AFD类型的分解在描述本地细节时表现优于KL类型，尽管后者有证明的全局最佳性。
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-Loève (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.
