算子方程基本问题解的再生核稀疏表示

在一个Hilbert空间中通过内积核定义的线性算子对应一个自然的再生核Hilbert空间结构.本文将称其为H-HK结构.这个结构本身内蕴一个基方法,可以解答线性算子的若干最基本的问题,包括确定或刻画其值域空间、解算子方程及解Moore-Penrose伪-(广义-)逆算子问题.在对已存在结果的简要综述之后,本文的目的是建立H-HK结构下的预正交自适应Fourier分解(pre-orthogonal adaptive Fourier decomposition,POAFD)算法.在这个方法之下导出上述3个问题的解的稀疏表示.在逐次跟踪匹配的优化方法论中POAFD的优选原理保证了它在理论上和实用上的最优性.它也具有算法上的可行性.所提供的方法可有效地应用于具体实际问题,包括信号与图像重构、常微分方程、偏微分方程和优化问题的数值解等.

Sparse representations in reproducing kernels of operator equations

In a Hilbert space a linear operator defined through an inner product kernel has a natural reproducing kernel structure.In the present paper we first define what we call the H-HK formulation that stands as an axiomatic basis of the study.There is a built-in mechanism in the H-HK formulation that can straightforwardly solve three basic type problems,namely,the image function identification,inverse problem,and Moore-Penrose pseudo-inverse problem.After a summary on the classical basis method we introduce the POAFD(pre-orthogonal adaptive Fourier decomposition)“non-basis”method in the H-HK formulation.We give historical notes,theoretical foundations,as well as algorithm principles of POAFD.The maximal selection principle of POAFD makes itself be the best among all the existing matching pursuit methods in the one-step-optimal-selection category.It,therefore,can be used to give fast converging sparse numerical solutions of approximation,ordinary and partial differential equations,and optimization problems.