Regularization in a functional reproducing kernel Hilbert space
Regularization in a functional reproducing kernel Hilbert space
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DOI：
10.1016/j.jco.2021.101567
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发表时间：
2021-03
期刊：
影响因子：
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通讯作者：
Rui Wang;Yuesheng Xu
中科院分区：
文献类型：
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作者：
Rui Wang;Yuesheng Xu
考虑求解一类半离散第一类积分方程组，其右端为有限维采样值向量，并在泛函再生核Hilbert空间（FRKHS）中给出了该系统的正则化方法，其中定义半离散积分算子的线性泛函是连续的.建立了正则化方法的表示定理，将无穷维问题化为有限维线性系统，并将其解表示为FRKHS核函数的线性组合.我们构建特定的FRKHS和他们的相关内核的功能从氡数据重建，并开发相关的正则化方法重建。我们提出的数值结果表明，建议的正则化方法优于传统的Tikhonov正则化在L2空间或正则化在经典的再生核希尔伯特空间。
We consider solving a system of semi-discrete first kind integral equations with a right-hand-side being a finite dimensional vector of sampling values and propose a regularization method for the system in a functional reproducing kernel Hilbert space (FRKHS), where the linear functionals that define the semi-discrete integral operator are continuous. A representer theorem for the regularization method is established, which reduces the infinite dimensional problem to a finite dimensional linear system and expresses its solution as a linear combination of the FRKHS kernel sessions. We construct specific FRKHSs and their associated kernels for reconstruction of a function from Radon data, and develop related regularization methods for the reconstruction. We present numerical results which demonstrate that the proposed regularization method outperforms either the traditional Tikhonov regularization in the L 2 space or the regularization in the classical reproducing kernel Hilbert space.