2023/03/24 by Fei Lu, Lu, Fei, Miao‐Jung Yvonne Ou +1 · 1 citation
Engineering · Mathematics · #FOS: Mathematics #Microwave Imaging and Scattering Analysis #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.2303.13737
openalex publication_date 2023/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Regularization is a long-standing challenge for ill-posed linear inverse problems, and a prototype is the Fredholm integral equation of the first kind with additive Gaussian measurement noise. We introduce a new RKHS regularization adaptive to measurement data and the underlying linear operator. This RKHS arises naturally in a variational approach, and its closure is the function space in which we can identify the true solution. Also, we introduce a small noise analysis to compare regularization norms by sharp convergence rates in the small noise limit. Our analysis shows that the RKHS- and L2-regularizers yield the same convergence rate when their optimal hyper-parameters are selected using the true solution, and the RKHS-regularizer has a smaller multiplicative constant. However, in computational practice, the RKHS regularizer significantly outperforms the L2-and l2-regularizers in producing consistently converging estimators when the noise level decays or the observation mesh refines.