2014/07/08 by Julianne Chung, Chung, Julianne, Malena I. Español +3
Computer Science · Mathematics · #FOS: Mathematics #Image and Signal Denoising Methods #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.1407.1911
openalex publication_date 2014/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In this work we consider the problem of finding optimal regularization parameters for general-form Tikhonov regularization using training data. We formulate the general-form Tikhonov solution as a spectral filtered solution using the generalized singular value decomposition of the matrix of the forward model and a given regularization matrix. Then, we find the optimal regularization parameter by minimizing the average of the errors between the filtered solutions and the true data. We extend the approach to the multi-parameter Tikhonov problem for the case where all the matrices involved are simultaneously diagonalizable. For problems where this is not the case, we describe an approach to compute optimal or near-optimal regularization parameters by using operator approximations for the original problem. Several tests are performed for 1D and 2D examples using different norms on the errors, showing the effectiveness of this approach.