2020/12/29 by Daniel Gerth, Gerth, Daniel, Ronny Ramlau +1
Engineering · Mathematics · #65F22 #65R30 #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Reservoir Engineering and Simulation Methods #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.2012.14875
openalex publication_date 2020/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A main drawback of classical Tikhonov regularization is that often the\nparameters required to apply theoretical results, e.g., the smoothness of the\nsought-after solution and the noise level, are unknown in practice. In this\npaper we investigate in new detail the residuals in Tikhonov regularization\nviewed as functions of the regularization parameter. We show that the residual\ncarries, with some restrictions, the information on both the unknown solution\nand the noise level. By calculating approximate solutions for a large range of\nregularization parameters, we can extract both parameters from the residual\ngiven only one set of noisy data and the forward operator. The smoothness in\nthe residual allows to revisit parameter choice rules and relate a-priori,\na-posteriori, and heuristic rules in a novel way that blurs the lines between\nthe classical division of the parameter choice rules. All results are\naccompanied by numerical experiments.\n