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Register a new userEstimating solution smoothness and data noise with Tikhonov regularization
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Added by
Daniel Gerth
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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A main drawback of classical Tikhonov regularization is that often the parameters required to apply theoretical results, e.g., the smoothness of the sought-after solution and the noise level, are unknown in practice. In this paper we investigate in new detail the residuals in Tikhonov regularization viewed as functions of the regularization parameter. We show that the residual carries, with some restrictions, the information on both the unknown solution and the noise level. By calculating approximate solutions for a large range of regularization parameters, we can extract both parameters from the residual given only one set of noisy data and the forward operator. The smoothness in the residual allows to revisit parameter choice rules and relate a-priori, a-posteriori, and heuristic rules in a novel way that blurs the lines between the classical division of the parameter choice rules. All results are accompanied by numerical experiments.
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With the rapid growth of data, how to extract effective information from data is one of the most fundamental problems. In this paper, based on Tikhonov regularization, we propose an effective method for reconstructing the function and its derivative from scattered data with random noise. Since the noise level is not assumed small, we will use the amount of data for reducing the random error, and use a relatively small number of knots for interpolation. An indicator function for our algorithm is constructed. It indicates where the numerical results are good or may not be good. The corresponding error estimates are obtained. We show how to choose the number of interpolation knots in the reconstruction process for balancing the random errors and interpolation errors. Numerical examples show the effectiveness and rapidity of our method. It should be remarked that the algorithm in this paper can be used for on-line data.
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