A weighted online regularization for a fully nonparametric model with heteroscedasticity

In this paper, combining B-spline function and Tikhonov regularization, we propose an online identification approach for reconstructing a smooth function and its derivative from scattered data with heteroscedasticity. Our methodology offers the unique advantage of enabling real-time updates based on new input data, eliminating the reliance on historical information. First, to address the challenge of heteroscedasticity and computation cost, we employ weight coefficients along with a judiciously chosen set of knots for interpolation. Second, a reasonable approach is provided to select weight coefficients and the regularization parameter in objective functional. Finally, We substantiate the efficacy of our approach through a numerical example and demonstrate its applicability in solving inverse problems. It is worth mentioning that the algorithm not only ensures the calculation efficiency, but also trades the data accuracy through the data volume.