Nonparametric regression in a statistical modified Helmholtz equation using the Fourier spectral regularization |
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Authors: | Dang Duc Trong To Duc Khanh Nguyen Huy Tuan Nguyen Dang Minh |
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Affiliation: | 1. Faculty of Maths and Computer Science, University of Science, HoChiMinh City, Vietnamddtrong@hcmus.edu.vn;3. Faculty of Maths and Computer Science, University of Science, HoChiMinh City, Vietnam;4. Centre for Mathematical Science, University of Science, HoChiMinh City, Vietnam |
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Abstract: | In this paper, we deal with the Cauchy problem for the modified Helmholtz equation. We consider two models of data: the bounded variance model and the i.i.d. model. The trigonometric estimators of nonparametric regression is applied to solve the problem. In addition, the general forms of regularization parameter corresponding to the pointwise mean squared error and the mean integrated squared error are discussed in detail. The minimax rate convergence corresponding to the bounded variance model is also presented. In the i.i.d. model, we construct the asymptotic confidence interval for the solution of the problem. Finally, we give some numerical experiments and discuss the obtained results. |
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Keywords: | statistical inverse problems modified Helmholtz equation nonparametric regression ill-posed problem |
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