Nonparametric regression in a statistical modified Helmholtz equation using the Fourier spectral regularization

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.

Keywords:

• statistical inverse problems
• modified Helmholtz equation
• nonparametric regression
• ill-posed problem

AMS Subject Classification::

• 62G08
• 15A29
• 47A52

Acknowledgements

The authors would like to thank the anonymous referees for the kind remarks leading to the new version of our paper. The authors wishes to express his gratitude to Mr Gary White for the typos paper.

Funding

This work is supported by National Foundation of Science and Technology Development (NAFOSTED Project 101.01.2012.07).