Two-parameter regularization of ill-posed spherical pseudo-differential equations in the space of continuous functions
This work provides a more accurate regularization technique for solving ill-posed spherical pseudo-differential equations, which is relevant for geophysics and related fields.
The paper proposes a two-step regularization method for solving ill-posed spherical pseudo-differential equations with noisy data, achieving an error bound in the uniform norm that is potentially smaller than using either step alone. Numerical experiments support the method's superiority.
In this paper, a two-step regularization method is used to solve an ill-posed spherical pseudo-differential equation in the presence of noisy data. For the first step of regularization we approximate the data by means of a spherical polynomial that minimizes a functional with a penalty term consisting of the squared norm in a Sobolev space. The second step is a regularized collocation method. An error bound is obtained in the uniform norm, which is potentially smaller than that for either the noise reduction alone or the regularized collocation alone. We discuss an a posteriori parameter choice, and present some numerical experiments, which support the claimed superiority of the two-step method.