Two-parameter regularization of ill-posed spherical pseudo-differential equations in the space of continuous functions

TL;DR

This paper introduces a two-step regularization approach for solving ill-posed spherical pseudo-differential equations with noisy data, combining polynomial approximation and collocation to improve accuracy.

Contribution

The paper presents a novel two-step regularization method that enhances solution stability and accuracy for ill-posed spherical equations, outperforming single-step techniques.

Findings

Error bounds in the uniform norm demonstrate improved accuracy.

Numerical experiments support the effectiveness of the two-step method.

The approach effectively handles noisy data in spherical pseudo-differential equations.

Abstract

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.