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

2015/01/02 by Hui Cao, Sergei V. Pereverzyev, Cao, Hui +5
Mathematics · Computer Science · #Numerical methods in inverse problems #Image and Signal Denoising Methods #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.1501.00362

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.

Related

• Parameter choice strategies for least-squares approximation of noisy smooth functions on the sphere
• Truncated Kernel Stochastic Gradient Descent on Spheres
• Parameter choice strategies for regularized least squares approximation of noisy continuous functions on the unit circle
• Methods of Solving Ill-Posed Problems
• Regularization of linear and nonlinear ill-posed problems by mollification