Convergence of Tikhonov regularization for solving ill-posed operator equations with solutions defined on surfaces

TL;DR

This paper analyzes Tikhonov regularization for ill-posed equations with solutions on surfaces, considering surface perturbations and applications like gravimetry and vector field denoising.

Contribution

It provides an error analysis accounting for surface perturbations and extends regularization analysis to functions in vector bundles over surfaces.

Findings

Error bounds for surface-perturbed Tikhonov regularization

Numerical verification in gravimetry inverse problems

Effective denoising of vector fields on surfaces

Abstract

We study Tikhonov regularization for solving ill--posed operator equations where the solutions are functions defined on surfaces. One contribution of this paper is an error analysis of Tikhonov regularization which takes into account perturbations of the surfaces, in particular when the surfaces are approximated by spline surfaces. Another contribution is that we highlight the analysis of regularization for functions with range in vector bundles over surfaces. We also present some practical applications, such as an inverse problem of gravimetry and an imaging problem for denoising vector fields on surfaces, and show the numerical verification.