Convergence of Tikhonov regularization for solving ill-posed operator equations with solutions defined on surfaces
It offers theoretical guarantees for regularization methods on surfaces, benefiting applied mathematicians and engineers working with inverse problems on curved domains.
This paper provides an error analysis of Tikhonov regularization for ill-posed operator equations with solutions on surfaces, accounting for surface perturbations and vector bundle ranges, and demonstrates numerical verification on gravimetry and denoising problems.
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.