A two-stage method for reconstruction of parameters in diffusion equations

TL;DR

This paper introduces a two-stage method combining total variation regularization and K-means clustering to efficiently reconstruct conductivity parameters in steady-state diffusion equations, with proven convergence and demonstrated effectiveness.

Contribution

It presents a novel two-stage approach that integrates regularization and clustering for improved conductivity reconstruction in diffusion equations.

Findings

The scheme converges theoretically.

Numerical examples show accurate reconstruction.

Effective handling of 'blocky' conductivity functions.

Abstract

Parameter reconstruction for diffusion equations has a wide range of applications. In this paper, we proposed a two-stage scheme to efficiently solve conductivity reconstruction problems for steady-state diffusion equations with solution data measured inside the domain. The first stage is based on total variation regularization of the log diffusivity and the split Bregman iteration method. In the second stage, we apply the K-means clustering for the reconstruction of ``blocky'' conductivity functions. The convergence of the scheme is theoretically proved and extensive numerical examples are shown to demonstrate the performance of the scheme.