Solving a Linear Inverse Problem by Using an $L^{\infty}$ Fitting and a Hybrid Penalty with an Application to an Inverse Heat Conduction Problem

Abstract

In this paper, we aim at solving a linear inverse problem in the case of measurement data contaminated by uniform noises. Due to ill-posedness, we use a regularization method to solve this problem. The proposed method combines an L 1 fitting with a hybrid penalty which consists of a total variation penalty and an L 2 penalty. An L 1 fitting is utilized to denoise uniform noises. We reconstruct a piecewise constant function, and hence we employ a total variation penalty to identify sharp boundaries, and use an L 2 penalty to improve the stability of the numerical algorithm. We apply the proposed method to an inverse heat conduction problem for testing the validity of the method. The numerical experiments show that the proposed method is accurate and robust.