An improved Tikhonov regularization method combined with the exponential filter function

Tikhonov regularization is one of the most popular methods for solving linear discrete ill-posed problems. This approach involves transforming the original problem into a penalized least-squares problem, yielding a solution that exhibits greater robustness against data inaccuracies and computational errors that may occur during the solving process. The choice of the regularization matrix significantly influences the accuracy of the resulting solution. In this paper, we propose a novel method for selecting the regularization matrix based on exponential filter functions, which have a unique connection with Tikhonov filter functions. Our proposed Tikhonov-exponential regularization method only requires one parameter, similar to the traditional Tikhonov regularization method. Computational examples demonstrate the advantages of our proposed method.