On the Regularizing Properties of Truncated Cholesky Factorization

Due to the principle of regularization by restricting the number of degrees of freedom, truncating the Cholesky factorization of a symmetric positive definite matrix can be expected to have a stabilizing effect. Based on this idea, we consider four different approaches for regularizing ill-posed linear operator equations. Convergence in the noise free case as well as — with an appropriate a priori truncation rule — in the situation of noisy data is analyzed. Moreover, we propose an a posteriori truncation rule and characterize its convergence. Numerical tests illustrate the theoretical results. Both analysis and computations suggest one of the four variants to be favorable to the others.