{'fraction_non_alphanumeric': 0.015665796344647518, 'fraction_numerical': 0.0013054830287206266, 'mean_word_length': 5.185483870967742, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the Regularizing Properties of Truncated Cholesky Factorization 1

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.