Convergence Rate of Runge–Kutta-Type Regularization for Nonlinear Ill-Posed Problems under Logarithmic Source Condition

We prove the logarithmic convergence rate of the families of usual and modified iterative Runge–Kutta methods for nonlinear ill-posed problems between Hilbert spaces under the logarithmic source condition, and numerically verify the obtained results. The iterative regularization is terminated by the a posteriori discrepancy principle

Decision Tree Modeling for Osteoporosis Screening in Postmenopausal Thai Women

Osteoporosis is still a serious public health issue in Thailand, particularly in postmenopausal women; meanwhile, new effective screening tools are required for rapid diagnosis. This study constructs and confirms an osteoporosis screening tool-based decision tree (DT) model. Four DT algorithms, namely, classification and regression tree; chi-squared automatic interaction detection (CHAID); quick, unbiased, efficient statistical tree; and C4.5, were implemented on 356 patients, of whom 266 were abnormal and 90 normal. The investigation revealed that the DT algorithms have insignificantly different performances regarding the accuracy, sensitivity, specificity, and area under the curve. Each algorithm possesses its characteristic performance. The optimal model is selected according to the performance of blind data testing and compared with traditional screening tools: Osteoporosis Self-Assessment for Asians and the Khon Kaen Osteoporosis Study. The Decision Tree for Postmenopausal Osteoporosis Screening (DTPOS) tool was developed from the best performance of CHAID’s algorithms. The age of 58 years and weight at a cutoff of 57.8 kg were the essential predictors of our tool. DTPOS provides a sensitivity of 92.3% and a positive predictive value of 82.8%, which might be used to rule in subjects at risk of osteopenia and osteoporosis in a community-based screening as it is simple to conduct

A Modified Asymptotical Regularization of Nonlinear Ill-Posed Problems

In this paper, we investigate the continuous version of modified iterative Runge–Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of ∥ F ( x δ ( T ) ) − y δ ∥ = τ δ + for some δ + > δ , and an appropriate source condition. We yield the optimal rate of convergence