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The high-index saddle dynamics (HiSD) method (SIAM J Sci Comput 41:A3576–A3595, 2019) serves as an efficient tool for computing index-k saddle points and constructing solution landscapes. Nevertheless, the conventional HiSD method often encounters slow convergence rates on ill-conditioned problems. To address this challenge, we propose an accelerated high-index saddle dynamics (A-HiSD) by incorporating the heavy ball method. We prove the linear stability theory of the continuous A-HiSD, and subsequently estimate the local convergence rate for the discrete A-HiSD. Our analysis demonstrates that the A-HiSD method exhibits a faster convergence rate compared to the conventional HiSD method, especially when dealing with ill-conditioned problems. We also perform various numerical experiments including the loss function of neural network to substantiate the effectiveness and acceleration of the A-HiSD method. [ABSTRACT FROM AUTHOR]

This work develops a formula for the large threshold limit of multivariate Normal tail probability when at least one of the normalised thresholds grows indefinitely. Derived using integration by parts, the formula expresses the tail probability in terms of conditional probabilities involving one less variate, thereby reducing the problem dimension by 1. The formula is asymptotic to Ruben’s formula under Salvage’s condition. It satisfies Plackett’s identity exactly or approximately, depending on the correlation parameter being differentiated. A recursive algorithm is proposed that allows the tail probability limit to be calculated in terms of univariate Normal probabilities only. The algorithm shows promise in numerical examples to offer a semi-analytical approximation under non-asymptotic situations to within an order of magnitude. The number of univariate Normal probability evaluations is at least n!, however, and in this sense the algorithm suffers from the curse of dimension. [ABSTRACT FROM AUTHOR]

This paper presents a two-parameter extension of the classical Poisson distribution, specifically tailored for rare event modeling. The proposed model is constructed as the sum of two independent Poisson random variables, using a convolution method. Some properties of the distribution, including the probability mass function (PMF), moment-generating function (MGF), mean, variance, higher-order moments, Skewness, and kurtosis, are derived.

Rationale: Zero-event counts are common in clinical studies, particularly when assessing rare adverse events. These occurrences can result from low event rates, short follow-up periods, and small sample sizes. When both intervention and control groups report zero events in a clinical trial, the study is referred to as a double-zero-event study, which presents methodological challenges for evidence synthesis. There has been ongoing debate about whether these studies should be excluded from evidence synthesis, as traditional two-stage meta-analysis methods may not estimate an effect size for them. Recent research suggests that these studies may still contain valuable clinical and statistical information., Aims and Objectives: This study examines the role of double-zero-event studies from the perspective of the fragility index (FI), a popular metric for assessing the robustness of clinical results. We aim to determine how including or excluding double-zero-event studies affects FI derivations in meta-analyses., Methods: We conducted an illustrative case study to demonstrate how double-zero-event studies can impact FI derivations. Additionally, we performed a large-scale analysis of 12,184 Cochrane meta-analyses involving zero-event studies to assess the prevalence and effect of double-zero-event studies on FI calculations., Results: Our analysis revealed that FI derivations in 6608 (54.2%) of these meta-analyses involved double-zero-event studies. Excluding double-zero-event studies could lead to artificially inflated FI values, potentially misrepresenting the results as more robust than they are., Conclusions: We advocate for retaining double-zero-event studies in meta-analyses and emphasise the importance of carefully considering their role in FI assessments. Including these studies ensures a more accurate evaluation of the robustness of clinical results in evidence synthesis., (© 2025 John Wiley & Sons Ltd.)