Johannes Vincent Meo | Searcharxiv

arXiv2026

A Priori Adaptive Numerical Methods for Estimating Blow-up Times of Autonomous ODEs

We present effective a priori adaptive numerical methods for estimating the blow-up time for solutions of autonomous ODEs. The novelty of our approach is to base our adaptive steps on the sensitivity of an auxiliary hitting time. We provide results on the theoretical error rates, and show that there is a benefit in terms of computational effort in choosing our adaptive algorithm over alternative approaches. Numerical experiments support our theoretical results and show how the methods perform in practice.

math.NA↗

arXiv2024

A Fast and Accurate Numerical Method for the Left Tail of Sums of Independent Random Variables

We present a flexible, deterministic numerical method for computing left-tail rare events of sums of non-negative, independent random variables. The method is based on iterative numerical integration of linear convolutions by means of Newtons-Cotes rules. The periodicity properties of convoluted densities combined with the Trapezoidal rule are exploited to produce a robust and efficient method, and the method is flexible in the sense that it can be applied to all kinds of non-negative continuous RVs. We present an error analysis and study the benefits of utilizing Newton-Cotes rules versus the fast Fourier transform (FFT) for numerical integration, showing that although there can be efficiency-benefits to using FFT, Newton-Cotes rules tend to preserve the relative error better, and indeed do so at an acceptable computational cost. Numerical studies on problems with both known and unknown rare-event probabilities showcase the method's performance and support our theoretical findings.

stat.CO↗