Rare event simulation of dynamic stochastic systems

Sammanfattning

Estimating rare event probabilities is a fundamental challenge across scientificand engineering disciplines, including structural reliability, finance, andclimate science. In these regimes, naive Monte Carlo sampling becomescomputationally infeasible, as it requires very large sample sizes to observesufficiently many rare event occurrences. This limitation has driven thedevelopment of advanced simulation techniques based on variance reductionand adaptive sampling strategies.This thesis investigates the implementation and performance of the AcceleratedWeight Histogram (AWH) method for rare event simulations in dynamicalsystems. AWH is an adaptive importance sampling technique originallydeveloped for exploring complex free energy landscapes in statisticalphysics. The method introduces a discrete control parameter 𝜆 defininga family of intermediate distributions that smoothly connect a referencestate to a rare event region. Through iterative bias refinement and MarkovChain Monte Carlo sampling, AWH estimates conditional probabilities andconstructs an adaptive bias potential that enhances exploration across the fullparameter space, thereby overcoming sampling barriers that limit conventionalapproaches.The AWH method is evaluated on a number of increasingly complex models,including a static Gaussian model, one-dimensional random walks, a double-wellpotential, and a two-dimensional three-hole potential landscape.The results demonstrate that AWH accurately recovers rare event probabilitiesspanning several orders of magnitude, showing excellent agreement withanalytical benchmarks. The method is also shown to effectively capture bothequilibrium distributions and path-dependent rare events, including barrier crossingdynamics. Mean first passage time analysis indicates that adaptivetrajectory lengths preserve kinetic information and can be used for analyzingof system dynamics.Overall, these findings show AWH as a robust and efficient framework forrare event estimation, with promising potential for applications to real-worldproblems such as structural failure analysis, extreme weather prediction,and financial risk assessment, where efficient sampling in high-dimensionalsystems remains a critical challenge.