Multilevel Monte Carlo Simulation of Diffusively Scaled Kinetic Equations

Explore a comprehensive lecture on advanced computational methods for simulating kinetic equations using multilevel Monte Carlo techniques. Learn how kinetic equations model particle dynamics through time-dependent probability densities in position-velocity phase space, with applications spanning fusion research, radiation therapy, and medical imaging. Discover the challenges posed by multiscale behavior in kinetic systems, where collision time scales are significantly shorter than evolution time scales of quantities of interest, leading to computational cost blowup in classical explicit time-integration methods. Understand the concept of asymptotic-preserving numerical schemes that can resolve limiting model dynamics without computational cost explosion. Examine how the multilevel Monte Carlo framework combines with asymptotic-preserving Monte Carlo schemes to efficiently simulate diffusively scaled Boltzmann-BGK equations through a hierarchy of simulations where coarser levels follow macroscopic limiting dynamics while finer levels resolve true high-collisional kinetic dynamics. Gain insights into the construction of correlated particle trajectories across the multilevel hierarchy and review numerical results demonstrating significant computational effort reduction while maintaining accuracy in kinetic dynamics estimation.