Toon Ingelaere | Searcharxiv

arXiv2024

Single-ensemble multilevel Monte Carlo for discrete ensemble Kalman methods

Ensemble Kalman methods solve problems in domains such as filtering and inverse problems with interacting particles that evolve over time. For computationally expensive problems, the cost of attaining a high accuracy quickly becomes prohibitive. We exploit a hierarchy of approximations to the underlying forward model and apply multilevel Monte Carlo (MLMC) techniques, improving the asymptotic cost-to-error relation. More specifically, we use MLMC at each time step to estimate the interaction term in a single, globally-coupled ensemble. This technique was proposed by Hoel et al. for the ensemble Kalman filter; our goal is to study its applicability to a broader family of ensemble Kalman methods.

math.NA↗

arXiv2024

Cell Escape Probabilities for Markov Processes on a Grid

Kinetic equations describe physical processes in a high-dimensional phase space and are often simulated using Markov process-based Monte Carlo routines. The quantities of interest are typically defined on the lower-dimensional position space and estimated on a grid (histogram). In several applications, such as the construction of diffusion Monte Carlo-like techniques and variance prediction for particle tracing Monte Carlo methods, the cell escape probabilities, i.e., the probabilities with which particles escape a grid cell during one step of the Markov process, are of interest. In this paper, we derive formulas to calculate the cell escape probabilities for common mesh elements in one, two, and three dimensions. Deterministic calculation of cell escape probabilities in higher dimensions becomes expensive and prone to quadrature errors due to the involved high-dimensional integrals. We therefore also introduce a stochastic Monte Carlo algorithm to calculate the escape probabilities, which is more robust at the cost of a statistical error. The code used to perform the numerical experiments and accompanying GeoGebra tutorials are openly available at https://gitlab.kuleuven.be/numa/public/escape-probabilities-markov-processes.

math-ph↗