핵심 개념
Efficient Multilevel MCMC method for solving Bayesian inverse problems in Navier-Stokes equations with Lagrangian observations.
초록
The content discusses the application of Multilevel Markov Chain Monte Carlo (MCMC) for Bayesian inverse problems in recovering initial velocity and random forcing in Navier-Stokes equations. It addresses the challenges of sampling posterior probabilities efficiently, especially when coupled with tracer equations. The MLMCMC method optimally approximates expectations by solving forward equations at different resolutions, reducing computational complexity significantly. The theoretical convergence rate is verified through numerical experiments. Extensions to include Lagrangian observations are discussed, along with detailed mathematical formulations and error analyses.
Introduction:
- Bayesian inverse problem for Navier-Stokes equation.
- Importance in weather forecasting, ocean modeling, aerospace engineering.
- Challenges of expensive MCMC sampling due to high complexity.
Parametric Navier Stokes equation:
- Definition of function spaces and weak formulation.
- Existence and uniqueness of solutions ensured by theorems.
- Regularity assumptions on forcing and initial conditions.
Bayesian inverse problem:
- Observation model for drifting tracers.
- Posterior probability measure and mismatch function defined.
- Continuity of forward map proven.
Posterior approximation:
- Truncation of forcing and initial condition series.
- Error estimates for approximated posterior measure.
FE Approximation:
- Finite Element approximation of truncated problem.
- Time discretization schemes and error analysis.
Multilevel MCMC:
- Application of MLMCMC method to solve Bayesian inverse problems efficiently.
- Sampling strategies, acceptance probabilities, and error estimates discussed.
Numerical experiments:
- Implementation details using Q1-Q2/Q1 elements and Euler implicit/explicit scheme.
- Reference posterior expectation computation and Gaussian prior generation.
인용구
"Sampling the posterior probability...leads to high complexity."
"The convergence of the method is rigorously proved."
"Numerical experiments verify the theoretical convergence rate."