Multilevel Markov Chain Monte Carlo for Bayesian Inverse Problems in Navier-Stokes Equation with Lagrangian Observations

Alapfogalmak

Efficient Multilevel MCMC method for solving Bayesian inverse problems in Navier-Stokes equations with Lagrangian observations.

Kivonat

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.

Statisztikák

None

Idézetek

"Sampling the posterior probability...leads to high complexity."
"The convergence of the method is rigorously proved."
"Numerical experiments verify the theoretical convergence rate."

Főbb Kivonatok

Multilevel Markov Chain Monte Carlo for Bayesian inverse problems for Navier Stokes equation with Lagrangian Observations

by Juntao Yang : arxiv.org 03-20-2024

https://arxiv.org/pdf/2403.12501.pdf

Mélyebb kérdések

How does the MLMCMC method compare to other sampling techniques

The Multilevel Markov Chain Monte Carlo (MLMCMC) method offers several advantages compared to other sampling techniques. One key benefit is its ability to reduce computational complexity by approximating the posterior expectation with a smaller number of samples while maintaining accuracy. This efficiency is achieved through a hierarchical approach that leverages different levels of resolution in solving the forward equations, leading to significant time savings in comparison to traditional MCMC methods. Additionally, MLMCMC has been rigorously proven for convergence and error estimation, providing confidence in its results. The method also allows for scalability, making it suitable for handling high-dimensional problems efficiently.

What are the implications of assuming more regularity on forcing and initial conditions

Assuming more regularity on the forcing and initial conditions in Bayesian inverse problems for fluid dynamics can have important implications. By imposing additional constraints on the smoothness and decay rates of these parameters, we enhance the well-posedness of the inverse problem and enable more accurate numerical approximations. Regularity assumptions ensure that solutions are within appropriate function spaces, facilitating error analysis and convergence proofs. Moreover, increased regularity can lead to improved stability and robustness of numerical schemes when solving complex fluid dynamics equations like Navier-Stokes.

How can this approach be extended to more complex fluid dynamics problems

This approach can be extended to tackle more complex fluid dynamics problems by adapting the MLMCMC framework to higher-dimensional systems or incorporating additional physical phenomena into the model. For instance, extending the methodology to three-dimensional Navier-Stokes equations or including multiphase flows would require adjusting the discretization schemes and sampling strategies accordingly. Furthermore, integrating turbulence models or considering non-Newtonian fluids could expand the applicability of MLMCMC in capturing diverse flow behaviors accurately. Overall, by tailoring the methodological aspects to specific challenges posed by complex fluid dynamics scenarios, MLMCMC can offer valuable insights into a wide range of real-world applications across various industries such as aerospace engineering or climate modeling.