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

Juntao Yang

TL;DR

The theory for the case of the uniform prior where the forcing and the initial condition depend linearly on a countable set of random variables which are uniformly distributed in a compact interval is developed.

Abstract

In this paper, we extend our work to the Bayesian inverse problems for inferring unknown forcing and initial condition of the forward Navier-Stokes equation coupled with tracer equation with noisy Lagrangian observation on the positions of the tracers. We consider the Navier-Stokes equations in the two dimensional periodic torus with a tracer equation which is a simple ordinary differential equation. We developed rigorously the theory for the case of the uniform prior where the forcing and the initial condition depend linearly on a countable set of random variables which are uniformly distributed in a compact interval. Numerical experiment using the MLMCMC method produces approximations for posterior expectation of quantities of interest which are in agreement with the theoretical optimal convergence rate established.

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

TL;DR

The theory for the case of the uniform prior where the forcing and the initial condition depend linearly on a countable set of random variables which are uniformly distributed in a compact interval is developed.

Abstract

In this paper, we extend our work to the Bayesian inverse problems for inferring unknown forcing and initial condition of the forward Navier-Stokes equation coupled with tracer equation with noisy Lagrangian observation on the positions of the tracers. We consider the Navier-Stokes equations in the two dimensional periodic torus with a tracer equation which is a simple ordinary differential equation. We developed rigorously the theory for the case of the uniform prior where the forcing and the initial condition depend linearly on a countable set of random variables which are uniformly distributed in a compact interval. Numerical experiment using the MLMCMC method produces approximations for posterior expectation of quantities of interest which are in agreement with the theoretical optimal convergence rate established.
Paper Structure (7 sections, 7 theorems, 52 equations, 2 figures, 2 tables)

This paper contains 7 sections, 7 theorems, 52 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Parametric Navier Stokes equation
  3. Bayesian inverse problem
  4. Posterior approximation by finite truncation of the forcing and the initial condition
  5. FE Approximation of the truncated problem
  6. Multilevel MCMC
  7. Numerical experiments

Key Result

Lemma 3.1

Under assumption random field, the forward map $\mathcal{G}(\zeta, \xi): U \rightarrow [\mathbb{R}^2]^{JK}$ is continuous as a mapping from the measurable space $(U, \Theta)$ to $([\mathbb{R}^2]^{JK}, \mathcal{B}([\mathbb{R}^2]^{JK}))$.

Figures (2)

  • Figure 1: MLMCMC error for 2D Navier-Stokes equation with uniform prior, a=2
  • Figure 2: MLMCMC error for 2D Navier-Stokes equation with uniform prior, a=3

Theorems & Definitions (11)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • Proposition 3.3
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • Proposition 5.2
  • proof
  • Lemma 5.3
  • ...and 1 more