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.

翻译：本文扩展了先前工作，针对带噪声拉格朗日示踪物位置观测的、耦合示踪物方程的纳维-斯托克斯方程正向问题，研究其未知强迫项和初始条件的贝叶斯反问题。我们考虑二维周期环面上的纳维-斯托克斯方程，并耦合一个简单常微分方程形式的示踪物方程。针对先验均匀分布的情形，我们严格建立了理论框架——其中强迫项和初始条件线性依赖于紧致区间上均匀分布的可数随机变量集。采用多级马尔可夫链蒙特卡洛方法的数值实验，产生了目标量后验期望的近似结果，该结果与已建立的理论最优收敛速率一致。