  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.
