  We consider the inverse problem of estimating parameters of a driven
diffusion (e.g., the underlying fluid flow, diffusion coefficient, or source
terms) from point measurements of a passive scalar (e.g., the concentration of
a pollutant). We present two particle methods that leverage the structure of
the inverse problem to enable efficient computation of the forward map, one for
time evolution problems and one for a Dirichlet boundary-value problem. The
methods scale in a natural fashion to modern computational architectures,
enabling substantial speedup for applications involving sparse observations and
high-dimensional unknowns. Numerical examples of applications to Bayesian
inference and numerical optimization are provided.
