  This paper presents a numerical method to implement the parameter estimation
method using response statistics that was recently formulated by the authors.
The proposed approach formulates the parameter estimation problem of It\^o
drift diffusions as a nonlinear least-squares problem. To avoid solving the
model repeatedly when using an iterative scheme in solving the resulting
least-squares problems, a polynomial surrogate model is employed on appropriate
response statistics with smooth dependence on the parameters. The existence of
minimizers of the approximate polynomial least-squares problems that converge
to the solution of the true least square problem is established under
appropriate regularity assumption of the essential statistics as functions of
parameters. Numerical implementation of the proposed method is conducted on two
prototypical examples that belong to classes of models with wide range of
applications, including the Langevin dynamics and the stochastically forced
gradient flows. Several important practical issues, such as the selection of
the appropriate response operator to ensure the identifiability of the
parameters and the reduction of the parameter space, are discussed. From the
numerical experiments, it is found that the proposed approach is superior
compared to the conventional approach that uses equilibrium statistics to
determine the parameters.
