A Parameter Estimation Method Using Linear Response Statistics: Numerical Scheme
This work provides a practical implementation of a previously formulated parameter estimation framework for stochastic differential equations, offering improved accuracy over existing methods.
The authors propose a numerical method for parameter estimation of Itô drift diffusions using linear response statistics, avoiding repeated model solves via a polynomial surrogate. Numerical experiments on Langevin dynamics and gradient flows show superiority over conventional equilibrium-statistics-based methods.
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 Itô 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.