A Parameter Estimation Method Using Linear Response Statistics: Numerical Scheme

TL;DR

This paper introduces a numerical scheme for parameter estimation in Itô drift diffusions using response statistics, employing polynomial surrogates to efficiently solve nonlinear least-squares problems with convergence guarantees.

Contribution

The authors develop a polynomial surrogate-based numerical method for parameter estimation that avoids repeated model solving and proves convergence to true solutions.

Findings

The method successfully estimates parameters in Langevin dynamics and gradient flows.

It outperforms conventional equilibrium statistic-based approaches.

Practical issues like response operator selection are addressed.

Abstract

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.