Fast derivatives of likelihood functionals for ODE based models using adjoint-state method

Abstract

We consider time series data modeled by ordinary differential equations (ODEs), widespread models in physics, chemistry, biology and science in general. The sensitivity analysis of such dynamical systems usually requires calculation of various derivatives with respect to the model parameters.We employ the adjoint state method (ASM) for efficient computation of the first and the second derivatives of likelihood functionals constrained by ODEs with respect to the parameters of the underlying ODE model. Essentially, the gradient can be computed with a cost (measured by model evaluations) that is independent of the number of the ODE model parameters and the Hessian with a linear cost in the number of the parameters instead of the quadratic one. The sensitivity analysis becomes feasible even if the parametric space is high-dimensional.The main contributions are derivation and rigorous analysis of the ASM in the statistical context, when the discrete data are coupled with the continuous ODE model. Further, we present a highly optimized implementation of the results and its benchmarks on a number of problems.The results are directly applicable in (e.g.) maximum-likelihood estimation or Bayesian sampling of ODE based statistical models, allowing for faster, more stable estimation of parameters of the underlying ODE model.

Adjoint Hamiltonian Monte Carlo algorithm for the estimation of elastic modulus through the inversion of elastic wave propagation data

“…Direct differentiation method is the most common method to evaluate gradients in the statistical literature . DDM involves a direct computation of the gradient of the state vector m k with respect to the parameters, that is, .…”

“…Although adjoint methods are ubiquitous in the optimization literature (specifically local optimization), [46][47][48] their use in statistical literature is rare. 49 A linearized forward-like and an adjoint-like partial differential equation (PDE) problem is solved to evaluate Hessian vector products in global seismic inversion using a Newton conjugate gradient method. 50 Adjoint methods were also used to evaluate gradients and Hessians in a stochastic Newton MCMC method 51 in a seismic inversion problem.…”

“…The adjoint method, through the introduction of an adjoint variable offers a way to bypass these expensive computations and requires the solution of just one equation, irrespective of the dimensionality of the parameter vector. Although adjoint methods are ubiquitous in the optimization literature (specifically local optimization), their use in statistical literature is rare . A linearized forward‐like and an adjoint‐like partial differential equation (PDE) problem is solved to evaluate Hessian vector products in global seismic inversion using a Newton conjugate gradient method .…”