# Fast Calibration of Car Following models to Trajectory data using the Adjoint Method

## TL;DR

This paper introduces an efficient adjoint method for calibrating car-following models to trajectory data, significantly reducing computational cost and improving accuracy compared to traditional gradient-free optimization techniques.

## Contribution

It develops an analytical gradient calculation using the adjoint method for car-following model calibration, enhancing efficiency and accuracy over existing methods.

## Key findings

- Adjoint method reduces calibration computational cost.
- Quasi-Newton algorithms outperform genetic algorithms.
- Calibration accuracy is improved with the adjoint approach.

## Abstract

Before a car-following model can be applied in practice, it must first be validated against real data in a process known as calibration. This paper discusses the formulation of calibration as an optimization problem, and compares different algorithms for its solution. The optimization consists of an arbitrary car following model, posed as either an ordinary or delay differential equation, being calibrated to an arbitrary source of trajectory data which may include lane changes. Typically, the calibration problem is solved using gradient free optimization. In this work, the gradient of the optimization problem is derived analytically using the adjoint method. The computational cost of the adjoint method does not scale with the number of model parameters, which makes it more efficient than evaluating the gradient numerically using finite differences. Numerical results are presented which show that quasi-newton algorithms using the adjoint method are significantly faster than a genetic algorithm, and also achieve slightly better accuracy of the calibrated model.