# Project : tropics

## Section: Application Domains

### Linearization

To simulate a complex system often requires solving a system of Partial Differential Equations.
This is sometimes too expensive, in particular in the context of real time.
When one wants to simulate the reaction of this complex system to small perturbations around a fixed
set of parameters, there is a very efficient approximate solution: just suppose that the system
is linear in a small neighborhood of the current set of parameter. The reaction of the system
is thus approximated by a simple product of the variation of the parameters with the
Jacobian matrix of the system. This Jacobian matrix can be obtained by AD.
This is especially cheap when the Jacobian matrix is sparse.
The simulation can be improved further by introducing higher-order derivatives, such as Taylor
expansions, which can also be computed through AD.
The result is often called a *reduced model*.