This module contains tools for performing tangent sensitivity analysis and adjoint sensitivity analysis.

The details are described in our paper "A mathematical analysis of the least squares sensitivity method" at arXiv

User should define two bi-variate functions, f and J.

f(u, s) defines a dynamical system u_{i+1} = f(u_i,s) parameterized by s

J(u, s) defines the objective function, whose ergodic long time average is the quantity of interest.

u0 = rand(m) # initial condition of m-degree-of-freedom system

n0, n = 1000, 1000 # spin-up time and trajectory length

tan = Tangent(f, u0, s, n0, n)

dJds = tan.dJds(J)

adj = Adjoint(f, u0, s, n0, n, J)

dJds = adj.dJds()