Suppose I have a nonlinear ordinary differential equation, in several variables, with a stated initial condition. How would I go about finding a nonlocal linear approximation? What is known about such approximations?
By nonlocal approximation I just mean that the would-be approximation minimizes the maximum distance between the solution of the approximation and the solution of the original, nonlinear initial value problem, or meets some other criterion such as minimizing the integral of the square of the distance.
At least in the case of minimizing the integral of squared distance, this problem amounts to a nonlinear least squares problem, and the criterion is a differentiable function of the parameters of the linear approximation, so maybe a Gauss-Newton type algorithm or even simple gradient descent (at this point I'm not worried about efficiency) could be applicable.
For the problem I have at hand, I constructed a linear approximation at the initial point, via the Jacobian of the original system of equations, but the solution of the local approximation is very different from the solution of the original nonlinear IVP at later times, which inspires me to look for a nonlocal approximation.