I have a general research problem which is as follows:
Suppose I have a set of coupled PDEs, $ \partial f(x) $, $\partial g(x, y)$, with some initial conditions $x_0, y_0$
I can solve these PDEs numerically and evaluate at a particular point $(x_1, y_1)$ as $F = f(x_1)$, $G = g(x_1,y_1)$.
I want to find the appropriate initial conditions $x_0, y_0$, such that $F$ and $G$ are equal to particular target values $F_{t}$, $G_{t}$
Does an analytical/numerical/algorithmic method exist for doing this?
I can see a general method of choosing the initial conditions, performing the integration, looking at the result, and then perturbing the initial conditions somehow, but wanted to know if a 'smart way' exists.
Thanks