Gateaux derivative
Recall that a functional takes functions to scalars.
1. example
We have an energy functional \(E[f] = \int_0^1 |f''(t)|^2 dt\) that measures who "wiggly" a function is. Let's think about the following optimization problem: find the function \(f\) that gives the minimum energy, subject to \(f(0) = a, f'(0) = b, f(1) = c, f'(1) = d\). That is, given the endpoints and the tangents at the endpoints, find the "smoothest" connecting function.
What do we want from a local optimum? Any small perturbation should result in a larger value. Then, let's consider a function \(f(t)\). Now, take another function \(p(t)\) such that \(p(0) = p(1) = p'(0) = p'(1) = 0\). That is, \(p\) does not mess with the constraints. The cartoon is that \(p(t)\) is maybe the graph of a little bump. Then, we define the Gateaux derivative \(dE_f[p] = \frac{d}{dh} E[f+hp] \big\vert_{h=0}\). This is read as "the derivative of energy at \(f\) in the \(p\) direction". By analogy, remember that the directional derivative for a multi-variate scalar function at a point \(x\) can be defined in an arbitrary direction \(v\) (see gradient). The LHS of the definition should be read as "how fast does the energy change with a infinitesimal amount of \(p\)?" Then, we try to make some argument about what sort of function \(f\) should be so that the Gateaux derivative is 0 in every direction.
2. notes
- when computing the Gateaux integral, you will often have to apply the leibniz rule and integration by parts