§ Gradient depends on norm

• consider a function $f: \mathbb R^n \to \mathbb R$ and an energy $e: (\mathbb R^n \to \mathbb R) \to \mathbb R$.
• We want to optimize $df/dt = - \nabla e(f)$.
• however, what even is $\nabla$?
• Recall that $de$ is the differential which at a point $f$ on the space $\mathbb R^n \to \mathbb R$, in a tangent direction $u \in \mathbb R^n \to \mathbb R$, computes $de|_f(u) \equiv \lim_{\epsilon \to 0} (e(f + \epsilon u) - e(f))/\epsilon$.
• Now the gradient is given by $\langle grad(e), u \rangle_X \equiv de(u)$. So the gradient is the unique vector such that the inner product with the gradient produces the value of the contangent evaluated in that direction.
• Said differently, $\langle nabla(e), -\rangle = de(-)$. This is a Reisez like representation theorem.
• Note that asking for an inner product means we need a hilbert space.
• One choice of inner product is given by $L^2$, where $\langle u, v \rangle_{L^2} \equiv \int \langle u(x), v(x) \rangle dx$.
• More generaly, we can use a Sobolev space, where we define the inner product given by $\langle u, v\rangle_{H^1} \equiv \langle \nabla u, \nabla v\rangle_{L^2}$, which can also be written as $\langle \nabla u, v\rangle_{L^2}$.
• Similarly, for the sobolev space $H^2$, we would use $\langle u, v\rangle_{H^2} \equiv \langle \nabla u, \nabla v\rangle_{L^2}$. which is equal to $\langle \nabla^2 u, v \rangle_{L^2}$.
• In general, we can write our inner product as something like $\langle Au, v\rangle_{L^2}$.

§ Solving heat equation with finite differences

• Solving $df/dt = \nabla f = d^2 f / dx^2$.

If we try to solve this equation using, say, explicit finite differences with grid spacing h, we will need a time step of size O(h 2) to remain stable—significantly slowing down computation as the grid is refined

§ Different norm is good for different situations

• TODO

§ Tangent point energy

• Key intuition: want energy that is small for points that are "close by" in terms of $t$ on the knot, want energy that repels points on the knot that are far away in terms of $t$ by close by in terms of $f(t)$.
• TODO