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Shang gave his talk today about the trajectory sampling method that Eric Vanden Eijden came up with. One problem he had was that a uniform distribution on the central sphere leads to very poor coverage of space because the trajectories diverge very fast in some regions and bunch up in other regions. This section will be about the idea of adaptively spawning new trajectories in regions where the trajectories are diverging.
note: $\vec{b}(x)$ is a vector field which is aligned with the gradient at $x$. I will use $b$ instead of the gradient because I don't fully understand the relationship between them. $b$ has units length / time. The gradient has units energy / length. At this point I think they are just related by a constant, but part of me also thinks the length of $b$ should not change throughout space $\vec{b} \sim \vec{\nabla} U / | \nabla U |$.
First, just to orient ourselves, here are some equations from Shangs's talk and Eric's notes.
\begin{equation}
V_{basin} = V_{ref} + F(\rho)
...
This is just $\rho$ integrated over the surface. This high dimensional integral will be done numerically.
\begin{equation}
\rho_i = \left| \vec{n}(x_{i}(0)) \cdot
\vec{\nabla} U(x_i(0)) \vec{b}(x_i(0)) \right|
\int J(x_i(t)) dt
\end{equation}
Where $\vec{n}(x_{i}(0))$ is the normal to the surface at $x_i(t=0)$. Time $t$ is a measure of the progress along the trajectory. At $t=0$ the trajectory is at the surface and it heads to the basin boundary for large $t$. $\nabla U$ is the gradient of the potential, which is always tangent to our steepest descent path.
The Jacobian is updated according to
\begin{equation}
J(x, t+dt) = J(x,t) \left[ 1 +
Tr \left( H(x(t) \right) \vec{\nabla} \cdot \vec{b}(x(t) dt \right]
\end{equation}
Where
$\vec{\nabla} \cdot \vec{b}$ is the divergence of $\vec{b}$ (written in Eric's notes as $Tr \vec{\nabla} \vec{b}$. This is ralated to the trace of the Hessian $H$
(they are simply proportional if $\vec{b}$ is
proportional to the
Hessian. gradient). I suppose that $J(x, t=0)$ is simply set equal to 1.
For a trajectory starting from $x(t=0)$ a small surface element of area $dS$ sweeps out a volume, $dS \int J(x(t)) dt$ (should check this).
The extra term in the definition of $\rho$ above is for the projection onto the reference surface.
...
The recursive equation for $\rho_i$ becomes
\begin{equation}
\rho_i = \left| \vec{n}(x_{i}(0)) \cdot
\vec{\nabla} U(x_i(0)) \vec{b}(x_i(0)) \right|
\left[
\int_0^{t_{death}} J(x_i(t), t) dt
+ J(x_i(t_{death}), t_{death}) F( \{ \rho_{DS} \} )
...
The messages from trajectory nodes to factor nodes will simply be the $\rho$ for that trajectory.
\begin{equation}
\mu_{t \to f} =
\left| \vec{n}(x(0)) \cdot
\vec{\nabla} U(x(0)) \vec{b}(x(0)) \right|
\left[
\int_0^{t_{death}} J(x(t), t) dt
+ J(x(t_{death}), t_{death}) F( \{ \rho_{DS} \} )
...
This message is then passed upstream to the trajectory which initiated the spawning event. This allows us to re-write the $\mu_{t \to f}$ in terms of the single incoming message
\begin{equation}
\mu_{t \to f} =
\left| \vec{n}(x(0)) \cdot
\vec{\nabla} U(x(0)) \vec{b}(x(0)) \right|
\left[
\int_0^{t_{death}} J(x(t), t) dt
+ J(x(t_{death}), t_{death}) \mu_{f \to t} )
