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diff --git a/Trajectory spawning.tex b/Trajectory spawning.tex
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+++ b/Trajectory spawning.tex
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This is just $\rho$ integrated over the surface. This high dimensional integral will be done numerically (probably by averaging).
\begin{equation}
\rho_i \rho(x) = \left|
\vec{n}(x_{i}(0)) \vec{n}(x) \cdot
\vec{b}(x_i(0)) \vec{b}(x) \right|
\int
J(x_i(t)) J(x, t) dt
\end{equation}
Where
$\vec{n}(x_{i}(0))$ $\vec{n}(x)$ is the normal to the surface at
$x_i(t=0)$. $x$. 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 + \vec{\nabla} \cdot
\vec{b}(x(t) \vec{b}(X(t, x)) dt \right]
\end{equation}
Where where $X(t, x)$ is the position along the trajectory starting from $x$ at time $t$.
Note that $\vec{\nabla} \cdot \vec{b}$ is the divergence of $\vec{b}$ (written in Eric's notes as $Tr \vec{\nabla} \vec{b}$. This is
ralated related to the trace of the Hessian $H$ (they are simply proportional if $\vec{b}$ is proportional to the gradient).
I suppose that $J(x, t=0)$ is simply set equal to 1.
For a trajectory starting from
$x(t=0)$ $x$ a small surface element of area $dS$
(normal to the trajectory) sweeps out a volume,
$dS $|\vec{b}(x)| dS \int
J(x(t)) J(x, t) dt$ (should check this).
The extra term in the definition of $\rho$ above is for the projection onto the reference surface.
If
$dt$ $|\vec{b}(x)| dt$ has units length (measuring the
length of distance along the trajectory), then
$dS
J(x(t),t))$ J(x,t))$ is the area of the surface element at time $t$, or
$J(x(t),t)$ \begin{equation}
\chi(x,t) = J(x,t)
\end{equation}
is the factor by which the surface element has increased or decreased by time $t$.
(the factors of $b$ seem a bit wrong here. Maybe it be $\chi(x,t) = J(x,t) |b(X(x,t)|/|b(x)|$)
The idea was to spawn new trajectories when the density of trajectories becomes very low. This can be implemented by spawning new trajectories when alpha is larger than a factor $\alpha > 1$. At this point the local density of trajectories has decreased by a factor of $\alpha$.
At time $t$ into trajectory
$i$, starting from $x_i$, we can spawn $N$ new trajectories from a surface of size $dS_i
J(x_i(t), \chi(x_i, t)$ that is perpendicular to the trajectory. If we want to keep the density constant we can choose $N \sim \alpha^{-1}$. The starting points for these trajectories are chosen somehow (randomly) from he surface. At this point trajectory $i$ dies and is replace by $N$ new ones. Those propagate forward and potentially spawn new trajectories.
This is describing a tree structure. Each trajectory has the potential to spawn children and the children can spawn new children. There is never any interaction between trajectories, everything is computed from local quantities, so the computation is still fairly trivially parallelizable.
How do we reconstruct the total volume from all these spawning events? To do so, we use a local variant of eq. \ref{eqn:basin_volume_from_traj}. If trajectory
$i$ spawns children at time $t$ then the surface element will have transformed to be size $S_i^{end} = S_i^{start}
J(x_i(t), \chi(x_i, t)$. From this surface we spawn new trajectories. The total downstream volume will be
\begin{equation}
V_i^{DS} = \int_{S_{i}^{end}} \rho(x) d \sigma(x) = S_{i}^{end} \left< \rho(x) \right>_{S_{i}^{end}}
\end{equation}
...
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)) \vec{n}(x) \cdot
\vec{b}(x(0)) \vec{b}(x) \right|
\left[
\int_0^{t_{death}}
J(x(t), J(x, t) dt
+
J(x(t_{death}), J(x, t_{death}) F( \{ \rho_{DS} \} )
\right]
\end{equation}
At a factor node (spawn surface) you combine all the incoming messages to estimate the relative downstream volume.
