deletions | additions       diff --git a/Trajectory spawning.tex b/Trajectory spawning.tex  index 13b371a..a2aada2 100644  --- a/Trajectory spawning.tex  +++ b/Trajectory spawning.tex 
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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.  First, just to orient ourselves, here are some equations from Shangs's talk. talk and Eric's notes.  \begin{equation}  V_{basin} = V_{ref} + S_{ref} \left< \frac{1}{\rho} \right>^{-1} F(\rho)  \label{eqn:basin_volume_from_traj}  \end{equation}  where $S_{ref}$ $V_{ref}$  is the area volume  of the spherical reference surface near enclosing  the minimum. When averaging over points generated uniformly in the volume $F(\rho ) = S_{ref} \left< \rho^{-1} \right>^{-1}$. For our purposes, when generating points uniformly on the surface, we will use  \begin{equation}  F(\rho) = \int_{surface} \rho (x) d \sigma (x)  \end{equation}  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)) \right|  \int J(x_i(t)) dt