Data: a 3D volume Ω with density field ρ(v), and the desired number of vertices
Result: an adaptive sampling V of Ω
Initialize particle locations V;
While  stopping condition not satisfied  do
  Update the - tree [22] data structure for the current sampling locations V;
  for  each particle  do
    Obtain particle ’s -nearest neighbors within five standard deviations (5σ) from - tree;
    for each particle do
      Compute by (2);
      Compute by (4);
    end
    Compute the total force by (5);
  end
  Compute the total energy by (3);
  Run L-BFGS algorithm with and , to update the particle locations V;
  Project V onto the 3D volume surface, if V jumps out of the volume boundary;
end
Algorithm 1: Particle optimization with density field ρ(v).