  We propose a new and simple discretization, named the Modified Virtual Grid
Difference (MVGD), for numerical approximation of the Laplace-Beltrami (LB)
operator on manifolds sampled by point clouds. The key observation is that both
the manifold and a function defined on it can both be parametrized in a local
Cartesian coordinate system and approximated using least squares. Based on the
above observation, we first introduce a local virtual grid with a scale adapted
to the sampling density centered at each point. Then we propose a modified
finite difference scheme on the virtual grid to discretize the LB operator.
Instead of using the local least squares values on all virtual grid points like
the typical finite difference method, we use the function value explicitly at
the grid located at the center (coincided with the data point). The new
discretization provides more diagonal dominance to the resulting linear system
and improves its conditioning. We show that the linear system can be robustly,
efficiently and accurately solved by existing fast solver such as the Algebraic
Multigrid (AMG) method. We will present numerical tests and comparison with
other exiting methods to demonstrate the effectiveness and the performance of
the proposed approach.
