Q:

Questions about Laplacian Surface Editing

I read the paper called Laplacian Surface Editing, but I'm confused about how does the author solve the formula (4). The paper says that 'Solving this quadratic minimization problem results in a sparse linear system of equations', so I have tried to use the quadprog function in matlab, but when the size of the nodes is growing up, my result is becoming messy. And I read the 'MATLAB 2D demo' code of this paper (http://igl.ethz.ch/projects/Laplacian-mesh-processing/Laplacian-mesh-editing/), it seems that they solve the question just by solving a linear system instead of using method such like 'interior-point' or 'active-set'. If someone has read this paper, Could you please tell me how on earth they solve the formula(4) ? Thanks a lot!

A:

I'll state this in terms of an analogous problem in a domain where you are looking for a function $u(x)$ (namely the Laplace equation): If you are looking to minimize the problem
$$
  \min_{u \in H^1} \frac 12 \|\nabla u\|^2_{L_2(\Omega)} - \int f\; u \; dx
$$
then this is equivalent to solving the partial differential equation
$$
  -\Delta u = f.
$$
This is because the PDE are the optimality conditions for the optimization problem and, because they are linear, allows for the direct solution.
In the same way, the minimization problem given by equation (4) of the paper leads to a partial differential equation -- here posed on the surface of a body -- as the optimality conditions, and this is what the authors solve.

