cortex.polyutils.Surface.create_biharmonic_solver

Surface.create_biharmonic_solver(boundary_verts, clip_D=0.1)

Set up biharmonic equation with Dirichlet boundary conditions on the cortical mesh and precompute Cholesky factorization for solving it. The vertices listed in boundary_verts are considered part of the boundary, and will not be included in the factorization.

To facilitate Cholesky decomposition (which requires a symmetric matrix), the squared Laplace-Beltrami operator is separated into left-hand-side (L2) and right-hand-side (Dinv) parts. If we write the L-B operator as the product of the stiffness matrix (V-W) and the inverse mass matrix (Dinv), the biharmonic problem is as follows (with u denoting non-boundary vertices)

\begin{eqnarray} L^2_{u} \phi &=& -\rho_{u} \\ \left[ D^{-1} (V-W) D^{-1} (V-W) \right]_{u} \phi &=& -\rho_{u} \\ \left[ (V-W) D^{-1} (V-W) \right]_{u} \phi &=& -\left[D \rho\right]_{u} \end{eqnarray}

Parameters:

boundary_vertslist or ndarray of length V

Indices of vertices that will be part of the Dirichlet boundary.

Returns:

lhssparse matrix

Left side of biharmonic problem, (V-W) D^{-1} (V-W)

rhssparse matrix, dia

Right side of biharmonic problem, D

Dinvsparse matrix, dia

Inverse mass matrix, D^{-1}

lhsfaccholesky Factor object

Factorized left side, solves biharmonic problem

notboundaryndarray, int

Indices of non-boundary vertices