lapy.heat¶

Functions for computing heat kernel and diffusion.

Inputs are eigenvalues and eigenvectors (for heat kernel) and the mesh geometries (tet or tria mesh) for heat diffusion.

lapy.heat.diagonal(t, x, evecs, evals, n)[source]¶

Compute heat kernel diagonal K(t,x,x).

For a given time t (can be a vector) using only the first n smallest eigenvalues and eigenvectors.

Parameters:

tfloat or np.ndarray: Time or array of time values, shape (n_times,).
xnp.ndarray: Vertex indices for the positions of K(t,x,x), shape (n_vertices,).
evecsnp.ndarray: Eigenvectors matrix, shape (n_vertices, n_eigenvectors).
evalsnp.ndarray: Vector of eigenvalues, shape (n_eigenvalues,).
nint: Number of eigenvectors and eigenvalues to use (smaller or equal to length).

Returns:

np.ndarray: Heat kernel diagonal values. Shape (n_vertices, n_times) if t is array, or (n_vertices, 1) if t is scalar. Rows correspond to vertices selected in x, columns to times in t.

Raises:

lapy.heat.diffusion(geometry, vids, m=1.0, aniso=None, use_cholmod=False)[source]¶

Compute the heat diffusion from initial vertices in vids.

Uses the backward Euler solution with time \(t = m l^2\), where l describes the average edge length.

Parameters:

geometryTriaMesh or TetMesh: Geometric object on which to run diffusion.
vidsint or np.ndarray: Vertex index or indices where initial heat is applied.
mfloat, default=1.0: Factor to compute time of heat evolution.
anisoint, default=None: Number of smoothing iterations for curvature computation on vertices.
use_cholmodbool, default=False: Which solver to use. If True, use Cholesky decomposition from scikit-sparse cholmod. If False, use spsolve (LU decomposition).

Returns:

Raises:

lapy.heat.kernel(t, vfix, evecs, evals, n)[source]¶

Compute heat kernel from all points to a fixed point.

For a given time t, computes K_t(p,q) using only the first n smallest eigenvalues and eigenvectors:

\[K_t (p,q) = \sum_j \exp(-\lambda_j t) \phi_j(p) \phi_j(q)\]

where \(\lambda_j\) are eigenvalues and \(\phi_j\) are eigenvectors.

Parameters:

tfloat or np.ndarray: Time (can also be array, if passing multiple times), shape (n_times,).
vfixint: Fixed vertex index.
evecsnp.ndarray: Matrix of eigenvectors, shape (n_vertices, n_eigenvectors).
evalsnp.ndarray: Column vector of eigenvalues, shape (n_eigenvalues,).
nint: Number of eigenvalues/vectors used in heat kernel (n <= n_eigenvectors).

Returns:

np.ndarray: Heat kernel values. Shape (n_vertices, n_times) if t is array, or (n_vertices, 1) if t is scalar. Rows correspond to all vertices, columns to times in t.

Raises: