Mesh Interpolator

class torchpme.lib.MeshInterpolator(cell: Tensor, ns_mesh: Tensor, interpolation_nodes: int, method: str)

Class for handling all steps related to interpolations in the context of a mesh based Ewald summation.

In particular, this includes two core functionalities: 1. “forwards” interpolation, in which the “charges” or more general “particle weights” of atoms are assigned to grid points of a mesh. This is done in the points_to_mesh() function. 2. “backwards” interpolation, in which values defined on a mesh are interpolated to arbitrary positions typically lying between mesh points. This is done in the mesh_to_points() function.

Since the computation of the interpolation weights for both of the above types of calculations is identical, this is performed in a separate function called compute_weights().

See also the Examples of the MeshInterpolator class for a demonstration of the functionalities of this class.

Parameters:

• cell (Tensor) – torch.tensor of shape (3, 3), where cell[i] is the i-th basis vector of the unit cell

• ns_mesh (Tensor) – toch.tensor of shape (3,) Number of mesh points to use along each of the three axes

• interpolation_nodes (int) – int The number n of nodes used in the interpolation per coordinate axis. The total number of interpolation nodes in 3D will be n^3. In general, for n nodes, the interpolation will be performed by piecewise polynomials of degree n - 1 (e.g. n = 4 for cubic interpolation). For Lagrange interpolation, only the values 3, 4, 5, 6, 7 are supported. For P3M interpolation, only the values 1, 2, 3, 4, 5 are supported.

• method (str) – str The interpolation method to use. Either “Lagrange” or “P3M”.

update(cell: Tensor | None = None, ns_mesh: Tensor | None = None) → None

Update buffers and derived attributes of the instance.

Call this to reuse a MeshInterpolator object when the cell parameters or the mesh resolution changes. If neither cell nor ns_mesh are passed there is nothing to be done.

Parameters:

• cell (Tensor | None) – torch.tensor of shape (3, 3), where cell[i] is the i-th basis vector of the unit cell

• ns_mesh (Tensor | None) – toch.tensor of shape (3,) Number of mesh points to use along each of the three axes

Return type:

None

get_mesh_xyz() → Tensor

Returns the Cartesian positions of the mesh points.

Returns:

torch.tensor of shape (nx, ny, nz, 3) containing the positions of the grid points

Return type:

Tensor

compute_weights(positions: Tensor)

Compute the interpolation weights of each atom for a given cell (specified during initialization of this class). The weights are not returned, but are used when calling the forward (points_to_mesh()) and backward (mesh_to_points()) interpolation functions.

Parameters:

positions (Tensor) – torch.tensor of shape (N, 3) containing the Cartesian coordinates of the N particles within the supercell.

points_to_mesh(particle_weights: Tensor) → Tensor

Generate a discretized density from interpolation weights. It assumes that compute_weights() has been called before to compute all the necessary weights and indices.

Parameters:

particle_weights (Tensor) – torch.tensor of shape (n_points, n_channels) particle_weights[i,a] is the weight (charge) that point (atom) i has to generate the “a-th” potential. In practice, this can be used to compute e.g. the Na and Cl contributions to the potential separately by using a one-hot encoding of the types.

Returns:

torch.tensor of shape (n_channels, n_mesh, n_mesh, n_mesh) Discrete density

Return type:

Tensor

mesh_to_points(mesh_vals: Tensor) → Tensor

Take a function defined on a mesh and interpolate its values on arbitrary positions.

Parameters:

mesh_vals (Tensor) – torch.tensor of shape (n_channels, nx, ny, nz) The tensor contains the values of a function evaluated on a three-dimensional mesh. (nx, ny, nz) are the number of points along each of the three directions, while n_channels provides the number of such functions that are treated simulateously for the present system.

Returns:

interpolated_values: torch.tensor of shape (n_points, n_channels) Values of the interpolated function.

Return type:

Tensor