darsia.MG#

class MG(depth=2, smoother_iterations=5, maxiter=100, tol=None, dim=2, mass_coeff=None, diffusion_coeff=None, verbose=False)[source]#

Bases: Solver

Muiltilvel solver for the problem: mass_coeff * x - diffusion_coeff * laplace(x) = rhs. For arrays with odd number of elements the last element is dropped in the refinement stages.

V-cycle.

NOTE: Add possibility to actually have a heterogeneous diffusion coefficient. As of now it is assumed to be constant, or multiplied after Laplacian.

Initialize the solver.

Parameters:

depthint: Depth of the multigrid hierarchy.
smoother_iterationsint: Number of iterations for the smoother.
maxiterint: Maximum number of iterations.
tolOptional[float]: Tolerance.
dimint: Dimension of the problem.
mass_coeffnp.ndarray or float: Mass coefficient.
diffusion_coeffnp.ndarray or float: Diffusion coefficient.
verbosebool: Print information.

Methods

`base_V_Cycle`(x0, rhs, depth[, h])	Base V-Cycle (recursive function)
`operator`(x, h)	The solution operator for the problem
`prolongate_parameters`(pad_tuple)	In case of heterogeneous parameters the parameters are prolongated by averaging.
`prolongation`(x)	Prolongate x, i.e., refine it by a factor 2.
`restrict_parameters`()	In case of heterogeneous parameters the parameters are restricted by averaging.
`restriction`(x)	Restrict x, i.e., coarsen it by a factor 2.
`update_params`([dim, mass_coeff, diffusion_coeff])	Update parameters of the solver.

Attributes

`smoother`	Smoother used in the MG cycle.
`smoother_iterations`	Number of smoother iterations per step.
`depth`	Depth of V-cycle.
`heterogeneous`	Flag controlling whether the material paremeters are heterogeneous.
`dim`	Spatial dimension of the problem.
`maxiter`	Maximum number of iterations.
`tol`	Tolerance for convergence.
`mass_coeff`	Mass coefficient.
`diffusion_coeff`	Diffusion coefficient.
`verbose`	Verbosity.

base_V_Cycle(x0, rhs, depth, h=1)[source]#

Base V-Cycle (recursive function)

Parameters:

x0np.ndarray: Initial guess.
rhsnp.ndarray: Right hand side.
depthint: Depth of the V-Cycle.
hfloat: Grid spacing.

Returns:

X (np.ndarray): solution.

operator(x, h)[source]#

The solution operator for the problem

Parameters:

xnp.ndarray: Input.
hfloat: Grid spacing.

Returns:

Output (np.ndarray).

prolongate_parameters(pad_tuple)[source]#

In case of heterogeneous parameters the parameters are prolongated by averaging.

Parameters:

pad_tupletuple: Tuple of tuples, each tuple contains the number of elements to be padded before and after the corresponding axis.

prolongation(x)[source]#

Prolongate x, i.e., refine it by a factor 2. All values are repeated twice. NOTE: if odd number of indices are restricted then prolongated, the boundary information is lost. Should probably be imporved upon in the future.

Parameters:

xnp.ndarray: Input.

Returns:

Output (np.ndarray).

restrict_parameters()[source]#: In case of heterogeneous parameters the parameters are restricted by averaging.

restriction(x)[source]#

Restrict x, i.e., coarsen it by a factor 2. Even and odd indices are averaged. Last index is dropped if odd.

Parameters:

xnp.ndarray: Input.

Returns:

Output (np.ndarray).

update_params(dim=None, mass_coeff=None, diffusion_coeff=None)[source]#

Update parameters of the solver.

Parameters:

dimint, optional: Spatial dimension of the problem.
mass_coefffloat or array, optional: Mass coefficient.
diffusion_coefffloat or array, optional: Diffusion coefficient.

depth#: Depth of V-cycle.

heterogeneous#: Flag controlling whether the material paremeters are heterogeneous.

smoother#: Smoother used in the MG cycle.

smoother_iterations#: Number of smoother iterations per step.