darsia.BeckmannGproxPGHDSolver#

class BeckmannGproxPGHDSolver(grid, weight=None, options={})[source]#

Bases: BeckmannProblem

This contains the implementation of the GproxPDHG algorithm described in “SOLVING LARGE-SCALE OPTIMIZATION PROBLEMS WITH A CONVERGENCE RATE INDEPENDENT OF GRID SIZE”

Methods

`broken_darcy_with_custom_flux_block`(...)	Construct the broken Darcy operator with given flux-flux block.
`cell_weighted_flux`(cell_flux)	Compute the cell-weighted flux.
`compute_dual`(p, gradient_poisson)	Compute the value of the dual functional.
`compute_kantorovich_potential`(...[, tol])	Compute the kantorovich potential from the normal flux
`compute_primal`(flux)	Compute the value of the primal functional $int_{Domain} \| flux \| $
`distance_matrix`(images)	Compute the distance between each iteam of a list.
`eliminate_flux`(jacobian, residual)	Eliminate the flux block from the jacobian and residual.
`eliminate_lagrange_multiplier`(...)	Eliminate the lagrange multiplier from the reduced system.
`exact_linearization`(solution)	Compute the exact linearization of the constrained minimization problem.
`flat_view`(img)	Flatten the image to a vector.
`flux_view`(vector)	Extract the flux from the vector.
`l1_dissipation`(flat_flux)	Compute the l1 dissipation of the solution.
`leray_projection`(p)	Leray projection of a vector fiels
`linear_solve`(matrix, rhs[, ...])	Solve the linear system.
`ndofs`()	Return the total number of degrees of freedom.
`optimality_conditions`(solution, ...)	Evaluate optimality conditions of the constrained minimization problem.
`pressure_view`(vector)	Extract the pressure from the vector.
`rescaled_flux_optimality_conditions`(solution)	Evaluate scaled optimality conditions of the constrained minimization problem.
`setup_amg_options`()	Setup the infrastructure for multilevel solvers.
`setup_poisson_solver`(solver_prefix[, rtol, ...])	Return the Poisson solver.
`solve_beckmann_problem`(flat_mass_diff)	Solve the Beckmann problem using GproxPDHG.
`transport_density`(flat_flux[, weighted, flatten])	Compute the transport density from the solution.
`transport_density_faces`(flat_flux)	Compute the transport density from the solution.

Attributes

`grid`	darsia.Grid: grid
`voxel_size`	np.ndarray: voxel size
`weight`	Optional[darsia.Image]: weight
`options`	dict: options for the solver
`verbose`	bool: verbosity
`callbacks`	list: list of callbacks to be called at each iteration
`norm_mode`	str: mode for computing the norm: l2 or manhattan
`regularization`	float: regularization parameter
`mobility_mode`	str: mode for computing the mobility

compute_dual(p, gradient_poisson)[source]#: Compute the value of the dual functional.

\[\int_\Omega \mathrm{pot}\,(f^+ - f^-) = \int_\Omega \mathrm{pot}\,(-\operatorname{div} \mathrm{poisson}) = \int_\Omega \nabla \mathrm{pot} \cdot \nabla \mathrm{poisson} = \int_\Omega p \cdot \nabla \mathrm{poisson}\]

compute_kantorovich_potential(flat_mass_diff, flux, tol=1e-06)[source]#

Compute the kantorovich potential from the normal flux

Parameters:

flat_mass_diffnp.ndarray: Difference of mass distributions.
fluxnp.ndarray: Flux on the faces.

Returns:

np.ndarray: Kantorovich potential.

compute_primal(flux)[source]#: Compute the value of the primal functional $int_{Domain} | flux | $

leray_projection(p)[source]#

Leray projection of a vector fiels

Parameters:

pnp.ndarray: Pressure.

Returns:

np.ndarray: Divergence free flux.

setup_amg_options()[source]#

Setup the infrastructure for multilevel solvers.

Basic default setup based on jacobi and block Gauss-Seidel smoothers. User-defined options can be passed via the options dictionary, using the key “amg_options”. The options follow the pyamg interface.

setup_poisson_solver(solver_prefix, rtol=1e-06, permeability_faces=None)[source]#

Return the Poisson solver.

Parameters:

permeability_facesnp.ndarray, optional: Permeability faces. Defaults to None.

Returns:

darsia.linalg.KSP: Poisson solver.

solve_beckmann_problem(flat_mass_diff)[source]#

Solve the Beckmann problem using GproxPDHG.

Parameters:

flat_mass_diffnp.ndarray: Difference of mass distributions.

Returns:

tuple: Distance, solution, info.

callbacks#: list: list of callbacks to be called at each iteration

grid#: darsia.Grid: grid

norm_mode#: str: mode for computing the norm: l2 or manhattan

options#: dict: options for the solver

verbose#: bool: verbosity

voxel_size#: np.ndarray: voxel size

weight#: Optional[darsia.Image]: weight