darsia.BeckmannNewtonSolver

class BeckmannNewtonSolver(grid, weight=None, options={})

Bases: BeckmannProblem

Class to determine the L1 EMD/Wasserstein distance solved with Newton’s method.

Implements the darsia.BeckmannProblem interface by specifying the method darsia.BeckmannProblem.solve_beckmann_problem().

Solves the Beckmann problem in mixed form with a Newton-like method. The linearization of the optimality conditions is relaxed with a cut-off parameter L for the mobility. In addition, to circumvent singular Jacobians, merely a fixed-point type linearization is employed. This results in a method that is in between a pure Newton method and a fixed-point iteration.

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_jacobian(solution)	Compute the Jacobian of the optimality conditions.
compute_residual(solution, beckmann_problem_rhs)	Compute the residual of the solution - the optimality conditions.
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.
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.
solve_beckmann_problem(mass_diff)	Solve the Beckmann problem using Newton's method.
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

convergence_criteria	"Convergence criteria for the Newton iteration.
grid	darsia.Grid: grid
voxel_size	np.ndarray: voxel size
options	dict: options for the solver
regularization	float: regularization parameter
verbose	bool: verbosity
mobility_mode	str: mode for computing the mobility
weight	Weight defined on cells
callbacks	list: list of callbacks to be called during the optimization

compute_jacobian(solution)

Compute the Jacobian of the optimality conditions.

Parameters:

solutionnp.ndarray

Solution.

Returns:

sps.linalg.splu

LU factorization of the jacobian.

compute_residual(solution, beckmann_problem_rhs)

Compute the residual of the solution - the optimality conditions.

Parameters:

solutionnp.ndarray

Solution.

Returns:

np.ndarray

Residual.

solve_beckmann_problem(mass_diff)

Solve the Beckmann problem using Newton’s method.

Parameters:

mass_diffnp.ndarray

Difference of mass distributions.

Returns:

tuple

Distance, solution, info.

convergence_criteria

“Convergence criteria for the Newton iteration.