Computes semi-discrete optimal transport maps. More...

#include <exploragram/optimal_transport/optimal_transport.h>

Inheritance diagram for GEO::OptimalTransportMap:

Classes
class	Callback
	Base class for the callbacks executed for each intersection between a Laguerre cell and a simplex of the background mesh. More...

Public Member Functions
	OptimalTransportMap (index_t dimension, Mesh *mesh, const std::string &delaunay="default", bool BRIO=false)
	OptimalTransportMap constructor. More...

virtual	~OptimalTransportMap ()
	OptimalTransportMap destructor.

index_t	dimension () const
	Gets the dimension. More...

Mesh &	mesh ()
	Gets the mesh. More...

void	set_Newton (bool x)
	Sets whether Newton algorithm should be used. More...

void	set_points (index_t nb_points, const double *points, index_t stride=0)
	Sets the points that define the target distribution. More...

void	set_air_particles (index_t nb_air_particles, const double *air_particles, index_t stride, double air_fraction)
	Sets the air particles that define the volume occupied by the free space. More...

double	air_fraction () const
	Gets the air fraction. More...

index_t	nb_air_particles () const
	Gets the number of air particles. More...

void	set_nu (index_t i, double nu)
	Sets the desired mass at one of the Diracs. More...

void	set_Laguerre_centroids (double *x)
	Specifies a user vector where the centroids of the Laguerre cells will be stored after computing transport. More...

void	set_epsilon (double eps)
	Sets the maximum error. More...

void	set_linsolve_epsilon (double eps)
	Sets the tolerance for linear solve. More...

void	set_linsolve_maxiter (index_t maxiter)
	Sets the maximum number of iterations for linear solve. More...

void	set_linesearch_maxiter (index_t maxiter)
	Sets the maximum number of line search iterations. More...

void	set_linesearch_init_iter (index_t init_iter)
	Sets the number of steplength reductions to be done at the first iteration. More...

void	set_regularization (double eps_reg)
	Sets the weight of the regularization term. More...

void	set_linear_solver (OTLinearSolver solver)
	Specifies whether a direct solver should be used. More...

void	optimize (index_t max_iterations)
	Computes the weights that realize the optimal transport map between the source mesh and the target pointset. More...

void	set_verbose (bool x)
	Enable/disable messages during optimization. More...

void	optimize_full_Newton (index_t max_iterations, index_t n=0)
	Computes the weights that realize the optimal transport map between the source mesh and the target pointset. More...

void	optimize_level (index_t b, index_t e, index_t max_iterations)
	Optimizes one level of the multilevel algorithm. More...

void	optimize_levels (const vector< index_t > &levels, index_t max_iterations)
	Multi-level optimization. More...

index_t	nb_points () const
	Gets the number of points. More...

const double *	point_ptr (index_t i) const
	Gets a point. More...

double	weight (index_t i) const
	Gets weight of a point. More...

void	set_weight (index_t i, double val)
	Sets a weight of a point. More...

double	potential (index_t i) const
	Gets the d+1-th coordinate of the embedding for a point. More...

RestrictedVoronoiDiagram *	RVD ()
	Gets the restricted Voronoi diagram. More...

void	set_save_RVD_iter (bool x, bool show_RVD_seed=false, bool last_iter_only=false)
	Sets whether the restricted Voronoi diagram at each iteration should be saved. More...

virtual void	get_RVD (Mesh &M)=0
	Computes a mesh with the restricted Voronoi diagram. More...

virtual void	compute_Laguerre_centroids (double *centroids)=0
	Computes the centroids of the Laguerre cells. More...

void	update_sparsity_pattern ()
	Updates the sparsity pattern of the Hessian right after a new Laguerre diagram was computed.

void	new_linear_system (index_t n, double *x)
	Starts a new linear system. More...

void	add_ij_coefficient (index_t i, index_t j, double a)
	Adds a coefficient to the matrix of the system. More...

void	add_i_right_hand_side (index_t i, double a)
	Adds a coefficient to the right hand side. More...

void	solve_linear_system ()
	Solves a linear system. More...

void	set_initial_weight (index_t i, double w)
	Sets the initial value of the weight associated with one of the points. More...

double	total_mass () const
	Gets the total mass of the domain. More...

void	compute_P1_Laplacian (const double weights, NLMatrix Laplacian, double measures)
	Computes the P1 Laplacian of the Laguerre cells. More...

Static Public Member Functions
static void	funcgrad_CB (index_t n, double x, double &f, double g)
	Callback for the numerical solver. More...

static void	newiteration_CB (index_t n, const double x, double f, const double g, double gnorm)
	Callback for the numerical solver. More...

Protected Member Functions
double	nu (index_t p) const
	Gets the mass of the Dirac associated with point p. More...

virtual void	newiteration ()
	Callback for the numerical solver.

void	save_RVD (index_t id)
	Saves the RVD at each iteration if specified on command line (just for debugging/ explaining the algorithm). More...

void	funcgrad (index_t n, double w, double &f, double g)
	Computes the objective function and its gradient. More...

virtual void	call_callback_on_RVD ()=0
	Calls the callback for each intersection between a Laguerre cell and a simplex of the background mesh.

void	eval_func_grad_Hessian (index_t n, const double w, double &f, double g)
	Computes the objective function, its gradient and its Hessian. More...

double	gradient_threshold (index_t n) const
	Computes the stopping criterion of the solver. More...

Protected Attributes
index_t	dimension_

index_t	dimp1_
	dimension_ + 1

Mesh *	mesh_

Delaunay_var	delaunay_

RestrictedVoronoiDiagram_var	RVD_

vector< double >	points_dimp1_

vector< double >	weights_

double	total_mass_

double	constant_nu_
	Value of one of the Diracs if cte.

vector< double >	nu_
	Value of all the Diracs.

double	epsilon_
	Acceptable relative deviation for the measure of a cell.

index_t	current_call_iter_

Callback *	callback_

std::string	last_stats_

bool	pretty_log_

index_t	level_

bool	save_RVD_iter_

bool	save_RVD_last_iter_

bool	show_RVD_seed_

index_t	current_iter_

bool	newton_

bool	verbose_

double	epsilon_regularization_
	Add a regularization term to remove translational degree of freedom for the weights.

index_t	nbZ_
	Number of empty cells in last iteration.

double	g_norm_
	Norm of the gradient in last iteration.

double	measure_of_smallest_cell_
	Measure of the smallest Laguerre cell.

bool	w_did_not_change_
	True if w did not change, thus there is no need to recompute the power diagram.

double *	Laguerre_centroids_
	If user-specified, then Laguerre centroids are output here.

double	linsolve_epsilon_
	maximum value of \( \\| Ax - b \\| / \\| b \\| \)

index_t	linsolve_maxiter_
	maximum number of iterations for linear solve

index_t	linesearch_maxiter_
	maximum number of steplength divisions

index_t	linesearch_init_iter_
	starting number of steplength divisions

OTLinearSolver	linear_solver_
	one of OT_PRECG, OT_SUPERLU, OT_CHOLMOD.

const double *	air_particles_
	if set, pointer to the air particles.

index_t	nb_air_particles_
	if non-zero, number of air particles.

index_t	air_particles_stride_
	Number of doubles between two consecutive air particles in air_particles_.

double	air_fraction_
	The fraction of the total mass occupied by air.

bool	clip_by_balls_
	Enabled if air fraction is specified without any air particles.

bool	user_H_g_
	True if class is just used by user to compute Hessian and gradient instead of doing full computation.

NLMatrix	user_H_
	User-defined Hessian matrix.

Static Protected Attributes
static OptimalTransportMap *	instance_

Detailed Description

Computes semi-discrete optimal transport maps.

Computes an optimal transport map between two distributions. The first distribution is represented by a simplicial mesh. The second distribution is a sum of Diracs. The algorithm is described in the following references:

3D algorithm: http://arxiv.org/abs/1409.1279
Earlier 2D version by Quentin M\'erigot: Q. Merigot. A multiscale approach to optimal transport. Computer Graphics Forum 30 (5) 1583–1592, 2011 (Proc SGP 2011).
Earlier article on OT and power diagrams: F. Aurenhammer, F. Hoffmann, and B. Aronov. Minkowski-type theorems and least-squares clustering. Algorithmica, 20:61-76, 1998.

Definition at line 91 of file optimal_transport.h.

Constructor & Destructor Documentation

◆ OptimalTransportMap()

GEO::OptimalTransportMap::OptimalTransportMap	(	index_t	dimension,
		Mesh *	mesh,
		const std::string &	delaunay = `"default"`,
		bool	BRIO = `false`
	)

OptimalTransportMap constructor.

Parameters

[in]	mesh	the source distribution, represented as a nd mesh.
[in]	delaunay	factory name of the Delaunay triangulation.
[in]	BRIO	true if vertices are already ordered using BRIO

Member Function Documentation

◆ add_i_right_hand_side()

void GEO::OptimalTransportMap::add_i_right_hand_side	(	index_t	i,
		double	a
	)

inline

Adds a coefficient to the right hand side.

Parameters

[in]	i	the index of the coefficient
[in]	a	the value to be added to the coefficient

Definition at line 493 of file optimal_transport.h.

◆ add_ij_coefficient()

void GEO::OptimalTransportMap::add_ij_coefficient	(	index_t	i,
		index_t	j,
		double	a
	)

inline

Adds a coefficient to the matrix of the system.

Parameters

[in]	i,j	the indices of the coefficient
[in]	a	the value to be added to the coefficient

Definition at line 477 of file optimal_transport.h.

◆ air_fraction()

double GEO::OptimalTransportMap::air_fraction ( ) const

inline

Gets the air fraction.

Returns: the air fraction previously specified by set_air_particles()

Definition at line 181 of file optimal_transport.h.

◆ compute_Laguerre_centroids()

virtual void GEO::OptimalTransportMap::compute_Laguerre_centroids ( double * centroids )

pure virtual

Computes the centroids of the Laguerre cells.

Parameters

[out] centroids a pointer to the dimension()*nb_points coordinates of the centroids.

Implemented in GEO::OptimalTransportMapOnSurface, GEO::OptimalTransportMap3d, and GEO::OptimalTransportMap2d.

◆ compute_P1_Laplacian()

void GEO::OptimalTransportMap::compute_P1_Laplacian	(	const double *	weights,
		NLMatrix	Laplacian,
		double *	measures
	)

Computes the P1 Laplacian of the Laguerre cells.

Parameters

[in]	Omega	the domain, either a surfacic or a volumetric mesh.
[in]	weights	the weights of the Laguerre diagram.
[out]	Laplacian	P1 Laplacian of the Laguerre diagram or nullptr if not needed.
[out]	measures	optional measures the measures of all Laguerre cells, or nullptr if not needed.

◆ dimension()

index_t GEO::OptimalTransportMap::dimension ( ) const

inline

Gets the dimension.

Returns: 2 for 2d, 3 for 3d.

Definition at line 115 of file optimal_transport.h.

◆ eval_func_grad_Hessian()

void GEO::OptimalTransportMap::eval_func_grad_Hessian	(	index_t	n,
		const double *	w,
		double &	f,
		double *	g
	)

protected

Computes the objective function, its gradient and its Hessian.

Gradient and Hessian are used to solve a Newton step H p = -g

Parameters

[in]	n	number of variables
[in]	w	current value of the variables
[out]	f	current value of the objective function
[out]	g	gradient of the objective function

◆ funcgrad()

void GEO::OptimalTransportMap::funcgrad	(	index_t	n,
		double *	w,
		double &	f,
		double *	g
	)

protected

Computes the objective function and its gradient.

Parameters

[in]	n	number of variables
[in]	w	current value of the variables
[out]	f	current value of the objective function
[out]	g	gradient of the objective function

◆ funcgrad_CB()

static void GEO::OptimalTransportMap::funcgrad_CB	(	index_t	n,
		double *	x,
		double &	f,
		double *	g
	)

static

Callback for the numerical solver.

Evaluates the objective function and its gradient.

Parameters

[in]	n	number of variables
[in]	x	current value of the variables
[out]	f	current value of the objective function
[out]	g	gradient of the objective function

◆ get_RVD()

virtual void GEO::OptimalTransportMap::get_RVD ( Mesh & M )

pure virtual

Computes a mesh with the restricted Voronoi diagram.

Parameters

[out] M a reference to the computed restricted Voronoi diagram.

Implemented in GEO::OptimalTransportMapOnSurface, GEO::OptimalTransportMap3d, and GEO::OptimalTransportMap2d.

◆ gradient_threshold()

double GEO::OptimalTransportMap::gradient_threshold ( index_t n ) const

inlineprotected

Computes the stopping criterion of the solver.

The stopping criterion is determined from the user-specified epsilon, number of samples and target measure of a cell (constant_nu_).

Parameters

n	number of samples

Returns: the gradient threshold

See also: set_epsilon()

Definition at line 602 of file optimal_transport.h.

◆ mesh()

Mesh& GEO::OptimalTransportMap::mesh ( )

inline

Gets the mesh.

Returns: a reference to the mesh

Definition at line 123 of file optimal_transport.h.

◆ nb_air_particles()

index_t GEO::OptimalTransportMap::nb_air_particles ( ) const

inline

Gets the number of air particles.

Returns: the air fraction previously specified by set_air_particles()

Definition at line 189 of file optimal_transport.h.

◆ nb_points()

index_t GEO::OptimalTransportMap::nb_points ( ) const

inline

Gets the number of points.

Returns: The number of points, that was previously defined by set_points()

Definition at line 345 of file optimal_transport.h.

◆ new_linear_system()

void GEO::OptimalTransportMap::new_linear_system	(	index_t	n,
		double *	x
	)

Starts a new linear system.

Parameters

[in]	n	the dimension of the system
[in]	x	pointer to a contiguous array of `n` doubles, where the solution will be stored.

◆ newiteration_CB()

static void GEO::OptimalTransportMap::newiteration_CB	(	index_t	n,
		const double *	x,
		double	f,
		const double *	g,
		double	gnorm
	)

static

Callback for the numerical solver.

Parameters

[in]	n	number of variables
[in]	x	current value of the variables
[in]	f	current value of the objective function
[in]	g	gradient of the objective function
[in]	gnorm	norm of the gradient of the objective function

◆ nu()

double GEO::OptimalTransportMap::nu ( index_t p ) const

inlineprotected

Gets the mass of the Dirac associated with point p.

Returns: the desired mass at point p.

Definition at line 545 of file optimal_transport.h.

◆ optimize()

void GEO::OptimalTransportMap::optimize ( index_t max_iterations )

Computes the weights that realize the optimal transport map between the source mesh and the target pointset.

Parameters

[in] max_iterations maximum number of solver iterations.

◆ optimize_full_Newton()

void GEO::OptimalTransportMap::optimize_full_Newton	(	index_t	max_iterations,
		index_t	n = `0`
	)

Computes the weights that realize the optimal transport map between the source mesh and the target pointset.

The algorithm is described in http://arxiv.org/abs/1603.05579 Kitawaga, Merigot, Thibert, A Newton Algorithm for semi-discrete OT.

Parameters

[in]	max_iterations	maximum number of solver iterations.
[in]	n	number of weights to optimize, used in hierarchical mode. If zero, optimizes all the weights.

◆ optimize_level()

void GEO::OptimalTransportMap::optimize_level	(	index_t	b,
		index_t	e,
		index_t	max_iterations
	)

Optimizes one level of the multilevel algorithm.

The function supposes that the sequence [0,b) has been previously optimized. It is used to initialize the sequence [b,e). The whole sequence [0,e) is then optimized.

Parameters

[in]	b	index fo the first point in the level
[in]	e	one position past the last index of the level
[in]	max_iterations	maximum number of iterations

◆ optimize_levels()

void GEO::OptimalTransportMap::optimize_levels	(	const vector< index_t > &	levels,
		index_t	max_iterations
	)

Multi-level optimization.

The points specified by set_points() need to have a hierarchical structure. They can be constructed by compute_hierarchical_sampling().

Parameters

[in]	levels	sample indices that correspond to level l are in the range levels[l] (included) ... levels[l+1] (excluded)
[in]	max_iterations	maximum number of iterations

See also: compute_hierarchical_sampling()

◆ point_ptr()

const double* GEO::OptimalTransportMap::point_ptr ( index_t i ) const

inline

Gets a point.

Parameters

[in] i index of the point

Returns: a const pointer to the coordinates of the (dimension()+1)d point i

Definition at line 355 of file optimal_transport.h.

◆ potential()

double GEO::OptimalTransportMap::potential ( index_t i ) const

inline

Gets the d+1-th coordinate of the embedding for a point.

Parameters

[in] i index of the point

Returns: the d+1-th coordinate that was computed for point i

Definition at line 383 of file optimal_transport.h.

◆ RVD()

RestrictedVoronoiDiagram* GEO::OptimalTransportMap::RVD ( )

inline

Gets the restricted Voronoi diagram.

Returns: a pointer to the restricted Voronoi diagram

Definition at line 415 of file optimal_transport.h.

◆ save_RVD()

void GEO::OptimalTransportMap::save_RVD ( index_t id )

protected

Saves the RVD at each iteration if specified on command line (just for debugging/ explaining the algorithm).

Parameters

[in] id index to be used for the file, that will be named RVD_id.meshb

◆ set_air_particles()

void GEO::OptimalTransportMap::set_air_particles	(	index_t	nb_air_particles,
		const double *	air_particles,
		index_t	stride,
		double	air_fraction
	)

inline

Sets the air particles that define the volume occupied by the free space.

If air particles are used, then set_air_particles() needs to be called before set_points().

Parameters

[in]	nb_air_particles	number of air particles.
[in]	air_particles	a pointer to the array of doubles with the coordinates of the air particles.
[in]	stride	number of doubles between two consecutive air particles in the array, or 0 if tightly packed.
[in]	air_fraction	the fraction of the total mass occupied by air.

Definition at line 164 of file optimal_transport.h.

◆ set_epsilon()

void GEO::OptimalTransportMap::set_epsilon ( double eps )

inline

Sets the maximum error.

Parameters

eps	acceptable relative deviation for the measure of a Voronoi cell.

Definition at line 218 of file optimal_transport.h.

◆ set_initial_weight()

void GEO::OptimalTransportMap::set_initial_weight	(	index_t	i,
		double	w
	)

inline

Sets the initial value of the weight associated with one of the points.

Parameters

[in]	i	index of the point, in 0..nb_points-1, where np_points corresponds to the parameter of set_points.
[in]	w	the value of the weight.

Definition at line 513 of file optimal_transport.h.

◆ set_Laguerre_centroids()

void GEO::OptimalTransportMap::set_Laguerre_centroids ( double * x )

inline

Specifies a user vector where the centroids of the Laguerre cells will be stored after computing transport.

Parameters

[out] x a vector of nb points * dimension() doubles.

Definition at line 209 of file optimal_transport.h.

◆ set_linear_solver()

void GEO::OptimalTransportMap::set_linear_solver ( OTLinearSolver solver )

inline

Specifies whether a direct solver should be used.

Parameters

[in] solver one of OT_PRECG (default), OT_SUPERLU, OT_CHOLMOD.

The direct solvers (OT_SUPERLU, OT_CHOLMOD) are recommended only for surfacic data, since the sparse factors become not so sparse when volumetric meshes are considered.

Definition at line 281 of file optimal_transport.h.

◆ set_linesearch_init_iter()

void GEO::OptimalTransportMap::set_linesearch_init_iter ( index_t init_iter )

inline

Sets the number of steplength reductions to be done at the first iteration.

Parameters

[in] init_iter the number of steplength reductions to be apllied at the first iteration. If left 0, start with Newton step at each iteration, else do tentative steplength prediction.

Definition at line 258 of file optimal_transport.h.

◆ set_linesearch_maxiter()

void GEO::OptimalTransportMap::set_linesearch_maxiter ( index_t maxiter )

inline

Sets the maximum number of line search iterations.

Parameters

[in] maxiter the maximum number of step length reduction for line search.

Definition at line 246 of file optimal_transport.h.

◆ set_linsolve_epsilon()

void GEO::OptimalTransportMap::set_linsolve_epsilon ( double eps )

inline

Sets the tolerance for linear solve.

Parameters

[in] eps the maximum value of \( \| Ax - b \| / \| b \| \)b

Definition at line 228 of file optimal_transport.h.

◆ set_linsolve_maxiter()

void GEO::OptimalTransportMap::set_linsolve_maxiter ( index_t maxiter )

inline

Sets the maximum number of iterations for linear solve.

Parameters

[in] maxiter the maximum number of iterations.

Definition at line 237 of file optimal_transport.h.

◆ set_Newton()

void GEO::OptimalTransportMap::set_Newton ( bool x )

inline

Sets whether Newton algorithm should be used.

It is (for now) incompatible with multilevel.

Parameters

[in] x if set, Newton algorithm is used instead of BFGS.

Definition at line 133 of file optimal_transport.h.

◆ set_nu()

void GEO::OptimalTransportMap::set_nu	(	index_t	i,
		double	nu
	)

Sets the desired mass at one of the Diracs.

If unspecified, then default value is total mass divided by number of point. Note that the sum of all specified masses should match the total mass.

Parameters

[in]	i	the index of the dirac, in 0 .. nb_points-1
[in]	nu	the desired mass at point i

◆ set_points()

void GEO::OptimalTransportMap::set_points	(	index_t	nb_points,
		const double *	points,
		index_t	stride = `0`
	)

Sets the points that define the target distribution.

If air particles are used, then set_air_particles() needs to be called before set_points().

Parameters

[in]	nb_points	number of points in the target distribution
[in]	points	an array of size nb_points * dimension() with the coordinates of the Diracs centers in the target distribution.
[in]	stride	number of doubles between two consecutive points. If 0 (default), then point coordinates are considered to be packed.

◆ set_regularization()

void GEO::OptimalTransportMap::set_regularization ( double eps_reg )

inline

Sets the weight of the regularization term.

The regularization term (norm of the weight vector) cancels the translational degree of freedom of the weights.

Parameters

[in] eps_reg the weight of the regularization term. Use 0.0 for no regularization.

Definition at line 269 of file optimal_transport.h.

◆ set_save_RVD_iter()

void GEO::OptimalTransportMap::set_save_RVD_iter	(	bool	x,
		bool	show_RVD_seed = `false`,
		bool	last_iter_only = `false`
	)

inline

Sets whether the restricted Voronoi diagram at each iteration should be saved.

If flag is set, then each iteration is saved in file "RVD_nnn.geogram".

Parameters

[in]	x	true if each iteration should be saved, false otherwise.
[in]	show_RVD_seed	if true, the seed associated with each restricted Voronoi cell is connected to it
[in]	last_iter_only	if true, only the last iteration is saved

Definition at line 431 of file optimal_transport.h.

◆ set_verbose()

void GEO::OptimalTransportMap::set_verbose ( bool x )

inline

Enable/disable messages during optimization.

Parameters

[in] x true if messages should be displayed, false otherwise.

Definition at line 298 of file optimal_transport.h.

◆ set_weight()

void GEO::OptimalTransportMap::set_weight	(	index_t	i,
		double	val
	)

inline

Sets a weight of a point.

Parameters

[in]	i	index of the point
[in]	val	new value of the weight

Definition at line 374 of file optimal_transport.h.

◆ solve_linear_system()

void GEO::OptimalTransportMap::solve_linear_system ( )

Solves a linear system.

The solution is stored in the vector that was previously specified to new_linear_system().

◆ total_mass()

double GEO::OptimalTransportMap::total_mass ( ) const

inline

Gets the total mass of the domain.

Returns: the total mass.

Definition at line 521 of file optimal_transport.h.

◆ weight()

double GEO::OptimalTransportMap::weight ( index_t i ) const

inline

Gets weight of a point.

Parameters

[in] i index of the point

Returns: the weight that was computed for point i

Definition at line 365 of file optimal_transport.h.

The documentation for this class was generated from the following file:

exploragram/optimal_transport/optimal_transport.h

Classes

Public Member Functions

Static Public Member Functions

Protected Member Functions

Protected Attributes

Static Protected Attributes

Detailed Description

Constructor & Destructor Documentation

◆ OptimalTransportMap()

Member Function Documentation

◆ add_i_right_hand_side()

◆ add_ij_coefficient()

◆ air_fraction()

◆ compute_Laguerre_centroids()

◆ compute_P1_Laplacian()

◆ dimension()

◆ eval_func_grad_Hessian()

◆ funcgrad()

◆ funcgrad_CB()

◆ get_RVD()

◆ gradient_threshold()

◆ mesh()

◆ nb_air_particles()

◆ nb_points()

◆ new_linear_system()

◆ newiteration_CB()

◆ nu()

◆ optimize()

◆ optimize_full_Newton()

◆ optimize_level()

◆ optimize_levels()

◆ point_ptr()

◆ potential()

◆ RVD()

◆ save_RVD()

◆ set_air_particles()

◆ set_epsilon()

◆ set_initial_weight()

◆ set_Laguerre_centroids()

◆ set_linear_solver()

◆ set_linesearch_init_iter()

◆ set_linesearch_maxiter()

◆ set_linsolve_epsilon()

◆ set_linsolve_maxiter()

◆ set_Newton()

◆ set_nu()

◆ set_points()

◆ set_regularization()

◆ set_save_RVD_iter()

◆ set_verbose()

◆ set_weight()

◆ solve_linear_system()

◆ total_mass()

◆ weight()