wmtk::components::topological_offset::OffsetEnergy< DIM > Class Template Reference

The offset term of the smoothing objective: w * (Phi(x) - c)^2. More...

#include <OffsetPotential.hpp>

Inheritance diagram for wmtk::components::topological_offset::OffsetEnergy< DIM >:

Public Types
using	VecD = Eigen::Matrix< double, DIM, 1 >
using	MatD = Eigen::Matrix< double, DIM, DIM >

Public Member Functions
	OffsetEnergy (const std::shared_ptr< const OffsetPotential< DIM > > &potential, double weight=1., bool gauss_newton=true, bool distance_residual=false)
double	value (const TVector &x) override
void	gradient (const TVector &x, TVector &gradv) override
void	hessian (const TVector &x, THessian &hessian) override
void	hessian (const TVector &x, MatrixXd &hessian) override
void	solution_changed (const TVector &new_x) override

Private Member Functions
bool	root_distance (const VecD &p, double &s, VecD &n) const
void	residual (const VecD &p, double &r, VecD &dr) const

Private Attributes
std::shared_ptr< const OffsetPotential< DIM > >	m_potential
double	m_weight
bool	m_gauss_newton
bool	m_distance_residual
double	m_last_root = 0.
	warm start for root_distance(), see there
bool	m_cache_valid = false
VecD	m_cache_p
double	m_cache_r = 0.
VecD	m_cache_dr

Detailed Description

template<int DIM>
class wmtk::components::topological_offset::OffsetEnergy< DIM >

The offset term of the smoothing objective: w * (Phi(x) - c)^2.

A polysolve::nonlinear::Problem in the shape of ExactDistanceEnergy2D/3D, so the shared smoother composes it into its EnergySum beside AMIPS with no special case, which is what lets a front vertex take the same path as every other vertex.

The residual form rather than Phi itself: Phi is a barrier, so minimising it would drive the vertex to infinity and maximising it into the complex, while the squared residual has its minimum exactly on the level set, where the front belongs.

Value, gradient and Hessian all follow from Phi, grad Phi and hess Phi by the chain rule:

E     = w (Phi - c)^2
grad  = 2 w (Phi - c) grad Phi
hess  = 2 w [ grad Phi grad Phi^T + (Phi - c) hess Phi ]

The second Hessian term changes sign with the residual and can make H indefinite far from the level set; gauss_newton drops it, leaving the always-PSD outer product. On by default: the dropped term vanishes at the solution, so it costs nothing at convergence and buys a descent direction everywhere.

Constructor & Destructor Documentation

OffsetEnergy()

template<int DIM>

wmtk::components::topological_offset::OffsetEnergy< DIM >::OffsetEnergy	(	const std::shared_ptr< const OffsetPotential< DIM > > &	potential,
		double	weight = 1.,
		bool	gauss_newton = true,
		bool	distance_residual = false
	)

distance_residual charges the signed distance from x to the level set along the field's normal, over delta – what (d - delta)/delta already is for the Euclidean field – instead of the value ratio (Phi - c)/c, which for the smooth field is a barrier value and not a length, and pulls far harder where two fronts are pressed together. Same level set and same root either way; only the charge changes. 3D keeps the value ratio (false).

Member Function Documentation

residual()

template<int DIM>

void wmtk::components::topological_offset::OffsetEnergy< DIM >::residual ( const VecD &  p, double &  r, VecD &  dr  ) const

private

r and its gradient under either residual (see the constructor). The last point's answer is cached: value, gradient and Hessian are asked at the same x in one iteration.

root_distance()

template<int DIM>

bool wmtk::components::topological_offset::OffsetEnergy< DIM >::root_distance ( const VecD &  p, double &  s, VecD &  n  ) const

private

The signed distance s from p to the level set along the field's normal n at p (n points toward the input; s > 0: the level set lies outward of p). Safeguarded Newton. false when no root brackets within 2 dhat, e.g. outside the support.

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

• /home/runner/work/wildmeshing-toolkit/wildmeshing-toolkit/components/topological_offset/wmtk/components/topological_offset/OffsetPotential.hpp
• /home/runner/work/wildmeshing-toolkit/wildmeshing-toolkit/components/topological_offset/wmtk/components/topological_offset/OffsetPotential.cpp