The nonlinear solver solves a nonlinear system of equations. More...

Detailed Description

The nonlinear solver solves a nonlinear system of equations.

The Newton-Raphson method is used to iteratively update the initial guess until the residual becomes zero within specified tolerances.

Each update step is obtained by solving a trust-region subproblem, i.e. a quadratic bound-constrained problem.

The trust-region outer loop is implemented following Nocedal and Wright, section 4.1.

The subproblem implemented here is an alternative linearization of the original subproblem, introduced in

‍Yuan, Ya-xiang. Trust region algorithms for nonlinear equations. Hong Kong Baptist University, Department of Mathematics, 1994.

#include <NewtonWithTrustRegion.h>

Inheritance diagram for NewtonWithTrustRegion:

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Public Member Functions
	NewtonWithTrustRegion (const OptionSet &options)

Public Member Functions inherited from Newton
	Newton (const OptionSet &options)

Result	solve (NonlinearSystem &system, const NonlinearSystem::Sol< false > &x0) override
	Solve the given nonlinear system.

Public Member Functions inherited from NonlinearSolver
	NonlinearSolver (const OptionSet &options)
	Construct a new NonlinearSolver object.

Public Member Functions inherited from Solver
	Solver (const OptionSet &options)
	Construct a new Solver object.

Public Member Functions inherited from NEML2Object
	NEML2Object ()=delete

	NEML2Object (NEML2Object &&)=delete

	NEML2Object (const NEML2Object &)=delete

NEML2Object &	operator= (const NEML2Object &)=delete

NEML2Object &	operator= (NEML2Object &&)=delete

virtual	~NEML2Object ()=default

	NEML2Object (const OptionSet &options)
	Construct a new NEML2Object object.

const OptionSet &	input_options () const

virtual void	setup ()
	Setup this object.

const std::string &	name () const
	A readonly reference to the object's name.

const std::string &	type () const
	A readonly reference to the object's type.

const std::string &	path () const
	A readonly reference to the object's path.

const std::string &	doc () const
	A readonly reference to the object's docstring.

template<typename T = NEML2Object>
const T *	host () const
	Get a readonly pointer to the host.

template<typename T = NEML2Object>
T *	host ()
	Get a writable pointer to the host.

Static Public Member Functions
static OptionSet	expected_options ()

Static Public Member Functions inherited from Newton
static OptionSet	expected_options ()

Static Public Member Functions inherited from NonlinearSolver
static OptionSet	expected_options ()

Static Public Member Functions inherited from Solver
static OptionSet	expected_options ()

Static Public Member Functions inherited from NEML2Object
static OptionSet	expected_options ()

Protected Member Functions
OptionSet	subproblem_options (const OptionSet &) const
	Extract options for the subproblem.

OptionSet	subproblem_solver_options (const OptionSet &) const
	Extract options for the subproblem solver.

void	prepare (const NonlinearSystem &system, const NonlinearSystem::Sol< true > &x) override
	Prepare solver internal data before the iterative update.

void	update (NonlinearSystem &system, NonlinearSystem::Sol< true > &x, const NonlinearSystem::Res< true > &r, const NonlinearSystem::Jac< true > &J) override
	Update trial solution.

NonlinearSystem::Sol< true >	solve_direction (const NonlinearSystem::Res< true > &r, const NonlinearSystem::Jac< true > &J) override
	Find the current update direction.

Scalar	merit_function_reduction (const NonlinearSystem::Res< true > &r, const NonlinearSystem::Jac< true > &J, const NonlinearSystem::Sol< true > &p) const
	Reduction in the merit function.

Protected Member Functions inherited from Newton
virtual bool	converged (size_t itr, const torch::Tensor &nR, const torch::Tensor &nR0) const
	Check for convergence. The current iteration is said to be converged if the residual norm is below the absolute tolerance or or the ratio between the residual norm and the initial residual norm is below the relative tolerance.

virtual void	final_update (NonlinearSystem &system, NonlinearSystem::Sol< true > &x, const NonlinearSystem::Res< true > &r, const NonlinearSystem::Jac< true > &J)
	Do a final update to track AD function graph.

Protected Attributes
TrustRegionSubProblem	_subproblem
	Trust-region subproblem.

Newton	_subproblem_solver
	Solver used to solver the trust-region subproblem.

Scalar	_delta
	The trust region radius.

Real	_delta_0
	Initial size of the trust region.

Real	_delta_max
	Maximum size of the trust region.

Real	_reduce_criteria
	Criteria for reducing the trust region.

Real	_expand_criteria
	Criteria for expanding the trust region.

Real	_reduce_factor
	Cutback factor if we do reduce the trust region.

Real	_expand_factor
	Expansion factor if we do increase the trust region.

Real	_accept_criteria
	Acceptance criteria for a step.

Additional Inherited Members
Public Types inherited from NonlinearSolver
enum class	RetCode { SUCCESS = 0 , MAXITER = 1 , FAILURE = 2 }

Public Attributes inherited from NonlinearSolver
Real	atol
	Absolute tolerance.

Real	rtol
	Relative tolerance.

unsigned int	miters
	Maximum number of iterations.

Public Attributes inherited from Solver
const bool	verbose
	Whether to print additional (debugging) information during the solve.

Constructor & Destructor Documentation

◆ NewtonWithTrustRegion()

NewtonWithTrustRegion ( const OptionSet & options )

Member Function Documentation

◆ expected_options()

OptionSet expected_options ( )

static

◆ merit_function_reduction()

Scalar merit_function_reduction	(	const NonlinearSystem::Res< true > &	r,
		const NonlinearSystem::Jac< true > &	J,
		const NonlinearSystem::Sol< true > &	p ) const

protected

Reduction in the merit function.

◆ prepare()

void prepare	(	const NonlinearSystem &	,
		const NonlinearSystem::Sol< true > &	)

overrideprotectedvirtual

Prepare solver internal data before the iterative update.

Reimplemented from Newton.

◆ solve_direction()

NonlinearSystem::Sol< true > solve_direction	(	const NonlinearSystem::Res< true > &	r,
		const NonlinearSystem::Jac< true > &	J )

overrideprotectedvirtual

Find the current update direction.

Reimplemented from Newton.

◆ subproblem_options()

OptionSet subproblem_options ( const OptionSet & ) const

protected

Extract options for the subproblem.

◆ subproblem_solver_options()

OptionSet subproblem_solver_options ( const OptionSet & options ) const

protected

Extract options for the subproblem solver.

◆ update()

void update	(	NonlinearSystem &	system,
		NonlinearSystem::Sol< true > &	x,
		const NonlinearSystem::Res< true > &	r,
		const NonlinearSystem::Jac< true > &	J )

overrideprotectedvirtual

Update trial solution.

Reimplemented from Newton.

Member Data Documentation

◆ _accept_criteria

Real _accept_criteria

protected

Acceptance criteria for a step.

◆ _delta

Scalar _delta

protected

The trust region radius.

◆ _delta_0

Real _delta_0

protected

Initial size of the trust region.

◆ _delta_max

Real _delta_max

protected

Maximum size of the trust region.

◆ _expand_criteria

Real _expand_criteria

protected

Criteria for expanding the trust region.

◆ _expand_factor

Real _expand_factor

protected

Expansion factor if we do increase the trust region.

◆ _reduce_criteria

Real _reduce_criteria

protected

Criteria for reducing the trust region.

◆ _reduce_factor

Real _reduce_factor

protected

Cutback factor if we do reduce the trust region.

◆ _subproblem

TrustRegionSubProblem _subproblem

protected

Trust-region subproblem.

◆ _subproblem_solver

Newton _subproblem_solver

protected

Solver used to solver the trust-region subproblem.

Detailed Description

Public Member Functions

Static Public Member Functions

Protected Member Functions

Protected Attributes

Additional Inherited Members

Constructor & Destructor Documentation

◆ NewtonWithTrustRegion()

Member Function Documentation

◆ expected_options()

◆ merit_function_reduction()

◆ prepare()

◆ solve_direction()

◆ subproblem_options()

◆ subproblem_solver_options()

◆ update()

Member Data Documentation

◆ _accept_criteria

◆ _delta

◆ _delta_0

◆ _delta_max

◆ _expand_criteria

◆ _expand_factor

◆ _reduce_criteria

◆ _reduce_factor

◆ _subproblem

◆ _subproblem_solver