#include <deal.II/lac/solver_fire.h>

Inheritance diagram for SolverFIRE< VectorType >:

Classes
struct	AdditionalData

Public Member Functions
	SolverFIRE (SolverControl &solver_control, VectorMemory< VectorType > &vector_memory, const AdditionalData &data=AdditionalData())

	SolverFIRE (SolverControl &solver_control, const AdditionalData &data)

virtual	~SolverFIRE ()

template<typename PreconditionerType = DiagonalMatrix<VectorType>>
void	solve (const std::function< double(VectorType &, const VectorType &)> &compute, VectorType &x, const PreconditionerType &inverse_mass_matrix)

template<typename MatrixType , typename PreconditionerType >
void	solve (const MatrixType &A, VectorType &x, const VectorType &b, const PreconditionerType &preconditioner)

Public Member Functions inherited from Solver< VectorType >
	Solver (SolverControl &solver_control, VectorMemory< VectorType > &vector_memory)

	Solver (SolverControl &solver_control)

boost::signals2::connection	connect (const std::function< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType &current_iterate)> &slot)

Public Member Functions inherited from Subscriptor
	Subscriptor ()

	Subscriptor (const Subscriptor &)

	Subscriptor (Subscriptor &&) noexcept

virtual	~Subscriptor ()

Subscriptor &	operator= (const Subscriptor &)

Subscriptor &	operator= (Subscriptor &&) noexcept

void	subscribe (const char *identifier=nullptr) const

void	unsubscribe (const char *identifier=nullptr) const

unsigned int	n_subscriptions () const

void	list_subscribers () const

template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

Protected Member Functions
virtual void	print_vectors (const unsigned int, const VectorType &x, const VectorType &v, const VectorType &g) const

Protected Attributes
const AdditionalData	additional_data

Protected Attributes inherited from Solver< VectorType >
GrowingVectorMemory< VectorType >	static_vector_memory

VectorMemory< VectorType > &	memory

boost::signals2::signal< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType &current_iterate), StateCombiner >	iteration_status

Additional Inherited Members
Public Types inherited from Solver< VectorType >
typedef VectorType	vector_type

Static Public Member Functions inherited from Subscriptor
static ::ExceptionBase &	ExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBase &	ExcNoSubscriber (std::string arg1, std::string arg2)

Detailed Description

template<typename VectorType = Vector<double>>
class SolverFIRE< VectorType >

FIRE (Fast Inertial Relaxation Engine) for minimization of (potentially non-linear) objective function \(E(\mathbf x)\), \(\mathbf x\) is a vector of \(n\) variables ( \(n\) is the number of variables of the objective function). Like all other solver classes, it can work on any kind of vector and matrix as long as they satisfy certain requirements (for the requirements on matrices and vectors in order to work with this class, see the documentation of the Solver base class). The type of the solution vector must be passed as template argument, and defaults to Vector<double>.

FIRE is a damped dynamics method described in Structural Relaxation Made Simple by Bitzek et al. 2006, typically used to find stable equilibrium configurations of atomistic systems in computational material science. Starting from a given initial configuration of the atomistic system, the algorithm relies on inertia to obtain (nearest) configuration with least potential energy.

Notation:

The global vector of unknown variables: \(\mathbf x\).
Objective function: \(E(\mathbf x)\).
Rate of change of unknowns: \(\mathbf v\).
Gradient of the objective function w.r.t unknowns: \(\mathbf g = \nabla E(\mathbf x)\).
Mass matrix: \(\mathbf M\).
Initial guess of unknowns: \(\mathbf x_0\).
Time step: \(\Delta t\).

Given initial values for \(\Delta t\), \(\alpha = \alpha_0\), \(\epsilon\), \(\mathbf x = \mathbf x_0\) and \(\mathbf v= \mathbf 0\) along with a given mass matrix \(\mathbf M\), FIRE algorithm is as follows,

Calculate \(\mathbf g = \nabla E(\mathbf x)\) and check for convergence ( \(\mathbf g \cdot \mathbf g < \epsilon^2 \)).
Update \(\mathbf x\) and \(V\) using simple (forward) Euler integration step,
\(\mathbf x = \mathbf x + \Delta t \mathbf v\),
\(\mathbf v = \mathbf v + \Delta t \mathbf M^{-1} \cdot \mathbf g\).
Calculate \(p = \mathbf g \cdot \mathbf v\).
Set \(\mathbf v = (1-\alpha) \mathbf v + \alpha \frac{|\mathbf v|}{|\mathbf g|} \mathbf g\).
If \(p<0\) and number of steps since \(p\) was last negative is larger than certain value, then increase time step \(\Delta t\) and decrease \(\alpha\).
If \(p>0\), then decrease the time step, freeze the system i.e., \(\mathbf v = \mathbf 0\) and reset \(\alpha = \alpha_0\).
Return to 1.

Also see Energy-Minimization in Atomic-to-Continuum Scale-Bridging Methods by Eidel et al. 2011.

Author: Vishal Boddu, Denis Davydov, 2017

Definition at line 89 of file solver_fire.h.