template<typename VectorType = Vector<double>>
class SolverFIRE< VectorType >
FIRE (Fast Inertial Relaxation Engine) for minimization of (potentially non-linear) objective function , is a vector of variables ( 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: .
Objective function: .
Rate of change of unknowns: .
Gradient of the objective function w.r.t unknowns: .
Mass matrix: .
Initial guess of unknowns: .
Time step: .
Given initial values for , , , and along with a given mass matrix , FIRE algorithm is as follows,
Calculate and check for convergence ( ).
Update and using simple (forward) Euler integration step, , .
Calculate .
Set .
If and number of steps since was last negative is larger than certain value, then increase time step and decrease .
If , then decrease the time step, freeze the system i.e., and reset .
Obtain a set of variables x that minimize an objective function described by the polymorphic function wrapper compute, with a given preconditioner inverse_mass_matrix and initial x values. The function compute returns the objective function's value and updates the objective function's gradient (with respect to the variables) when passed in as first argument based on the second argument– the state of variables.
Interface for derived class. This function gets the current iteration x (variables), v (x's time derivative) and g (the gradient) in each step. It can be used for graphical output of the convergence history.
Connect a function object that will be called periodically within iterative solvers. This function is used to attach monitors to iterative solvers, either to determine when convergence has happened, or simply to observe the progress of an iteration. See the documentation of this class for more information.
Parameters
slot
A function object specified here will, with each call, receive the number of the current iteration, the value that is used to check for convergence (typically the residual of the current iterate with respect to the linear system to be solved) and the currently best available guess for the current iterate. Note that some solvers do not update the approximate solution in every iteration but only after convergence or failure has been determined (GMRES is an example); in such cases, the vector passed as the last argument to the signal is simply the best approximate at the time the signal is called, but not the vector that will be returned if the signal's return value indicates that the iteration should be terminated. The function object must return a SolverControl::State value that indicates whether the iteration should continue, has failed, or has succeeded. The results of all connected functions will then be combined to determine what should happen with the iteration.
Returns
A connection object that represents the connection from the signal to the function object. It can be used to disconnect the function object again from the signal. See the documentation of the BOOST Signals2 library for more information on connection management.
A signal that iterative solvers can execute at the end of every iteration (or in an otherwise periodic fashion) to find out whether we should continue iterating or not. The signal may call one or more slots that each will make this determination by themselves, and the result over all slots (function calls) will be determined by the StateCombiner object.
The arguments passed to the signal are (i) the number of the current iteration; (ii) the value that is used to determine convergence (oftentimes the residual, but in other cases other quantities may be used as long as they converge to zero as the iterate approaches the solution of the linear system); and (iii) a vector that corresponds to the current best guess for the solution at the point where the signal is called. Note that some solvers do not update the approximate solution in every iteration but only after convergence or failure has been determined (GMRES is an example); in such cases, the vector passed as the last argument to the signal is simply the best approximate at the time the signal is called, but not the vector that will be returned if the signal's return value indicates that the iteration should be terminated.