Broyden mixer. More...

#include <broyden2_mixer.hpp>

Inherits sirius::mixer::Mixer< FUNCS... >.

Public Member Functions
	Broyden2 (std::size_t max_history, double beta, double beta0, double beta_scaling_factor, double linear_mix_rmse_tol)

void	mix_impl () override

Public Member Functions inherited from sirius::mixer::Mixer< FUNCS... >
	Mixer (std::size_t max_history)
	Construct a mixer. Functions have to initialized individually. More...

void	initialize_function (const FunctionProperties< typename std::tuple_element< FUNC_INDEX, std::tuple< FUNCS... > >::type > &function_prop, const typename std::tuple_element< FUNC_INDEX, std::tuple< FUNCS... > >::type &init_value, ARGS &&... args)

void	set_input (const typename std::tuple_element< FUNC_INDEX, std::tuple< FUNCS... > >::type &input)
	Set input for next mixing step. More...

void	get_output (typename std::tuple_element< FUNC_INDEX, std::tuple< FUNCS... > >::type &output)
	Access last generated output. Mixing must have been performed at least once. More...

double	mix (double rms_min__)
	Mix input and stored history. Returns the root mean square error computed by inner products of residuals. More...

Private Attributes
double	beta_

double	beta0_

double	beta_scaling_factor_

double	linear_mix_rmse_tol_

sddk::mdarray< double, 2 >	S_

sddk::mdarray< double, 1 >	gamma_

Additional Inherited Members
Static Public Attributes inherited from sirius::mixer::Mixer< FUNCS... >
static constexpr std::size_t	number_of_functions

Protected Member Functions inherited from sirius::mixer::Mixer< FUNCS... >
virtual void	mix_impl ()=0

void	update_residual ()

void	update_rms ()

std::size_t	idx_hist (std::size_t step) const

double	inner_product (const std::tuple< std::unique_ptr< FUNCS >... > &x, const std::tuple< std::unique_ptr< FUNCS >... > &y)

void	scale (double alpha, std::tuple< std::unique_ptr< FUNCS >... > &x)

void	copy (const std::tuple< std::unique_ptr< FUNCS >... > &x, std::tuple< std::unique_ptr< FUNCS >... > &y)

void	axpy (double alpha, const std::tuple< std::unique_ptr< FUNCS >... > &x, std::tuple< std::unique_ptr< FUNCS >... > &y)

void	rotate (double c, double s, std::tuple< std::unique_ptr< FUNCS >... > &x, std::tuple< std::unique_ptr< FUNCS >... > &y)

Protected Attributes inherited from sirius::mixer::Mixer< FUNCS... >
std::size_t	step_

std::size_t	max_history_

std::vector< double >	rmse_history_

std::tuple< FunctionProperties< FUNCS >... >	functions_

std::tuple< std::unique_ptr< FUNCS >... >	input_

std::vector< std::tuple< std::unique_ptr< FUNCS >... > >	output_history_

std::vector< std::tuple< std::unique_ptr< FUNCS >... > >	residual_history_

Detailed Description

template<typename... FUNCS>
class sirius::mixer::Broyden2< FUNCS >

Broyden mixer.

Quasi-Newton, limited-memory method which updates \( x_{n+1} = x_n - G_nf_n \) where \( G_n \) is an approximate inverse Jacobian. Broyden 2 is derived by recursively taking rank-1 updates to the inverse Jacobian. It requires the secant equation \( G_{n+1}\Delta f_n = \Delta x_n\) to hold for the next approximate Jacobian \( G_{n+1}\), leading to the update:

\[ G_{n+1} := G_n + (\Delta x_n - G_n \Delta f_n)(\Delta f_n^*\Delta f_n)^{-1}\Delta f_n^*. \]

By induction we can show that \( G_n \) satisfies

\[ G_n = G_1\prod_{i=1}^{n-1}\left(I - \frac{\Delta f_i \Delta f_i^*}{\Delta f_i^*\Delta f_i}\right) + \sum_{i=1}^{n-1}\frac{\Delta x_i \Delta f_i^*}{\Delta f_i^*\Delta f_i}\prod_{j=i+1}^{n-1} \left(I - \frac{\Delta f_j \Delta f_j^*}{\Delta f_j^*\Delta f_j}\right) \]

which shows orthogonalization with respect to the \( \Delta f_i \) vectors. A practical implementation can be derived from the identity

\[ G_n = G_1 + \sum_{i=1}^{n-1}\left(\Delta x_i - G_1\Delta f_i\right)\gamma_{i,n}^* \]

where

\[ \gamma_{i,n}^* = \frac{\Delta f_i^*}{\Delta f_i^*\Delta f_i}\prod_{j=i+1}^{n-1} \left(I - \frac{\Delta f_j\Delta f_j^*}{\Delta f_j^*\Delta f_j}\right). \]

The \( \gamma_{i,n}\) vector satisfies the following recursion relation:

\[ \gamma_{i,n}^* = \frac{\Delta f_i^*}{\Delta f_i^*\Delta f_i}\left(I - \sum_{j=i+1}^{n-1} \Delta f_j \gamma_{j,n}^*\right) \]

When updating \( x_{n+1} = x_n - G_nf_n\) we only have to compute:

\[ \alpha_i := \gamma_i^*f_n = \frac{1}{\Delta f_i^*\Delta f_i}\left[\Delta f_i^*f_n - \sum_{j=i+1}^{n-1}\alpha_j\Delta f_i^* \Delta f_j \right] \]

for \( i \) from \( n-1\) down to \( 1 \) so that

\[\begin{aligned} x_{n+1} &= x_n - G_nf_n \\ &= x_n - G_1f_n + \sum_{i=1}^{n-1}\alpha_iG_1\Delta f_i - \sum_{i=1}^{n-1}\alpha_i\Delta x_i. \end{aligned}\]

In particular when \( G_1 = -\beta I\) we get the following update:

\[ x_{n+1} = x_n + \beta f_n - \sum_{i=1}^{n-1}\alpha_i \beta \Delta f_i - \sum_{i=1}^{n-1}\alpha_i\Delta x_i. \]

Finally, we store the vectors \( f_1, \cdots, f_n \) and \( x_1, \dots, x_n \) and update the Gram-matrix \( S_{ij} = f_i^*f_j \) in every iteration. The \( \alpha_i \) coefficients can be easily computed from \( S \).

Note: This class does not do anything to improve the stability of the Gram-Schmidt procedure (clearly visible in the first explicit expression for \( G_n \)) and can therefore be very unstable for larger history sizes.

Definition at line 99 of file broyden2_mixer.hpp.

Constructor & Destructor Documentation

◆ Broyden2()

template<typename... FUNCS>

sirius::mixer::Broyden2< FUNCS >::Broyden2	(	std::size_t	max_history,
		double	beta,
		double	beta0,
		double	beta_scaling_factor,
		double	linear_mix_rmse_tol
	)

inline

Definition at line 102 of file broyden2_mixer.hpp.

Member Function Documentation

◆ mix_impl()

template<typename... FUNCS>

void sirius::mixer::Broyden2< FUNCS >::mix_impl ( )

template<typename... FUNCS>

sddk::mdarray<double, 1> sirius::mixer::Broyden2< FUNCS >::gamma_

private

Definition at line 193 of file broyden2_mixer.hpp.

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

broyden2_mixer.hpp

Public Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ Broyden2()

Member Function Documentation

◆ mix_impl()

Member Data Documentation

◆ beta_

◆ beta0_

◆ beta_scaling_factor_

◆ linear_mix_rmse_tol_

◆ S_

◆ gamma_