douglas_rachford_pd_stepsize¶

odl.solvers.nonsmooth.douglas_rachford.douglas_rachford_pd_stepsize(L, tau=None, sigma=None)[source]¶

Default step sizes for douglas_rachford_pd.

Parameters:

Lsequence of Operator or float: The operators or the norms of the operators that are used in the douglas_rachford_pd method. For Operator entries, the norm is computed with Operator.norm(estimate=True).
taupositive float, optional: Use this value for tau instead of computing it from the operator norms, see Notes.
sigmatuple of float, optional: The sigma step size parameters for the dual update.

Returns:

taufloat: The tau step size parameter for the primal update.
sigmatuple of float: The sigma step size parameters for the dual update.

Notes

To guarantee convergence, the parameters $\tau$ , $\sigma_i$ and $L_i$ need to satisfy

$\tau \sum_{i=1}^n \sigma_i \|L_i\|^2 < 4.$

This function has 4 options, $\tau$ / $\sigma$ given or not given.

If neither $\tau$ nor $\sigma$ are given, they are chosen as:

$\tau = \frac{1}{\sum_{i=1}^n \|L_i\|}, \quad \sigma_i = \frac{2}{n \tau \|L_i\|^2}$
If only $\sigma$ is given, $\tau$ is set to:

$\tau = \frac{2}{\sum_{i=1}^n \sigma_i \|L_i\|^2}$
If only $\tau$ is given, $\sigma$ is set to:

$\sigma_i = \frac{2}{n \tau \|L_i\|^2}$
If both are given, they are returned as-is without further validation.