Module regpy.solvers.linear.landweber

Classes

class Landweber (setting, data, init, stepsize=None)

The linear Landweber method. Solves the linear, ill-posed equation T(x) = g^\delta, in Hilbert spaces by gradient descent for the residual \Vert T(x) - g^\delta\Vert^2, where (\Vert\cdot\Vert)\ is the Hilbert space norm in the codomain, and gradients are computed with respect to the Hilbert space structure on the domain.

The number of iterations is effectively the regularization parameter and needs to be picked carefully.

Parameters

setting : RegularizationSetting
The setting of the forward problem.
data : array-like
The measured data/right hand side.
init : array-like
The initial guess.
stepsize : float, optional
The step length; must be chosen not too large. If omitted, it is guessed from the norm of the derivative at the initial guess.
Expand source code 
class Landweber(RegSolver):
    r"""The linear Landweber method. Solves the linear, ill-posed equation
    \[
        T(x) = g^\delta,
    \]
    in Hilbert spaces by gradient descent for the residual
    \[
        \Vert T(x) - g^\delta\Vert^2,
    \]
    where \(\Vert\cdot\Vert)\ is the Hilbert space norm in the codomain, and gradients are computed with
    respect to the Hilbert space structure on the domain.

    The number of iterations is effectively the regularization parameter and needs to be picked
    carefully.

    Parameters
    ----------
    setting : regpy.solvers.RegularizationSetting
        The setting of the forward problem.
    data : array-like
        The measured data/right hand side.
    init : array-like
        The initial guess.
    stepsize : float, optional
        The step length; must be chosen not too large. If omitted, it is guessed from the norm of
        the derivative at the initial guess.
    """

    def __init__(self, setting, data, init, stepsize=None):
        super().__init__(setting)
        self.rhs = data
        """The right hand side gets initialized to measured data"""
        self.x = init
        self.y = self.op(self.x)
        norm = setting.op_norm()
        self.stepsize = stepsize or 1 / norm**2
        """The stepsize."""

    def _next(self):
        self._residual = self.y - self.rhs
        self._gy_residual = self.h_codomain.gram(self._residual)
        self._update = self.op.adjoint(self._gy_residual)
        self.x -= self.stepsize * self.h_domain.gram_inv(self._update)
        self.y = self.op(self.x)

        if self.log.isEnabledFor(logging.INFO):
            norm_residual = np.sqrt(np.real(np.vdot(self._residual, self._gy_residual)))
            self.log.info('|residual| = {}'.format(norm_residual))

Ancestors

Instance variables

var rhs

The right hand side gets initialized to measured data

var stepsize

The stepsize.

Inherited members