Lcurve

class cherab.inversion.Lcurve(*args, **kwargs)Source

Bases: _SVDBase

L-curve criterion for regularization parameter optimization.

The L-curve is a log-log plot of the residual norm versus the regularization norm. The L-curve criterion for tikhnov regularization gives the optimal regularization parameter \(\lambda\) as the corner point of the L-curve by maximizing the curvature of the L-curve.

Note

The theory and implementation of the L-curve criterion are described here.

Parameters:

*args, **kwargs

Parameters are the same as _SVDBase.

Examples

>>> lcurve = Lcurve(s, U, basis, data=data)

Methods

curvature(beta)	Calculate L-curve curvature.
eta(beta)	Calculate squared regularization norm \(\eta\).
eta_diff(beta)	Calculate differential of \(\eta\).
filter(beta)	Calculate the filter factors \(f_{\lambda, i}\).
plot_L_curve([fig, axes, bounds, n_beta, ...])	Plotting the L-curve in log-log scale.
plot_curvature([fig, axes, bounds, n_beta, ...])	Plotting the curvature of L-curve as function of regularization parameter.
regularization_norm(beta)	Return the regularization norm: \(\sqrt{\eta} = \|\mathbf{x}_\lambda\|_\mathbf{H}\).
residual_norm(beta)	Return the residual norm: \(\sqrt{\rho} = \|\mathbf{T}\mathbf{x}_\lambda - \mathbf{b}\|_{\mathbf{Q}}\).
rho(beta)	Calculate squared residual norm \(\rho\).
solution(beta)	Calculate the solution vector \(\mathbf{x}_\lambda\).
solve([bounds, stepsize])	Solve the ill-posed inversion equation.

Attributes

B	Matrix \(\mathbf{B}\) from \(\mathbf{Q} = \mathbf{B}^\mathsf{T}\mathbf{B}\).
U	Left singular vectors \(\mathbf{U}\).
basis	Inverted solution basis \(\tilde{\mathbf{V}}\).
bounds	Bound of log10 of regularization parameter \(\lambda\).
data	Given data for inversion calculation \(\mathbf{b}\).
lambda_opt	Optimal regularization parameter defined after solve is executed.
s	Singular values \(\mathbf{s}\).