cherab.phix.inversion.lcurve.Lcurve

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

Bases: _SVDBase

L-curve criterion optimization for regularization parameter.

L curve is the trajectory of the point \((\log||Ax_\lambda-b||, \log||L(x_\lambda-x_0)||)\), those of which mean the residual and the regularization norm, respectively. The “corner” of this curve is assosiated with optimized point of regularization parameter, where the curvature of the L curve is maximized. This theory is mentioned by P.C.Hansen [1].

Parameters:

• s (vector_like) – singular values of \(A\) like \(\sigma = (\sigma_1, \sigma_2, ...) \in \mathbb{R}^r\)

• u (array_like) – left singular vectors of \(A\) like \(U = (u_1, u_2, ...) \in \mathbb{R}^{m\times r}\)

• basis (array_like) – inverted solution basis \(\tilde{V} \in \mathbb{R}^{n\times r}\). Here, \(\tilde{V} = L^{-1}V\), where \(V\in\mathbb{R}^{n\times r}\) is the right singular vectors of \(A\) and \(L^{-1}\) is the inverse of regularization operator \(L \in \mathbb{R}^{n\times n}\).

• **kwargs (_SVDBase properties, optional) – kwargs are used to specify properties like a data

References

[1]

P.C.Hansen, “The L-curve and its use in the numerical treatment of inverse problems”, January 2001 In book: Computational Inverse Problems in Electrocardiology, Publisher: WIT Press

Methods

curvature(beta)	Calculate L-curve curvature.
eta(beta)	Calculate squared regularization norm: \(\eta = ||Lx_\lambda||^2\)
eta_diff(beta)	Calculate differential of eta: \(\eta' = \frac{d\eta}{d\lambda}\)
inverted_solution(beta)	Calculate the inverted solution using SVD components at given regularization parameter.
optimize([itemax, bounds])	Excute the optimization of L-curve regularization.
plot_L_curve([fig, axes, bounds, n_beta, ...])	Plotting the L curve in log-log scale.
plot_curvature([fig, axes, bounds, n_beta])	Plotting L-curve curvature vs regularization parameters.
regularization_norm(beta)	Return the residual norm: \(\sqrt{\eta} = ||L x_\lambda||\)
residual_norm(beta)	Return the residual norm: \(\sqrt{\rho} = ||Ax_\lambda - b||\)
rho(beta)	Calculate squared residual norm: \(\rho = ||Ax_\lambda - b||^2\).
solve([bounds, stepsize])	Solve the ill-posed inversion equation.
w(beta)	Calculate window function using regularization parameter \(\lambda\).

Attributes

basis	The inverted solution basis \(\tilde{V} \in \mathbb{R}^{n\times r}\).
data	Given data for inversion calculation.
lambda_opt	Optimal regularization parameter defined after solve is executed.
s	Singular values of \(A\)
u	Left singular vectors of \(A\).