cherab.phix.inversion.SVDInversionBase

class cherab.phix.inversion.SVDInversionBase(s, u, vh, data, inversion_base_vectors=None, L_inv=None)

Bases: object

Base class for inversion calculation based on singular value decomposition (eco algorithum i.e. not full matrices of u, vh).

This provides users useful tools for regularization computation using SVD components. The estimated solution \(x_\lambda\) is defined by the following linear equation:

\[Ax = b,\]

where \(A\) is a \(M \times N\) matrix. Adding the regularization term, the estimated solution \(x_\lambda\) is derived from the following equation:

\[ \begin{align}\begin{aligned}x_\lambda :&= \text{argmin} \{ ||Ax-b||^2 + \lambda ||L(x - x_0)||^2 \} \\\ &= ( A^\intercal A + \lambda L^\intercal L )^{-1} (A^\intercal\ b + \lambda L^\intercal Lx_0),\end{aligned}\end{align} \]

where \(\lambda\) is the reguralization parameter, \(L\) is a matrix operator in regularization term (e.g. laplacian) and \(x_0\) is a prior assumption.

The SVD components are based on the following equation:

\[U\Sigma V^\intercal = (u_1, u_2, ...) \cdot \text{diag}(\sigma_1, \sigma_2,...) \cdot (v_1, v_2, ...)^\intercal = AL^{-1}\]

Using this components, The \(x_\lambda\) can be reconstructed as follows:

\[x_\lambda = \sum_{i=0}^{K} w_i(\lambda)\frac{u_i \cdot b}{\sigma_i} L^{-1} v_i, \quad (K \equiv \min(M, N)),\]

\(w_i\) is defined as follows:

\[w_i \equiv \frac{1}{1 + \lambda / \sigma_i^2}.\]

Parameters:

• s (vector_like) – singular values \(\sigma_i\) in \(s\) vectors.

• u (array_like) – SVD left singular vectors forms as one matrix like \(u = (u_1, u_2, ...)\)

• vh (array_like) – SVD right singular vectors forms as one matrix like \(vh = (v_1, v_2, ...)^T\)

• data (vector_like) – given data for inversion calculation

• inversion_base_vectors (array-like, optional) – The components of inversions base represented as L_inv @ vh.T. This property is offered to speed up the calculation of inversions. If None, it is automatically computed when calculating the inverted solution.

• L_inv (array_like, optional) – inversion matrix in the regularization term. L_inv is \(L^{-1}\) in \(||L(x - x_0)||^2\)

Methods

eta([beta])	Calculate squared regularization norm \(\eta = ||L(x - x_0)||^2\)
eta_diff([beta])	Calculate differential of eta by regularization parameter.
inverted_solution([beta])	Calculate the inverted solution using given regularization parameter.
regularization_norm([beta])	Return the residual norm \(\sqrt{\eta} = ||L (x - x_0)||\)
residual_norm([beta])	Return the residual norm \(\sqrt{\rho} = ||Ax - b||\)
rho([beta])	Calculate squared residual norm \(\rho = ||Ax - b||^2\).
w([beta])	Calculate window function using regularization parameter as a valuable and using singular values.

Attributes

L_inv	Inversion matrix in the regularization term.
beta	Regularization parameter.
data	Given data for inversion calculation.
inversion_base_vectors	The components of inversions base represented as L_inv @ vh.T.
s	singular values \(\sigma_i\) in \(s\) vectors.
u	SVD left singular vectors forms as one matrix like \(u = (u_1, u_2, ...)\)
vh	SVD right singular vectors forms as one matrix like \(vh = (v_1, v_2, ...)^T\)

eta(beta=None)

Calculate squared regularization norm \(\eta = ||L(x - x_0)||^2\)

Parameters:

beta (float | None) – regularization parameter, by default self._beta

Returns:

squared regularization norm

Return type:

numpy.floating

eta_diff(beta=None)

Calculate differential of eta by regularization parameter.

Parameters:

beta (float | None) – regularization parameter, by default self._beta

Returns:

differential of squared regularization norm

Return type:

numpy.floating

inverted_solution(beta=None)

Calculate the inverted solution using given regularization parameter.

Parameters:

beta (float | None) – regularization parameter, by default None

Returns:

solution vector

Return type:

numpy.ndarray

regularization_norm(beta=None)

Return the residual norm \(\sqrt{\eta} = ||L (x - x_0)||\)

Parameters:

beta (float | None) – reguralization parameter, by default self._beta

Returns:

regularization norm

Return type:

float

residual_norm(beta=None)

Return the residual norm \(\sqrt{\rho} = ||Ax - b||\)

Parameters:

beta (float | None) – reguralization parameter, by default self._beta

Returns:

residual norm

Return type:

float

rho(beta=None)

Calculate squared residual norm \(\rho = ||Ax - b||^2\).

Parameters:

beta (float | None) – regularization parameter, by default None

Returns:

squared residual norm

Return type:

numpy.floating

w(beta=None)

Calculate window function using regularization parameter as a valuable and using singular values.

Parameters:

beta (float | None) – regularization parameter, by default None

Returns:

window function \(\frac{1}{1 + \lambda / \sigma_i^2}\)

Return type:

numpy.ndarray (N, )

property L_inv: NDArray[float64] | None

Inversion matrix in the regularization term.

L_inv is \(L^{-1}\) in \(||L(x - x_0)||^2\), by default numpy.identity(self._vh.shape[1])

property beta: float

Regularization parameter.

property data: ndarray[Any, dtype[float64]]

Given data for inversion calculation.

property inversion_base_vectors: NDArray[float64] | None

The components of inversions base represented as L_inv @ vh.T. This property is offered to speed up the calculation of inversions. If None, it is automatically computed when calculating the inverted solution.

Type:

array_like or None if not set

property s: ndarray[Any, dtype[float64]]

singular values \(\sigma_i\) in \(s\) vectors.

Type:

vector_like

property u: ndarray[Any, dtype[float64]]

SVD left singular vectors forms as one matrix like \(u = (u_1, u_2, ...)\)

Type:

array_like

property vh: ndarray[Any, dtype[float64]]

SVD right singular vectors forms as one matrix like \(vh = (v_1, v_2, ...)^T\)

Type:

array_like