pylops.optimization.leastsquares.RegularizedInversion

pylops.optimization.leastsquares.RegularizedInversion(Op, Regs, data, Weight=None, dataregs=None, epsRs=None, x0=None, returninfo=False, **kwargs_solver)[source]

Regularized inversion.

Solve a system of regularized equations given the operator Op, a data weighting operator Weight, and a list of regularization terms Regs.

Parameters:
Op : pylops.LinearOperator

Operator to invert

Regs : list

Regularization operators (None to avoid adding regularization)

data : numpy.ndarray

Data

Weight : pylops.LinearOperator, optional

Weight operator

dataregs : list, optional

Regularization data (if None a zero data will be used for every regularization operator in Regs)

epsRs : list, optional

Regularization dampings

x0 : numpy.ndarray, optional

Initial guess

returninfo : bool, optional

Return info of LSQR solver

**kwargs_solver

Arbitrary keyword arguments for chosen solver (scipy.sparse.linalg.lsqr and pylops.optimization.solver.cgls are used as default for numpy and cupy data, respectively)
Returns:
xinv : numpy.ndarray

Inverted model.

istop : int

Gives the reason for termination

1 means \(\mathbf{x}\) is an approximate solution to \(\mathbf{d} = \mathbf{Op}\,\mathbf{x}\)

2 means \(\mathbf{x}\) approximately solves the least-squares problem

itn : int

Iteration number upon termination

r1norm : float

\(||\mathbf{r}||_2^2\), where \(\mathbf{r} = \mathbf{d} - \mathbf{Op}\,\mathbf{x}\)

r2norm : float

\(\sqrt{\mathbf{r}^T\mathbf{r} + \epsilon^2 \mathbf{x}^T\mathbf{x}}\). Equal to r1norm if \(\epsilon=0\)

See also

RegularizedOperator
Regularized operator
NormalEquationsInversion
Normal equations inversion
PreconditionedInversion
Preconditioned inversion

Notes

Solve the following system of regularized equations given the operator \(\mathbf{Op}\), a data weighting operator \(\mathbf{W}^{1/2}\), a list of regularization terms \(\mathbf{R}_i\), the data \(\mathbf{d}\) and regularization damping factors \(\epsilon_\mathbf{I}\): and \(\epsilon_{\mathbf{R}_i}\):

\[\begin{split}\begin{bmatrix} \mathbf{W}^{1/2} \mathbf{Op} \\ \epsilon_{\mathbf{R}_1} \mathbf{R}_1 \\ \vdots \\ \epsilon_{\mathbf{R}_N} \mathbf{R}_N \end{bmatrix} \mathbf{x} = \begin{bmatrix} \mathbf{W}^{1/2} \mathbf{d} \\ \epsilon_{\mathbf{R}_1} \mathbf{d}_{\mathbf{R}_1} \\ \vdots \\ \epsilon_{\mathbf{R}_N} \mathbf{d}_{\mathbf{R}_N} \\ \end{bmatrix}\end{split}\]

where the Weight provided here is equivalent to the square-root of the weight in pylops.optimization.leastsquares.NormalEquationsInversion. Note that this system is solved using the scipy.sparse.linalg.lsqr and an initial guess x0 can be provided to this solver, despite the original solver does not allow so.