pylops.optimization.leastsquares.RegularizedInversion

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

Arbitrary keyword arguments for scipy.sparse.linalg.lsqr solver
Returns:
xinv : numpy.ndarray

Inverted model \(\mathbf{Op}\)

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

\(||\mathbf{r}||_2\), 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 \(\epsilon_I\): and \(\epsilon_{{R}_i}\):

\[\begin{split}\begin{bmatrix} \mathbf{W}^{1/2} \mathbf{Op} \\ \epsilon_{R_1} \mathbf{R}_1 \\ ... \\ \epsilon_{R_N} \mathbf{R}_N \end{bmatrix} \mathbf{x} = \begin{bmatrix} \mathbf{W}^{1/2} \mathbf{d} \\ \epsilon_{R_1} \mathbf{d}_{R_1} \\ ... \\ \epsilon_{R_N} \mathbf{d}_{R_N} \\ \end{bmatrix}\end{split}\]

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.

Examples using pylops.optimization.leastsquares.RegularizedInversion