pylops.optimization.leastsquares.regularized_inversion(Op, y, Regs, x0=None, Weight=None, dataregs=None, epsRs=None, engine='scipy', show=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.

Op : pylops.LinearOperator

Operator to invert of size \([N \times M]\)

y : numpy.ndarray

Data of size \([N \times 1]\)

Regs : list

Regularization operators (None to avoid adding regularization)

x0 : numpy.ndarray, optional

Initial guess of size \([M \times 1]\)

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

engine : str, optional

Solver to use (scipy or pylops)

show : bool, optional

Display normal equations solver log


Arbitrary keyword arguments for chosen solver (scipy.sparse.linalg.lsqr and pylops.optimization.solver.cgls are used for engine scipy and pylops, respectively)

xinv : numpy.ndarray

Inverted model.

istop : int

Gives the reason for termination

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

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

itn : int

Iteration number upon termination

r1norm : float

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

r2norm : float

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

See also

Regularized operator
Normal equations inversion
Preconditioned inversion


See pylops.optimization.cls_leastsquares.RegularizedInversion