pylops.optimization.leastsquares.PreconditionedInversion

pylops.optimization.leastsquares.PreconditionedInversion(Op, P, data, x0=None, returninfo=False, **kwargs_solver)[source]

Preconditioned inversion.

Solve a system of preconditioned equations given the operator Op and a preconditioner P.

Parameters:
Op : pylops.LinearOperator

Operator to invert

P : pylops.LinearOperator

Preconditioner

data : numpy.ndarray

Data

x0 : numpy.ndarray

Initial guess

returninfo : bool

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

RegularizedInversion
Regularized inversion
NormalEquationsInversion
Normal equations inversion

Notes

Solve the following system of preconditioned equations given the operator \(\mathbf{Op}\), a preconditioner \(\mathbf{P}\), the data \(\mathbf{d}\)

\[\mathbf{d} = \mathbf{Op}\,\mathbf{P} \mathbf{m}\]

where \(\mathbf{m}\) is the solution in the preconditioned space and \(\mathbf{x} = \mathbf{P}\mathbf{m}\) is the solution in the original space.

Examples using pylops.optimization.leastsquares.PreconditionedInversion