pylops.optimization.leastsquares.preconditioned_inversion¶

pylops.optimization.leastsquares.preconditioned_inversion(Op, y, P, x0=None, engine='scipy', show=False, **kwargs_solver)[source]¶

Preconditioned inversion.

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

Parameters:

Oppylops.LinearOperator: Operator to invert of size \([N \times M]\)
ynumpy.ndarray: Data of size \([N \times 1]\)
Ppylops.LinearOperator: Preconditioner
x0numpy.ndarray: Initial guess of size \([M \times 1]\)
enginestr, optional: Solver to use (scipy or pylops)
showbool, optional: Display normal equations solver log
**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:

xinvnumpy.ndarray

Inverted model.

istopint

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

itnint

Iteration number upon termination

r1normfloat

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

r2normfloat

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

Examples using `pylops.optimization.leastsquares.preconditioned_inversion`¶