pylops.optimization.leastsquares.NormalEquationsInversion(Op, Regs, data, Weight=None, dataregs=None, epsI=0, epsRs=None, x0=None, returninfo=False, **kwargs_cg)[source]

Inversion of normal equations.

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

Op : pylops.LinearOperator

Operator to invert

Regs : list

Regularization operators (None to avoid adding regularization)

data : numpy.ndarray


Weight : pylops.LinearOperator, optional

Weight operator

dataregs : list, optional

Regularization data (must have the same number of elements as Regs)

epsI : float, optional

Tikhonov damping

epsRs : list, optional

Regularization dampings (must have the same number of elements as Regs)

x0 : numpy.ndarray, optional

Initial guess

returninfo : bool, optional

Return info of CG solver


Arbitrary keyword arguments for solver

xinv : numpy.ndarray

Inverted model.

istop : int

Convergence information:

0: successful exit

>0: convergence to tolerance not achieved, number of iterations

<0: illegal input or breakdown

See also

Regularized inversion
Preconditioned inversion


Solve the following normal equations for a system of regularized equations given the operator \(\mathbf{Op}\), a data weighting operator \(\mathbf{W}\), a list of regularization terms \(\mathbf{R_i}\), the data \(\mathbf{d}\) and regularization damping factors \(\epsilon_I\) and \(\epsilon_{{R}_i}\):

\[( \mathbf{Op}^T \mathbf{W} \mathbf{Op} + \sum_i \epsilon_{{R}_i}^2 \mathbf{R}_i^T \mathbf{R}_i + \epsilon_I^2 \mathbf{I} ) \mathbf{x} = \mathbf{Op}^T \mathbf{W} \mathbf{d} + \sum_i \epsilon_{{R}_i}^2 \mathbf{R}_i^T \mathbf{d}_{R_i}\]

Examples using pylops.optimization.leastsquares.NormalEquationsInversion