# pylops.optimization.leastsquares.regularized_inversion¶

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.

Parameters: 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 **kwargs_solver 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 $$\epsilon=0$$

RegularizedOperator
Regularized operator
NormalEquationsInversion
Normal equations inversion
PreconditionedInversion
Preconditioned inversion

Notes

## Examples using pylops.optimization.leastsquares.regularized_inversion¶ Causal Integration MP, OMP, ISTA and FISTA Total Variation (TV) Regularization Wavelet estimation 03. Solvers 06. 2D Interpolation 16. CT Scan Imaging