pylops.optimization.cls_leastsquares.RegularizedInversion#
- class pylops.optimization.cls_leastsquares.RegularizedInversion(Op, callbacks=None)[source]#
Regularized inversion.
Solve a system of regularized equations given the operator
Op
, a data weighting operatorWeight
, and a list of regularization termsRegs
.- Parameters
- Op
pylops.LinearOperator
Operator to invert of size \([N \times M]\).
- Op
See also
RegularizedOperator
Regularized operator
NormalEquationsInversion
Normal equations inversion
PreconditionedInversion
Preconditioned inversion
Notes
Solve the following system of regularized equations given the operator \(\mathbf{Op}\), a data weighting operator \(\mathbf{W}^{1/2}\), a list of regularization terms \(\mathbf{R}_i\), the data \(\mathbf{y}\) and regularization data \(\mathbf{y}_{\mathbf{R}_i}\), and the damping factors \(\epsilon_\mathbf{I}\): and \(\epsilon_{\mathbf{R}_i}\):
\[\begin{split}\begin{bmatrix} \mathbf{W}^{1/2} \mathbf{Op} \\ \epsilon_{\mathbf{R}_1} \mathbf{R}_1 \\ \vdots \\ \epsilon_{\mathbf{R}_N} \mathbf{R}_N \end{bmatrix} \mathbf{x} = \begin{bmatrix} \mathbf{W}^{1/2} \mathbf{y} \\ \epsilon_{\mathbf{R}_1} \mathbf{y}_{\mathbf{R}_1} \\ \vdots \\ \epsilon_{\mathbf{R}_N} \mathbf{y}_{\mathbf{R}_N} \\ \end{bmatrix}\end{split}\]where the
Weight
provided here is equivalent to the square-root of the weight inpylops.optimization.leastsquares.NormalEquationsInversion
. Note that this system is solved using thescipy.sparse.linalg.lsqr
and an initial guessx0
can be provided to this solver, despite the original solver does not allow so.Methods
__init__
(Op[, callbacks])callback
(x, *args, **kwargs)Callback routine
finalize
(*args[, show])Finalize solver
run
(x[, engine, show])Run solver
setup
(y, Regs[, Weight, dataregs, epsRs, show])Setup solver
solve
(y, Regs[, x0, Weight, dataregs, ...])Run entire solver
step
()Run one step of solver