pylops.optimization.cls_leastsquares.RegularizedInversion

class pylops.optimization.cls_leastsquares.RegularizedInversion(Op, callbacks=None)

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 \([N \times M]\).

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 in pylops.optimization.leastsquares.NormalEquationsInversion. Note that this system is solved using the scipy.sparse.linalg.lsqr and an initial guess x0 can be provided to this solver, despite the original solver does not allow so.

Methods

__init__(Op[, callbacks])	Initialize self.
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

setup(y, Regs, Weight=None, dataregs=None, epsRs=None, show=False)

Setup solver

Parameters:

y : np.ndarray

Data of size \([N \times 1]\)

Regs : list

Regularization operators (None to avoid adding regularization)

Weight : pylops.LinearOperator, optional

Weight operator

dataregs : list, optional

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

epsRs : list, optional

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

show : bool, optional

Display setup log

step()

Run one step of solver

This method is used to run one step of the solver. Users can change the function signature by including any other input parameter required when applying one step of the solver

Parameters:

x : np.ndarray

Current model vector to be updated by a step of the solver

show : bool, optional

Display step log

run(x, engine='scipy', show=False, **kwargs_solver)

Run solver

Parameters:

x : np.ndarray

Current model vector to be updated by multiple steps of the solver. If None, x is assumed to be a zero vector

engine : str, optional

Solver to use (scipy or pylops)

show : bool, optional

Display iterations 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)

Returns:

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 \(\epsilon=0\)

solve(y, Regs, x0=None, Weight=None, dataregs=None, epsRs=None, engine='scipy', show=False, **kwargs_solver)

Run entire solver

Parameters:

y : np.ndarray

Data of size \([N \times 1]\)

Regs : list

Regularization operators (None to avoid adding regularization)

x0 : numpy.ndarray, optional

Initial guess

Weight : pylops.LinearOperator, optional

Weight operator

dataregs : list, optional

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

epsRs : list, optional

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

engine : str, optional

Solver to use (scipy or pylops)

show : bool, optional

Display 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)

Returns:

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 \(\epsilon=0\)