pylops.optimization.cls_basic.CGLS

class pylops.optimization.cls_basic.CGLS(Op, callbacks=None)

Conjugate gradient least squares

Solve an overdetermined system of equations given an operator Op and data y using conjugate gradient iterations.

Parameters:

Op : pylops.LinearOperator

Operator to invert of size \([N \times M]\)

Notes

Minimize the following functional using conjugate gradient iterations:

\[J = || \mathbf{y} - \mathbf{Op}\,\mathbf{x} ||_2^2 + \epsilon^2 || \mathbf{x} ||_2^2\]

where \(\epsilon\) is the damping coefficient.

Methods

__init__(Op[, callbacks])	Initialize self.
callback(x, *args, **kwargs)	Callback routine
finalize([show])	Finalize solver
run(x[, niter, show, itershow])	Run solver
setup(y[, x0, niter, damp, tol, show])	Setup solver
solve(y[, x0, niter, damp, tol, show, itershow])	Run entire solver
step(x[, show])	Run one step of solver

setup(y, x0=None, niter=None, damp=0.0, tol=0.0001, show=False)

Setup solver

Parameters:

y : np.ndarray

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

x0 : np.ndarray, optional

Initial guess of size \([M \times 1]\). If None, initialize internally as zero vector

niter : int, optional

Number of iterations (default to None in case a user wants to manually step over the solver)

damp : float, optional

Damping coefficient

tol : float, optional

Tolerance on residual norm

show : bool, optional

Display setup log

Returns:

x : np.ndarray

Initial guess of size \([N \times 1]\)

step(x, show=False)

Run one step of solver

Parameters:

x : np.ndarray

Current model vector to be updated by a step of CG

show : bool, optional

Display iteration log

run(x, niter=None, show=False, itershow=[10, 10, 10])

Run solver

Parameters:

x : np.ndarray

Current model vector to be updated by multiple steps of CGLS

niter : int, optional

Number of iterations. Can be set to None if already provided in the setup call

show : bool, optional

Display iterations log

itershow : list, optional

Display set log for the first N1 steps, last N2 steps, and every N3 steps in between where N1, N2, N3 are the three element of the list.

Returns:

x : np.ndarray

Estimated model of size \([M \times 1]\)

finalize(show=False)

Finalize solver

Parameters:

show : bool, optional

Display finalize log

solve(y, x0=None, niter=10, damp=0.0, tol=0.0001, show=False, itershow=[10, 10, 10])

Run entire solver

Parameters:

y : np.ndarray

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

x0 : np.ndarray

Initial guess of size \([M \times 1]\). If None, initialize internally as zero vector

niter : int, optional

Number of iterations (default to None in case a user wants to manually step over the solver)

damp : float, optional

Damping coefficient

tol : float, optional

Tolerance on residual norm

show : bool, optional

Display logs

itershow : list, optional

Display set log for the first N1 steps, last N2 steps, and every N3 steps in between where N1, N2, N3 are the three element of the list.

Returns:

x : np.ndarray

Estimated model of size \([M \times 1]\)

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

iit : int

Iteration number upon termination

r1norm : float

\(||\mathbf{r}||_2\), 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 \(\epsilon=0\)

cost : numpy.ndarray, optional

History of r1norm through iterations