pylops.optimization.solver.cgls(Op, y, x0, niter=10, damp=0.0, tol=0.0001, show=False, callback=None)[source]

Conjugate gradient least squares

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

Op : pylops.LinearOperator

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

y : np.ndarray

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

x0 : np.ndarray, optional

Initial guess

niter : int, optional

Number of iterations

damp : float, optional

Damping coefficient

tol : float, optional

Tolerance on residual norm

show : bool, optional

Display iterations log

callback : callable, optional

Function with signature (callback(x)) to call after each iteration where x is the current model vector

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 \(\mathbf{d} = \mathbf{Op}\,\mathbf{x}\)

2 means \(\mathbf{x}\) approximately solves the least-squares problem

iit : int

Iteration number upon termination

r1norm : float

\(||\mathbf{r}||_2\), where \(\mathbf{r} = \mathbf{d} - \mathbf{Op}\,\mathbf{x}\)

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


Minimize the following functional using conjugate gradient iterations:

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

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

Examples using pylops.optimization.solver.cgls