pylops.optimization.solver.cgls¶
-
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 datay
using conjugate gradient iterations.Parameters: - 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 wherex
is the current model vector
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 \(\mathbf{d} = \mathbf{Op}\,\mathbf{x}\)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
Notes
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.
- Op :