pylops.optimization.solver.cgls¶
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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
Opand datayusing 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 wherexis the current model vector
Returns: - x :
np.ndarray Estimated model of size \([M \times 1]\)
- istop :
int Gives the reason for termination
1means \(\mathbf{x}\) is an approximate solution to \(\mathbf{d} = \mathbf{Op}\,\mathbf{x}\)2means \(\mathbf{x}\) 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
r1normif \(\epsilon=0\)- 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 :
