pylops.optimization.leastsquares.NormalEquationsInversion¶
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pylops.optimization.leastsquares.NormalEquationsInversion(Op, Regs, data, Weight=None, dataregs=None, epsI=0, epsRs=None, x0=None, returninfo=False, **kwargs_cg)[source]¶ Inversion of normal equations.
Solve the regularized normal equations for a system of equations given the operator
Op, a data weighting operatorWeightand a list of regularization termsRegsParameters: - Op :
pylops.LinearOperator Operator to invert
- Regs :
list Regularization operators (
Noneto avoid adding regularization)- data :
numpy.ndarray Data
- Weight :
pylops.LinearOperator, optional Weight operator
- dataregs :
list, optional Regularization data (must have the same number of elements as
Regs)- epsI :
float, optional Tikhonov damping
- epsRs :
list, optional Regularization dampings (must have the same number of elements as
Regs)- x0 :
numpy.ndarray, optional Initial guess
- returninfo :
bool, optional Return info of CG solver
- **kwargs_cg
Arbitrary keyword arguments for
scipy.sparse.linalg.cgsolver
Returns: - xinv :
numpy.ndarray Inverted model.
- istop :
int Convergence information:
0: successful exit>0: convergence to tolerance not achieved, number of iterations<0: illegal input or breakdown
See also
RegularizedInversion- Regularized inversion
PreconditionedInversion- Preconditioned inversion
Notes
Solve the following normal equations for a system of regularized equations given the operator \(\mathbf{Op}\), a data weighting operator \(\mathbf{W}\), a list of regularization terms \(\mathbf{R_i}\), the data \(\mathbf{d}\) and regularization damping factors \(\epsilon_I\) and \(\epsilon_{{R}_i}\):
\[( \mathbf{Op}^T \mathbf{W} \mathbf{Op} + \sum_i \epsilon_{{R}_i}^2 \mathbf{R}_i^T \mathbf{R}_i + \epsilon_I^2 \mathbf{I} ) \mathbf{x} = \mathbf{Op}^T \mathbf{W} \mathbf{d} + \sum_i \epsilon_{{R}_i}^2 \mathbf{R}_i^T \mathbf{d}_{R_i}\]- Op :
