pylops.optimization.leastsquares.normal_equations_inversion¶

pylops.optimization.leastsquares.normal_equations_inversion(Op, y, Regs, x0=None, Weight=None, dataregs=None, epsI=0.0, epsRs=None, NRegs=None, epsNRs=None, engine='scipy', show=False, **kwargs_solver)[source]¶

Inversion of normal equations.

Solve the regularized normal equations for a system of equations given the operator Op, a data weighting operator Weight and optionally a list of regularization terms Regs and/or NRegs.

Parameters:

Oppylops.LinearOperator: Operator to invert of size \([N \times M]\)
ynumpy.ndarray: Data of size \([N \times 1]\)
Regslist: Regularization operators (None to avoid adding regularization)
x0numpy.ndarray, optional: Initial guess of size \([M \times 1]\)
Weightpylops.LinearOperator, optional: Weight operator
dataregslist, optional: Regularization data (must have the same number of elements as Regs)
epsIfloat, optional: Tikhonov damping
epsRslist, optional: Regularization dampings (must have the same number of elements as Regs)
NRegslist: Normal regularization operators (None to avoid adding regularization). Such operators must apply the chain of the forward and the adjoint in one go. This can be convenient in cases where a faster implementation is available compared to applying the forward followed by the adjoint.
epsNRslist, optional: Regularization dampings for normal operators (must have the same number of elements as NRegs)
enginestr, optional: Solver to use (scipy or pylops)
showbool, optional: Display normal equations solver log
**kwargs_solver: Arbitrary keyword arguments for chosen solver (scipy.sparse.linalg.cg and pylops.optimization.solver.cg are used for engine scipy and pylops, respectively)

Note

When user does not supply atol, it is set to “legacy”.

Returns: