lsmr

mrinufft.extras.optim.lsmr(operator, kspace_data, damp=0.0, atol=1e-06, btol=1e-06, conlim=100000000.0, max_iter=100, x0=None, x_init=None, callback=None, progressbar=True)

Solve a general regularized linear least-squares problem using the LSMR algorithm.

Solves problems of the form .. math:

\arg\min \|A x - b\|_2^2 + \gamma^2 \|x - x0\|_2^2

Stop iterating if:

• numerical convergence is reached: \(\|Ax-b\| <= atol \|A\| * \|x\| + btol * \|b\|\)

• estimation of the conditioning of the problem diverge: cond(A)>=conlim

• Maximum number of iteration reached.

Parameters:

• $base_params

• atol (float, optional) – Stopping tolerance on the absolute error. Default is 1e-6.

• btol (float, optional) – Stopping tolerance on the relative error. Default is 1e-6.

• conlim (float, optional) – Limit on condition number. Iteration stops if condition exceeds this value. Default is 1e8.

• $returns

• operator (FourierOperatorBase)

• kspace_data (ndarray[tuple[int, ...], dtype[_ScalarType_co]])

• damp (float)

• max_iter (int)

• x0 (ndarray[tuple[int, ...], dtype[_ScalarType_co]] | None)

• x_init (ndarray[tuple[int, ...], dtype[_ScalarType_co]] | None)

• callback (Callable | None)

• progressbar (bool)

References

[1]

D. C.-L. Fong and M. A. Saunders, “LSMR: An iterative algorithm for sparse least-squares problems”, SIAM J. Sci. Comput., vol. 33, pp. 2950-2971, 2011. :arxiv:`1006.0758`

[2]

LSMR Software, https://web.stanford.edu/group/SOL/software/lsmr/

Notes

• LSMR is generally more stable than LSQR, notably in term of image residual norm \(\|A^H(Ax-b)\|\), and is similar to the MINRES algorithm for least squares problems.

• It usually converges faster than LSQR and can stop in fewer iterations.

Note

This function uses numpy for all CPU arrays, and cupy for all on-gpu array. It will convert all its array argument to the respective array library. The outputs will be converted back to the original array module and device.