lsqr

mrinufft.extras.lsqr(operator: FourierOperatorBase, kspace_data: NDArray, damp: float = 0.0, atol: float = 1e-06, btol: float = 1e-06, conlim: float = 100000000.0, max_iter: int = 100, x0: NDArray | None = None, x_init: NDArray | None = None, callback: Callable | None = None, progressbar: bool | tqdm = True)

Solve a regularized linear least-squares problem with the LSQR algorithm [1].

This implementation follows the numerically stable formulation described in [2] and is adapted from the SciPy implementation [3].

Solves problems of the form

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

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

References

[1]

Paige, C. C., & Saunders, M. A. (1982). LSQR: An algorithm for sparse linear equations and sparse least squares. ACM Transactions on Mathematical Software, 8(1), 43-71.

[2]

S.-C. Choi, “Iterative Methods for Singular Linear Equations and Least-Squares Problems”, Dissertation, http://www.stanford.edu/group/SOL/dissertations/sou-cheng-choi-thesis.pdf

[3]

scipy/scipy

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.

See also

• This function is part of the optimizer registry. You can find other registered functions in this registry below:

Key	Function
"lsqr"	lsqr(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)
"lsmr"	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)
"cg"	cg(operator, kspace_data, damp=0.0, x0=None, x_init=None, max_iter=10, tol=0.0001, progressbar=True, callback=None)