lsqr

mrinufft.extras.optim.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)

Solve a general regularized linear least-squares problem using the LSQR 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]

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.