low_rank_toolbox.cssp.gpodr

low_rank_toolbox.cssp.gpodr(U, oversampling_size=None, tol=None, max_iter=None, return_projector=False, return_inverse=False, **extra_args)[source]

Gappy POD+R - QDEIM and randomized oversampling.

When tol is given, then p is ignored and the algorithm will select the number of rows to satisfy the condition: sigma_{min}(U[p, :])^{-1} <= tol

Reference

Stability of discrete empirical interpolation and gappy proper orthogonal decomposition with randomized and deterministic sampling points Benjamin Peherstorfer, Zlatko Drmač, and Serkan Gugercin SIAM Journal on Scientific Computing, 42(5), A2837-A2864.

Parameters:

  • U (ndarray) – Orthonormal matrix of size n x k

  • oversampling_size (int) – Oversampling size

  • tol (float) – Tolerance for the quantity sigma_{min}(U[p, :])^{-1}

  • return_projector (bool) – If True, return also the matrix U @ pinv(U[p, :])

  • return_inverse (bool) – If True, return also the inverse of U[p, :]

  • extra_args (dict) –

    Additional arguments:
    qr_kwargs: dict

    Additional arguments for the QR factorization

    lstsq_kwargs: dict

    Additional arguments for the lstsq function

  • max_iter (int | None)

Returns:

  • p (list) – Selection of m row indices

  • P_U (ndarray (n x k) (optional)) – Matrix U @ pinv(U[p, :])

  • inv_U (ndarray (k x k) (optional)) – Inverse of U[p, :]