`pymor.algorithms.lrradi`¶

Module Contents¶

pymor.algorithms.lrradi.hamiltonian_shifts(A, E, B, R, K, Z, shift_options)[source]¶

Compute further shift parameters for low-rank RADI iteration.

Compute Galerkin projection of Hamiltonian matrix on space spanned by last few columns of $Z$ and return the eigenvalue of the projected Hamiltonian with the most impact on convergence as the next shift parameter.

See [BBujanovicKurschnerS18], pp. 318-321.

Parameters

A: The Operator A from the corresponding Riccati equation.
E: The Operator E from the corresponding Riccati equation.
B: The VectorArray B from the corresponding Riccati equation.
R: A VectorArray representing the currently computed residual factor.
K: A VectorArray representing the currently computed iterate.
Z: A VectorArray representing the currently computed solution factor.
shift_options: The shift options to use (see ricc_lrcf_solver_options).

Returns

shifts: A NumPy array containing a set of stable shift parameters.

pymor.algorithms.lrradi.hamiltonian_shifts_init(A, E, B, C, shift_options)[source]¶

Compute initial shift parameters for low-rank RADI iteration.

Compute Galerkin projection of Hamiltonian matrix on space spanned by $C$ and return the eigenvalue of the projected Hamiltonian with the most impact on convergence as the next shift parameter.

See [BBujanovicKurschnerS18], pp. 318-321.

Parameters

A: The Operator A from the corresponding Riccati equation.
E: The Operator E from the corresponding Riccati equation.
B: The VectorArray B from the corresponding Riccati equation.
C: The VectorArray C from the corresponding Riccati equation.
shift_options: The shift options to use (see ricc_lrcf_solver_options).

Returns

shifts: A NumPy array containing a set of stable shift parameters.

pymor.algorithms.lrradi.ricc_lrcf_solver_options(lrradi_tol=1e-10, lrradi_maxiter=500, lrradi_shifts='hamiltonian_shifts', hamiltonian_shifts_init_maxiter=20, hamiltonian_shifts_subspace_columns=6)[source]¶

Returns available Riccati equation solvers with default solver options.

Parameters

lrradi_tol: See solve_ricc_lrcf.
lrradi_maxiter: See solve_ricc_lrcf.
lrradi_shifts: See solve_ricc_lrcf.
hamiltonian_shifts_init_maxiter: See hamiltonian_shifts_init.
hamiltonian_shifts_subspace_columns: See hamiltonian_shifts.

Returns

A dict of available solvers with default solver options.

pymor.algorithms.lrradi.solve_ricc_lrcf(A, E, B, C, R=None, S=None, trans=False, options=None)[source]¶

Compute an approximate low-rank solution of a Riccati equation.

See pymor.algorithms.riccati.solve_ricc_lrcf for a general description.

This function is an implementation of Algorithm 2 in [BBujanovicKurschnerS18].

Parameters

A: The Operator A.
E: The Operator E or None.
B: The operator B as a VectorArray from A.source.
C: The operator C as a VectorArray from A.source.
R: The matrix R as a 2D NumPy array or None.
S: The operator S as a VectorArray from A.source or None.
trans: Whether the first Operator in the Riccati equation is transposed.
options: The solver options to use. (see ricc_lrcf_solver_options)

Returns

Z: Low-rank Cholesky factor of the Riccati equation solution, VectorArray from A.source.

pymor.algorithms.lrradi¶

Module Contents¶

`pymor.algorithms.lrradi`¶