pymor.algorithms.lrradi

Module Contents

Functions

ricc_lrcf_solver_options	Returns available Riccati equation solvers with default solver options.
solve_ricc_lrcf	Compute an approximate low-rank solution of a Riccati equation.
hamiltonian_shifts_init	Compute initial shift parameters for low-rank RADI iteration.
hamiltonian_shifts	Compute further shift parameters for low-rank RADI iteration.

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_init_seed=None, hamiltonian_shifts_subspace_columns=6)

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_init_seed

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, trans=False, options=None)

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.

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.hamiltonian_shifts_init(A, E, B, C, shift_options)

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.hamiltonian_shifts(A, E, B, R, K, Z, shift_options)

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.