pymor.algorithms.lradi

Module Contents

Functions

lyap_lrcf_solver_options	Return available Lyapunov solvers with default options.
solve_lyap_lrcf	Compute an approximate low-rank solution of a Lyapunov equation.
projection_shifts_init	Find starting projection shifts.
projection_shifts	Find further projection shifts.

pymor.algorithms.lradi.lyap_lrcf_solver_options(lradi_tol=1e-10, lradi_maxiter=500, lradi_shifts='projection_shifts', projection_shifts_init_maxiter=20, projection_shifts_init_seed=None, projection_shifts_subspace_columns=6)

Return available Lyapunov solvers with default options.

Parameters

lradi_tol

See solve_lyap_lrcf.

lradi_maxiter

See solve_lyap_lrcf.

lradi_shifts

See solve_lyap_lrcf.

projection_shifts_init_maxiter

See projection_shifts_init.

projection_shifts_init_seed

See projection_shifts_init.

projection_shifts_subspace_columns

See projection_shifts.

Returns

A dict of available solvers with default solver options.

pymor.algorithms.lradi.solve_lyap_lrcf(A, E, B, trans=False, options=None)

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

See pymor.algorithms.lyapunov.solve_lyap_lrcf for a general description.

This function uses the low-rank ADI iteration as described in Algorithm 4.3 in [Kurschner16].

Parameters

A

The non-parametric Operator A.

E

The non-parametric Operator E or None.

B

The operator B as a VectorArray from A.source.

trans

Whether the first Operator in the Lyapunov equation is transposed.

options

The solver options to use (see lyap_lrcf_solver_options).

Returns

Z

Low-rank Cholesky factor of the Lyapunov equation solution, VectorArray from A.source.

pymor.algorithms.lradi.projection_shifts_init(A, E, B, shift_options)

Find starting projection shifts.

Uses Galerkin projection on the space spanned by the right-hand side if it produces stable shifts. Otherwise, uses a randomly generated subspace. See [Kurschner16], pp. 92-95.

Parameters

A

The Operator A from the corresponding Lyapunov equation.

E

The Operator E from the corresponding Lyapunov equation.

B

The VectorArray B from the corresponding Lyapunov equation.

shift_options

The shift options to use (see lyap_lrcf_solver_options).

Returns

shifts

A NumPy array containing a set of stable shift parameters.

pymor.algorithms.lradi.projection_shifts(A, E, V, Z, prev_shifts, shift_options)

Find further projection shifts.

Uses Galerkin projection on spaces spanned by LR-ADI iterates. See [Kurschner16], pp. 92-95.

Parameters

A

The Operator A from the corresponding Lyapunov equation.

E

The Operator E from the corresponding Lyapunov equation.

V

A VectorArray representing the currently computed iterate.

Z

A VectorArray representing the current approximate solution.

prev_shifts

A NumPy array containing the set of all previously used shift parameters.

shift_options

The shift options to use (see lyap_lrcf_solver_options).

Returns

shifts

A NumPy array containing a set of stable shift parameters.