pymor.reductors.h2

Reductors based on H2-norm.

Module Contents

class pymor.reductors.h2.GapIRKAReductor(fom, mu=None, solver_options=None)[source]

Bases: GenericIRKAReductor

Gap-IRKA reductor.

Parameters:

Methods

reduce

Reduce using gap-IRKA.
reduce(rom0_params, tol=0.0001, maxit=100, num_prev=1, conv_crit='sigma', projection='orth')[source]

Reduce using gap-IRKA.

See [BBG22] Algorithm 1.

Parameters:

  • rom0_params –

    Can be:

    • order of the reduced model (a positive integer),

    • initial interpolation points (a 1D NumPy array),

    • dict with 'sigma', 'b', 'c' as keys mapping to initial interpolation points (a 1D NumPy array), right tangential directions (VectorArray from fom.input_space), and left tangential directions (VectorArray from fom.output_space), all of the same length (the order of the reduced model),

    • initial reduced-order model (LTIModel).

    If the order of reduced model is given, initial interpolation data is generated randomly.

  • tol – Tolerance for the convergence criterion.

  • maxit – Maximum number of iterations.

  • num_prev – Number of previous iterations to compare the current iteration to. A larger number can avoid occasional cyclic behavior.

  • conv_crit –

    Convergence criterion:

    • 'sigma': relative change in interpolation points

    • 'htwogap': \(\mathcal{H}_2-gap\) distance of reduced-order models divided by \(\mathcal{L}_2\) norm of new reduced-order model

    • 'ltwo': relative \(\mathcal{L}_2\) distance of reduced-order models

  • projection –

    Projection method:

    • 'orth': projection matrix is orthogonalized with respect to the Euclidean inner product,

    • 'biorth': projection matrix is orthogonalized with respect to the E product.

Returns:

rom – Reduced LTIModel model.

class pymor.reductors.h2.GenericIRKAReductor(fom, mu=None)[source]

Bases: pymor.core.base.BasicObject

Generic IRKA related reductor.

Parameters:

Methods

reconstruct

Reconstruct high-dimensional vector from reduced vector u.
reconstruct(u)[source]

Reconstruct high-dimensional vector from reduced vector u.

class pymor.reductors.h2.IRKAReductor(fom, mu=None)[source]

Bases: GenericIRKAReductor

Iterative Rational Krylov Algorithm reductor.

Parameters:

Methods

reduce

Reduce using IRKA.
reduce(rom0_params, tol=0.0001, maxit=100, num_prev=1, force_sigma_in_rhp=False, projection='orth', conv_crit='sigma', compute_errors=False)[source]

Reduce using IRKA.

See [GAB08] (Algorithm 4.1) and [ABG10] (Algorithm 1).

Parameters:

  • rom0_params –

    Can be:

    • order of the reduced model (a positive integer),

    • initial interpolation points (a 1D NumPy array),

    • dict with 'sigma', 'b', 'c' as keys mapping to initial interpolation points (a 1D NumPy array), right tangential directions (NumPy array of shape (len(sigma), fom.dim_input)), and left tangential directions (NumPy array of shape (len(sigma), fom.dim_input)),

    • initial reduced-order model (LTIModel).

    If the order of reduced model is given, initial interpolation data is generated randomly.

  • tol – Tolerance for the convergence criterion.

  • maxit – Maximum number of iterations.

  • num_prev – Number of previous iterations to compare the current iteration to. Larger number can avoid occasional cyclic behavior of IRKA.

  • force_sigma_in_rhp – If False, new interpolation are reflections of the current reduced-order model’s poles. Otherwise, only poles in the left half-plane are reflected.

  • projection –

    Projection method:

    • 'orth': projection matrices are orthogonalized with respect to the Euclidean inner product

    • 'biorth': projection matrices are biorthogonalized with respect to the E product

    • 'arnoldi': projection matrices are orthogonalized using the Arnoldi process (available only for SISO systems).

  • conv_crit –

    Convergence criterion:

    • 'sigma': relative change in interpolation points

    • 'h2': relative \(\mathcal{H}_2\) distance of reduced-order models

  • compute_errors –

    Should the relative \(\mathcal{H}_2\)-errors of intermediate reduced-order models be computed.

    Warning

    Computing \(\mathcal{H}_2\)-errors is expensive. Use this option only if necessary.

Returns:

rom – Reduced LTIModel model.

class pymor.reductors.h2.OneSidedIRKAReductor(fom, version, mu=None)[source]

Bases: GenericIRKAReductor

One-Sided Iterative Rational Krylov Algorithm reductor.

Parameters:

  • fom – The full-order LTIModel to reduce.

  • version –

    Version of the one-sided IRKA:

    • 'V': Galerkin projection using the input Krylov subspace,

    • 'W': Galerkin projection using the output Krylov subspace.

  • mu – Parameter values.

Methods

reduce

Reduce using one-sided IRKA.
reduce(rom0_params, tol=0.0001, maxit=100, num_prev=1, force_sigma_in_rhp=False, projection='orth', conv_crit='sigma', compute_errors=False)[source]

Reduce using one-sided IRKA.

Parameters:

  • rom0_params –

    Can be:

    • order of the reduced model (a positive integer),

    • initial interpolation points (a 1D NumPy array),

    • dict with 'sigma', 'b', 'c' as keys mapping to initial interpolation points (a 1D NumPy array), right tangential directions (NumPy array of shape (len(sigma), fom.dim_input)), and left tangential directions (NumPy array of shape (len(sigma), fom.dim_input)),

    • initial reduced-order model (LTIModel).

    If the order of reduced model is given, initial interpolation data is generated randomly.

  • tol – Tolerance for the largest change in interpolation points.

  • maxit – Maximum number of iterations.

  • num_prev – Number of previous iterations to compare the current iteration to. A larger number can avoid occasional cyclic behavior.

  • force_sigma_in_rhp – If False, new interpolation are reflections of the current reduced-order model’s poles. Otherwise, only poles in the left half-plane are reflected.

  • projection –

    Projection method:

    • 'orth': projection matrix is orthogonalized with respect to the Euclidean inner product,

    • 'Eorth': projection matrix is orthogonalized with respect to the E product.

  • conv_crit –

    Convergence criterion:

    • 'sigma': relative change in interpolation points,

    • 'h2': relative \(\mathcal{H}_2\) distance of reduced-order models.

  • compute_errors –

    Should the relative \(\mathcal{H}_2\)-errors of intermediate reduced-order models be computed.

    Warning

    Computing \(\mathcal{H}_2\)-errors is expensive. Use this option only if necessary.

Returns:

rom – Reduced LTIModel model.

class pymor.reductors.h2.TFIRKAReductor(fom, mu=None)[source]

Bases: GenericIRKAReductor

Realization-independent IRKA reductor.

See [BG12].

Parameters:

  • fom – TransferFunction or Model with a transfer_function attribute, with eval_tf and eval_dtf methods that should be defined at least over the open right half of the complex plane.

  • mu – Parameter values.

Methods

reconstruct

Reconstruct high-dimensional vector from reduced vector u.

reduce

Reduce using TF-IRKA.
reconstruct(u)[source]

Reconstruct high-dimensional vector from reduced vector u.

reduce(rom0_params, tol=0.0001, maxit=100, num_prev=1, force_sigma_in_rhp=False, conv_crit='sigma', compute_errors=False)[source]

Reduce using TF-IRKA.

Parameters:

  • rom0_params –

    Can be:

    • order of the reduced model (a positive integer),

    • initial interpolation points (a 1D NumPy array),

    • dict with 'sigma', 'b', 'c' as keys mapping to initial interpolation points (a 1D NumPy array), right tangential directions (NumPy array of shape (len(sigma), fom.dim_input)), and left tangential directions (NumPy array of shape (len(sigma), fom.dim_input)),

    • initial reduced-order model (LTIModel).

    If the order of reduced model is given, initial interpolation data is generated randomly.

  • tol – Tolerance for the convergence criterion.

  • maxit – Maximum number of iterations.

  • num_prev – Number of previous iterations to compare the current iteration to. Larger number can avoid occasional cyclic behavior of TF-IRKA.

  • force_sigma_in_rhp – If False, new interpolation are reflections of the current reduced-order model’s poles. Otherwise, only poles in the left half-plane are reflected.

  • conv_crit –

    Convergence criterion:

    • 'sigma': relative change in interpolation points

    • 'h2': relative \(\mathcal{H}_2\) distance of reduced-order models

  • compute_errors –

    Should the relative \(\mathcal{H}_2\)-errors of intermediate reduced-order models be computed.

    Warning

    Computing \(\mathcal{H}_2\)-errors is expensive. Use this option only if necessary.

Returns:

rom – Reduced LTIModel model.

class pymor.reductors.h2.TSIAReductor(fom, mu=None)[source]

Bases: GenericIRKAReductor

Two-Sided Iteration Algorithm reductor.

Parameters:

Methods

reduce

Reduce using TSIA.
reduce(rom0_params, tol=0.0001, maxit=100, num_prev=1, projection='orth', conv_crit='sigma', compute_errors=False)[source]

Reduce using TSIA.

See [XZ11] (Algorithm 1) and [BKohlerS11].

In exact arithmetic, TSIA is equivalent to IRKA (under some assumptions on the poles of the reduced model). The main difference in implementation is that TSIA computes the Schur decomposition of the reduced matrices, while IRKA computes the eigenvalue decomposition. Therefore, TSIA might behave better for non-normal reduced matrices.

Parameters:

  • rom0_params –

    Can be:

    • order of the reduced model (a positive integer),

    • initial interpolation points (a 1D NumPy array),

    • dict with 'sigma', 'b', 'c' as keys mapping to initial interpolation points (a 1D NumPy array), right tangential directions (NumPy array of shape (len(sigma), fom.dim_input)), and left tangential directions (NumPy array of shape (len(sigma), fom.dim_input)),

    • initial reduced-order model (LTIModel).

    If the order of reduced model is given, initial interpolation data is generated randomly.

  • tol – Tolerance for the convergence criterion.

  • maxit – Maximum number of iterations.

  • num_prev – Number of previous iterations to compare the current iteration to. Larger number can avoid occasional cyclic behavior of TSIA.

  • projection –

    Projection method:

    • 'orth': projection matrices are orthogonalized with respect to the Euclidean inner product

    • 'biorth': projection matrices are biorthogonalized with respect to the E product

  • conv_crit –

    Convergence criterion:

    • 'sigma': relative change in interpolation points

    • 'h2': relative \(\mathcal{H}_2\) distance of reduced-order models

  • compute_errors –

    Should the relative \(\mathcal{H}_2\)-errors of intermediate reduced-order models be computed.

    Warning

    Computing \(\mathcal{H}_2\)-errors is expensive. Use this option only if necessary.

Returns:

rom – Reduced LTIModel.

class pymor.reductors.h2.VectorFittingReductor(s, Hs, weights=None, conjugate=True)[source]

Bases: pymor.core.base.BasicObject

Vector fitting reductor.

Only for single-input single-output (SISO) systems.

Parameters:

  • s – Sampling points in the complex plane as a 1D NumPy array.

  • Hs – Transfer function values at the sampling points s as a 1D NumPy array. Alternatively, TransferFunction or Model with transfer_function attribute.

  • weights – Weights in the weighted least squares error as a 1D NumPy array. If not given, it is set to a vector of ones.

  • conjugate – Whether to include conjugated data.

Methods

reduce

Reduce using vector fitting.
reduce(r=None, lambdas=None, tol=0.0001, maxit=100)[source]

Reduce using vector fitting.

Based on [DrmavcGB15].

Parameters:

  • r – Reduced order (if not given, it is len(lambdas)).

  • lambdas – Initial poles (if not given, it is set to -np.logspace(-1, 0, r)).

  • tol – Tolerance for the convergence criterion.

  • maxit – Maximum number of iterations.

Returns:

rom – Reduced-order LTIModel.