pymor.reductors.interpolation

Module Contents

class pymor.reductors.interpolation.DelayBHIReductor(fom, mu=None)

Bases: GenericBHIReductor

Bitangential Hermite interpolation for LinearDelayModels.

Parameters:

• fom – The full-order LinearDelayModel to reduce.

• mu – Parameter values.

class pymor.reductors.interpolation.GenericBHIReductor(fom, mu=None)

Bases: pymor.core.base.BasicObject

Generic bitangential Hermite interpolation reductor.

This is a generic reductor for reducing any linear Model that has a transfer function that is a FactorizedTransferFunction (see [BG09]). The interpolation here is limited to only up to the first derivative. Interpolation points are assumed to be pairwise distinct.

In particular, given:

• interpolation points \(\sigma_i\),

• right tangential directions \(b_i\), and

• left tangential directions \(c_i\),

for \(i = 1, 2, \ldots, r\), which are closed under conjugation (if \(\sigma_i\) is real, then so are \(b_i\) and \(c_i\); if \(\sigma_i\) is complex, there is \(\sigma_j\) such that \(\sigma_j = \overline{\sigma_i}\), \(b_j = \overline{b_i}\), \(c_j = \overline{c_i}\)), this reductor finds a transfer function \(\hat{H}\) such that

\[\begin{split}H(\sigma_i) b_i & = \hat{H}(\sigma_i) b_i, \\ c_i^T H(\sigma_i) & = c_i^T \hat{H}(\sigma_i), \\ c_i^T H'(\sigma_i) b_i & = c_i^T \hat{H}'(\sigma_i) b_i,\end{split}\]

for all \(i = 1, 2, \ldots, r\).

Parameters:

• fom – The full-order Model to reduce.

• mu – Parameter values.

Methods

reconstruct	Reconstruct high-dimensional vector from reduced vector u.
reduce	Bitangential Hermite interpolation.

reconstruct(u)

Reconstruct high-dimensional vector from reduced vector u.

reduce(sigma, b, c, projection='orth')

Bitangential Hermite interpolation.

Parameters:

• sigma – Interpolation points (closed under conjugation), sequence of length r.

• b – Right tangential directions, NumPy array of shape (r, fom.dim_input).

• c – Left tangential directions, NumPy array of shape (r, fom.dim_output).

• 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

Returns:

rom – Reduced-order model.

class pymor.reductors.interpolation.LTIBHIReductor(fom, mu=None)

Bases: GenericBHIReductor

Bitangential Hermite interpolation for LTIModels.

Parameters:

• fom – The full-order LTIModel to reduce.

• mu – Parameter values.

Methods

reduce

Bitangential Hermite interpolation.

reduce(sigma, b, c, projection='orth')

Bitangential Hermite interpolation.

Parameters:

• sigma – Interpolation points (closed under conjugation), sequence of length r.

• b – Right tangential directions, NumPy array of shape (r, fom.dim_input).

• c – Left tangential directions, NumPy array of shape (r, fom.dim_output).

• 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 rational Arnoldi process (available only for SISO systems).

Returns:

rom – Reduced-order model.

class pymor.reductors.interpolation.SOBHIReductor(fom, mu=None)

Bases: GenericBHIReductor

Bitangential Hermite interpolation for SecondOrderModels.

Parameters:

• fom – The full-order SecondOrderModel to reduce.

• mu – Parameter values.

class pymor.reductors.interpolation.TFBHIReductor(fom, mu=None)

Bases: pymor.core.base.BasicObject

Loewner bitangential Hermite interpolation reductor.

See [BG12].

Parameters:

• fom – TransferFunction or Model with a transfer_function attribute.

• mu – Parameter values.

Methods

reconstruct	Reconstruct high-dimensional vector from reduced vector u.
reduce	Realization-independent tangential Hermite interpolation.

reconstruct(u)

Reconstruct high-dimensional vector from reduced vector u.

reduce(sigma, b, c)

Realization-independent tangential Hermite interpolation.

Parameters:

• sigma – Interpolation points (closed under conjugation), sequence of length r.

• b – Right tangential directions, NumPy array of shape (r, fom.dim_input).

• c – Left tangential directions, NumPy array of shape (r, fom.dim_output).

Returns:

lti – The reduced-order LTIModel interpolating the transfer function of fom.