# pymor.reductors.interpolation¶

## Module Contents¶

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

Bitangential Hermite interpolation for LinearDelayModels.

Parameters

fom

The full-order LinearDelayModel to reduce.

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

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

Methods

 reconstruct Reconstruct high-dimensional vector from reduced vector u. reduce Bitangential Hermite interpolation.
reconstruct(u)[source]

Reconstruct high-dimensional vector from reduced vector u.

reduce(sigma, b, c, projection='orth')[source]

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 biorthogolized with respect to the E product

Returns

rom

Reduced-order model.

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

Bitangential Hermite interpolation for LTIModels.

Parameters

fom

The full-order LTIModel to reduce.

mu

Methods

 reduce Bitangential Hermite interpolation.
reduce(sigma, b, c, projection='orth')[source]

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 biorthogolized 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)[source]

Bitangential Hermite interpolation for SecondOrderModels.

Parameters

fom

The full-order SecondOrderModel to reduce.

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

Loewner bitangential Hermite interpolation reductor.

See [BG12].

Parameters

fom

TransferFunction or Model with a transfer_function attribute.

mu

Methods

 reconstruct Reconstruct high-dimensional vector from reduced vector u. reduce Realization-independent tangential Hermite interpolation.
reconstruct(u)[source]

Reconstruct high-dimensional vector from reduced vector u.

reduce(sigma, b, c)[source]

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.