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 $i=1,2,\dots ,r$, 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

$$\begin{array}{rl}H({\sigma }_i)b_i& =\hat{H}({\sigma }_i)b_i,\\ c_i^{T}H({\sigma }_i)& =c_i^T\hat{H}({\sigma }_i),\\ c_i^T{H}^{\prime }({\sigma }_i)b_i& =c_i^T{\hat{H}}^{\prime }({\sigma }_i)b_i,\end{array}$$

for all $i=1,2,\dots ,r$.

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.