pymor.reductors.loewner

Module Contents

class pymor.reductors.loewner.LoewnerReductor(s, Hs, partitioning='even-odd', ordering='regular', conjugate=True, mimo_handling='full')

Bases: pymor.core.cache.CacheableObject

Reductor based on Loewner interpolation framework.

The reductor implements interpolation based on the Loewner framework as in [ALI17].

Parameters:

• s – NumPy array of shape (n,) containing the frequencies.

• Hs – NumPy array of shape (n, p, m) for MIMO systems with p outputs and m inputs or NumPy array of shape (n,) for SISO systems where the NumPy arrays resemble the transfer function samples. Alternatively, TransferFunction or Model with transfer_function attribute.

• partitioning – str or tuple of length 2. Strings can either be ‘even-odd’ or ‘half-half’ defining the partitioning rule. A user-defined partitioning can be defined by passing a tuple of the left and right indices. Defaults to even-odd.

• ordering – The ordering with respect to which the partitioning rule is executed. Can be either ‘magnitude’, ‘random’ or ‘regular’. Defaults to ‘regular’.

• conjugate – Whether to guarantee realness of reduced LTIModel by keeping complex conjugates in the same partitioning or not. If True will automatically generate conjugate data if necessary.

• mimo_handling –

  Option indicating how to treat MIMO systems. Can be:

  • 'random' for using random tangential directions.

  • 'full' for fully interpolating all input-output pairs.

  • Tuple (ltd, rtd) where ltd corresponds to left and rtd to right tangential directions.

Methods

loewner_quadruple	Constructs a Loewner quadruple as NumPy arrays.
reduce	Reduce using Loewner framework.

cache_region = 'memory'

loewner_quadruple()

Constructs a Loewner quadruple as NumPy arrays.

The Loewner quadruple [ALI17]

\[(\mathbb{L},\mathbb{L}_s,V,W)\]

consists of the Loewner matrix \(\mathbb{L}\), the shifted Loewner matrix \(\mathbb{L}_s\), left interpolation data \(V\) and right interpolation data \(W\).

Returns:

• L – Loewner matrix as a NumPy array.

• Ls – Shifted Loewner matrix as a NumPy array.

• V – Left interpolation data as a NumPy array.

• W – Right interpolation data as a NumPy array.

reduce(r=None, tol=1e-12)

Reduce using Loewner framework.

Parameters:

• r – Integer for target order of reduced model. If an interpolant with order less than r exists then the output will have the minimal order of an interpolant. Otherwise, the output will be an LTIModel with order r. If None the order of the reduced model will be the minimal order of an interpolant.

• tol – Truncation tolerance for rank of Loewner matrices.

Returns:

rom – Reduced LTIModel.