pymor.reductors.aaa

Module Contents

class pymor.reductors.aaa.PAAAReductor(sampling_values, samples, force_real=True, nsp_tol=1e-16, post_process=True, L_rk_tol=1e-08, parameters=None)[source]

Bases: pymor.core.base.BasicObject

Reductor implementing the parametric AAA algorithm.

The reductor implements the parametric AAA algorithm using sampled data. Use generate_samples to generate data from a model beforehand. MIMO and non-parametric data is accepted. See [NSeteT18] for the non-parametric and [CRBG23] for the parametric version of the algorithm. The reductor computes approximations based on the multivariate barycentric form \(H(s,p,...)\) (see make_bary_func), where the inputs \(s,p,...\) are referred to as the variables. Further, \(s\) is the Laplace variable and \(p\) as well as other remaining inputs are parameters.

Note

The dimension of the Loewner matrix which will be assembled in the algorithm has len(sampling_values[0])*...*len(sampling_values[-1]) rows. This reductor should only be used with a low number of variables.

Parameters:

  • sampling_values – Values where sample data has been evaluated. Sampling values are represented as a list of NumPy arrays such that sampling_values[i] corresponds to sampling values of the i-th variable given as a NumPy array. The first variable is the Laplace variable. In the non-parametric case (i.e., the only variable is the Laplace variable) this can also be a NumPy array representing the sampling values.

  • samples – Data sampled at the values specified in sampling_values as a NumPy array. Samples are represented as a tensor S. E.g., for 3 inputs S[i,j,k] corresponds to the sampled value at (sampling_values[0][i],sampling_values[1][j],sampling_values[2][k]). The samples (i.e., S[i,j,k]) can be scalars or 2-dimensional NumPy arrays. E.g., in the MIMO case S[i,j,k] represents a matrix of dimension dim_output times dim_input.

  • force_real – Whether to compute complex conjugates of first sampling variables and enforce interpolation in complex conjugate pairs (allows for constructing real system matrices).

  • nsp_tol – Tolerance for null space of higher-dimensional Loewner matrix to check for interpolation or convergence.

  • post_process – Whether to do post-processing or not. If the Loewner matrix has a null space of dimension greater than 1, it is assumed that the algorithm converged to a non-minimal order interpolant which may cause numerical issues. In this case, the post-processing procedure computes an interpolant of minimal order.

  • L_rk_tol – Tolerance for ranks of 1-D Loewner matrices computed in post-processing.

  • parameters – Optional Parameters or parameter-dimension mapping for the resulting transfer function. Defaults to a single vector parameter named p. Parameter sampling axes follow the order used by to_numpy.

itpl_part[source]

A nested list such that itpl_part[i] corresponds to indices of interpolated values with respect to the i-th variable. I.e., self.sampling_values[i][itpl_part[i]] represents a list of all interpolated samples of the i-th variable.

Methods

reduce

Reduce using p-AAA.
generate_samples[source]
reduce(tol=1e-07, itpl_part=None, max_itpl=None)[source]

Reduce using p-AAA.

Parameters:

  • tol – Convergence tolerance for relative error of rom over the set of samples.

  • itpl_part – Initial partition for interpolation values. Should be None or a nested list such that itpl_part[i] corresponds to indices of interpolated values with respect to the i-th variable. I.e., self.sampling_values[i][itpl_part[i]] represents a list of all initially interpolated samples of the i-th variable. If None p-AAA will start with no interpolated values.

  • max_itpl – Maximum number of interpolation points to use with respect to each variable. Should be None or a list such that self.num_vars == len(max_itpl). If None max_itpl[i] will be set to len(self.sampling_values[i]) - 1.

Returns:

rom – Reduced TransferFunction model.

pymor.reductors.aaa.make_bary_func(itpl_nodes, itpl_vals, coefs, removable_singularity_tol=1e-14)[source]

Return function for (multivariate) barycentric form.

The multivariate barycentric form for two variables is given by

\[H(s,p) = \frac{\sum_{i=1}^{k}\sum_{j=1}^{q}\frac{\alpha_{ij}H(s_i,p_j)}{(s-s_i)(p-p_j)}} {\sum_{i=1}^{k}\sum_{j=1}^{q}\frac{\alpha_{ij}}{(s-s_i)(p-p_j)}}\]

where for \(i=1,\ldots,k\) and \(j=1,\ldots,q\) we have that \(s_i\) and \(p_j\) are interpolation nodes, \(H(s_i,p_j)\) are interpolation values and \(\alpha_{ij}\) represent the barycentric coefficients. This implementation can also handle single-variable barycentric forms as well as barycentric forms with more than two variables.

Parameters:

  • itpl_nodes – Nested list such that itpl_nodes[i] contains interpolation nodes of the i-th variable.

  • itpl_vals – Vector of interpolation values with len(itpl_nodes[0])*...*len(itpl_nodes[-1]) entries.

  • coefs – Vector of barycentric coefficients with len(itpl_nodes[0])*...*len(itpl_nodes[-1]) entries.

  • removable_singularity_tol – Tolerance for evaluating the barycentric function at a removable singularity and performing pole cancellation.

Returns:

bary_func – (Multi-variate) rational function in barycentric form.