n-D Loewner... Taming the curse of dimensionality
About and problem description
This page provides supplementary materials to the paper entitled "The Loewner framework for parametric systems: taming the curse of dimensionality", by A.C. Antoulas, I-V. Gosea and C. Poussot-Vassal (available as arXiv).
We provide additional (regularly updated) numerical benches and results, grounded on the above contribution. Next, we treat problems of three different nature:
Approximation from an available function (set #1):
the multi-variables function H(s1,s2,...sn) - either rational or non-rational / complex or real-valued - is available.Approximation from an available tensor data (set #2):
the (large-scale) n-D tensor data is available. In this configuration, the multi-variables function H(s1,s2,...sn) is unknown, but its evaluation at some points has been computed, and is provided. The n-D tensor denotes the evaluation of H at some points, thus leading to a n-dimensional matrix.Nonlinear eigenvalue approximation (set #3):
The objective, based on one of the above ingredients, is to construct a rational model that matches / approximates the data.
Data-sets description (Matlab format)
The proposed data-sets consist of a collection of examples, either constructed by us or proposed in the literature (other references, MOR Wiki...). For each case, we provide a ZIP file which contains data in Matlab format ".mat" files. More specifically ("X" denotes the name or the number of the experiment)
"tab_X.mat": contains the n-D tensor. More specifically, it contains
tab: the n-D tensor
p_c: the column interpolation points
p_r: the row interpolation points
"model_X.mat": contains the statistics and model, all gathered in the structure "stat". More specificalkly, it contains the following fields
H: the handle function
n: the number of variables
ord: the order of each variables
p_c: the interpolation points
w: the response of the original system at
c_full: the Lagrangian coefficients (obtained with the full n-D Loewner matrix null-space computation)
c_rec: the Lagrangian coefficients (obtained with the recursive 1-D Loewner matrix null-space computation)
flop_full: FLOPs (obtained with the full n-D Loewner matrix null-space computation)
flop_rec: FLOPs (obtained with the recursive 1-D Loewner matrix null-space computation)
time_full: CPU time (obtained with the full n-D Loewner matrix null-space computation)
time_rec: CPU time (obtained with the recursive 1-D Loewner matrix null-space computation)
Set #1: approximation from function
Here we consider first a collection of functions H(s1,s2,...,sn), that are rational / polynomial and accessible. We aim at demonstrating the scalability with respect to the number of variables "n".
>> Data (Matlab)
>> Our score sheet (PDF)
Additionally, we consider possible an other function set (possibly non-rational), obtained from literature
>> Data (Matlab)
>> Our score sheet (PDF)
Set #2: approximation from tensor
Here we consider examples where the tensor data is has been computed and we want to construct a rational parametric model. Most of the examples come from the MOR Wiki page. In most of these cases, the parametric realization is known but its parametric orders dependency is unknown.
>> Data (Matlab)
>> Our score sheet (PDF)
Set #3: NLEVP approximation
Here we consider a collection of data related to the non-linear eigenvalue problem.
>> Data (Matlab)
>> Our score sheet (PDF)