On adaptive sampling approaches for multivariate macromodeling of parameterized systems

Behavioral macromodels have gained considerable attention due to the difficulty to perform numerical simulations of complex systems, both due to the long runtime and the excessive memory requirements. Suitable Model Order Reduction techniques, either via truncation or projection of large-scale first-principle formulations, or via data-driven identification from input-output response measurements, have been proven adequate for the derivation of compact macromodels, especially in the field of Electronic Design Automation. Recently, also the use of parametrized macromodels has become popular, because it enables analysis and optimization with respect to physical or geometrical parameters. Macromodel extration in this case is particularly challenging, since model complexity scales exponentially with the number of parameters, resulting in an expensive increase of the costs. A solution is to develop a suitable model structure and an associated adaptive sampling algorithm, with the purpose of reducing the number of input-output responses that need to be collected for model training. Adaptive Sampling strategies aim to find new sub-optimal points where the full (large-scale) system is solved, in order to improve the model in an iterative manner. We focus this work on macromodeling of complex Linear and Time-Invariant systems. The modeling framework that is here considered to obtain the parametrized macromodel is purely data-driven and is known as the Parametrized Sanathanan-Koerner (PSK) Iteration. The framework is based on a representation of the model transfer function as a ration of two rational functions expressed in a pole-residue form, with constant basis poles and parameterized residues. In this setting, adaptive approaches were already presented for a small set of parameters, but they do not scale adequately when the number of parameters is high. The problem was mainly linked to an inadequate choice of the parametric basis used in the model prototype to approximate the parametric variation. In this context it was pointed out the good high-dimensional properties of the Gaussian Radial Basis Functions (RBFs) as parametric basis functions. In this case, the macromodel complexity can grow adaptively, e.g. the number of basis functions can increase and can be even adaptively optimized, e.g. the hyperparameters such as the RBFs centers and the width can be tuned. Therefore, one must more appropriately refer to this challenge as Adaptive Macromodeling. In this thesis, we investigated some techniques for adaptive macromodeling to be used for parametrized systems with a moderate number of parameters, but adopting a model structure that has the potential to scale favorably. We used the Gaussian RBFs as parametric basis functions with the same centers for the numerator and the denominator. We proposed four new adaptive procedures to be compared with an explorative procedure based on Sobol sequence, which it was already known before this work. These procedures were tested on a real parametrized system which is a typical system for electronic high-speed applications, and on synthetic examples which were created with similar features with respect to real systems. The results that has been obtained show the same performance in terms of accuracy for all the procedures presented. Therefore, it seems there is no need to use non-standard adaptive procedures for systems which are similar to the ones that were tested, in particular when the number of parameters is up to three.