# Interpolatory rational model order reduction of parametric problems lacking uniform inf-sup stability

## TL;DR

This paper introduces a minimal rational interpolation method for efficiently approximating parametric PDE solution maps with meromorphic structure, demonstrating convergence and adaptivity through theoretical analysis and numerical experiments.

## Contribution

The paper presents a novel minimal rational interpolation approach for model order reduction of parametric PDEs with meromorphic solutions, including convergence analysis and adaptive error estimation.

## Key findings

- The method accurately captures resonance features of the solution map.
- The approach converges under certain structural conditions.
- Numerical experiments confirm theoretical predictions and effectiveness.

## Abstract

We present a technique for the approximation of a class of Hilbert space-valued maps which arise within the framework of Model Order Reduction for parametric partial differential equations, whose solution map has a meromorphic structure. Our MOR stategy consists in constructing an explicit rational approximation based on few snapshots of the solution, in an interpolatory fashion. Under some restrictions on the structure of the original problem, we describe a priori convergence results for our technique, hereafter called minimal rational interpolation, which show its ability to identify the main features (e.g. resonance locations) of the target solution map. We also investigate some procedures to obtain a posteriori error indicators, which may be employed to adapt the degree and the sampling points of the minimal rational interpolant. Finally, some numerical experiments are carried out to confirm the theoretical results and the effectiveness of our technique.