# Rational kernel-based interpolation for complex-valued frequency response functions

arXiv: https://arxiv.org/abs/2307.13484 · HTML: https://mlsanity.com/paper/2307.13484/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 49 references that are in the map.
- First posted on arXiv 25 Jul 2023.

## Builds on (most cited first, 9 of 9)

- [Numerical issues in maximum likelihood parameter estimation for Gaussian process interpolation](https://mlsanity.com/paper/2101.09747/) (2021)
- [Complex-Valued Gaussian Processes for Regression](https://mlsanity.com/paper/1511.05710/) (2015)
- [Parameter selection in Gaussian process interpolation: an empirical study of selection criteria](https://mlsanity.com/paper/2107.06006/) (2021)
- [Surrogate modeling based on resampled polynomial chaos expansions](https://mlsanity.com/paper/1810.09116/) (2018)
- [The p-AAA algorithm for data driven modeling of parametric dynamical systems](https://mlsanity.com/paper/2003.06536/) (2020)
- [Complex-Valued Kernel Methods for Regression](https://mlsanity.com/paper/1610.09915/) (2016)
- [Sparse Bayesian Learning for Complex-Valued Rational Approximations](https://mlsanity.com/paper/2206.02523/) (2022)
- [A Blackbox Yield Estimation Workflow with Gaussian Process Regression Applied to the Design of Electromagnetic Devices](https://mlsanity.com/paper/2003.13278/) (2020)
- [Toward a certified greedy Loewner framework with minimal sampling](https://mlsanity.com/paper/2303.01015/) (2023)

## Built on by (most cited first, 1 of 1)

- [Maximum a Posteriori Estimation for Linear Structural Dynamics Models Using Bayesian Optimization with Rational Polynomial Chaos Expansions](https://mlsanity.com/paper/2408.03569/) (2024)

## Abstract (arXiv)

This work is concerned with the kernel-based approximation of a complex-valued function from data, where the frequency response function of a partial differential equation in the frequency domain is of particular interest. In this setting, kernel methods are employed more and more frequently, however, standard kernels do not perform well. Moreover, the role and mathematical implications of the underlying pair of kernels, which arises naturally in the complex-valued case, remain to be addressed. We introduce new reproducing kernel Hilbert spaces of complex-valued functions, and formulate the problem of complex-valued interpolation with a kernel pair as minimum norm interpolation in these spaces. Moreover, we combine the interpolant with a low-order rational function, where the order is adaptively selected based on a new model selection criterion. Numerical results on examples from different fields, including electromagnetics and acoustic examples, illustrate the performance of the method, also in comparison to available rational approximation methods.