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

TL;DR

This work introduces new reproducing kernel Hilbert spaces of complex-valued functions, and forms the problem ofcomplex-valued interpolation with a kernel pair as minimum norm interpolation in these spaces, and combines the interpolant with a low-order rational function.

Abstract

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.

Figures (14)

• Figure 1: Left: Illustrations with the test function $F_{\mathrm{rat}} \in H^2_\mathrm{sym}(\Gamma_{0.1 + \epsilon}), \epsilon >0$ defined in \ref{['eq:Frat']}. Right: Convergence of the RMSE as a function of the number of (equidistant) training points. Solid lines: complex/real interpolation with the Szegö kernel for $H^2(\Gamma_\alpha)$, combined with the zero pseudo-kernel (blue) and the pseudo-kernel \ref{['equ:pseudo-kern-symm']} (red). Dashed lines: interpolation with a Gaussian kernel for the real and imaginary part separately (green) and polynomial interpolation on Chebyshev nodes (purple).
• Figure 1: Top: Dashed lines show the function to approximate. Black dots indicate the training data. Solid lines represent a bad approximation model which, however, is selected by the LOO criterion. Zoomed plot (gray background) highlights the influence of a wrongly identified pole. Bottom: Leave-on-out predictions, which show strong local variations between 4500s and 4520s. However, these variations do not significantly affect the values at the respective training points.
• Figure 1: Left: Parallel connection of (underdamped) series RLC circuits. Right: Black crosses indicate the distribution of $2 N_1=2000$ poles of the circuit admittance $Y_1$ in the complex plane. Red crosses indicate the two additional poles considered for the circuit admittance $Y_2$ with $2 N_2=2004$ poles. Blue line indicates the considered frequency range.
• Figure 1: Log-normal prior on hyper-parameter $\alpha$ for $|\Omega|=1$. Left: log-normal prior density of $\alpha$. Note that the mode of the density is indeed at $\alpha = \left| \Omega \right| = 1$. Right: prior density of $\log_{10}(\alpha)$. This is a Gaussian density with mean $\mu_\alpha / \ln(10) \approx 3.91$ and standard deviation $\sigma_\alpha / \ln(10) \approx 1.30$.
• Figure 2: Comparison of different model selection criteria for two benchmark problems. $\epsilon_{\mathrm{LOO},1}$ and $\epsilon_{\mathrm{LOO},2}$ denote the leave-on-out residual without and with retuning of hyper-parameters, respectively. The stabilized criterion $\epsilon_{\mathrm{LOO}, \mathrm{stab}}$ (with retuning) defined in \ref{['eq:LOOstab']} gives the best results.

Theorems & Definitions (31)

• Remark 1.1
• Definition 2.1: Complex RKHS
• Theorem 2.2: Moore-Aronszajn
• Theorem 2.3: Interpolation
• Theorem 2.4
• Definition 2.5: Complex/real RKHS
• Remark 2.6
• Proposition 2.7
• Remark 2.8
• Proposition 2.9