## 3 Results
### Dielectric Inset Waveguide  
We consider a rectangular waveguide with a dispersive dielectric inset, similar to the one investigated in [53]. A 2D illustration of the waveguide is depicted in Figure 5, where \(w\) denotes its width, \(\ell\) is the length of the dielectric material, and \(d\) is a vacuum offset. The waveguide is further extended in the \(y\) direction by its height \(h\), which is not shown in Figure 5. The dielectric material has permittivity \(\varepsilon=\varepsilon_{0}\varepsilon_{\text{r}}\) and permeability \(\mu=\mu_{0}\mu_{\text{r}}\), where the subscripts \"0\" and \"r\" refer to the absolute value of the material property in vacuum and to its relative value for the given dielectric material,  
Figure 4: Cost-error relation for the approximations of the meromorphic function. The approximations are constructed with a dimension-adaptive weighted Leja interpolation algorithm [37] and with a degree-adaptive LAR-gPC algorithm [9; 50]. The size of the validation sample is \(Q=10^{5}\). The reference expected value is computed with quasi-MC integration based on a Sobol sample size equal to \(10^{8}\): (**a**) RMS validation error, (**b**) Expected value relative error. respectively. The relative material values are given by Debye relaxation models of second order [54], such that:  
\[\varepsilon_{\mathrm{r}} =\varepsilon_{\infty}+\frac{\varepsilon_{\mathrm{s,1}}-\varepsilon_ {\infty}}{1+(\iota\omega\tau_{\mathrm{c,1}})}+\frac{\varepsilon_{\mathrm{s,2}} -\varepsilon_{\infty}}{1+(\iota\omega\tau_{\mathrm{c,2}})},\] \[\mu_{\mathrm{r}} =\mu_{\infty}+\frac{\mu_{\mathrm{s,1}}-\mu_{\infty}}{1+\big{(} \iota\omega\tau_{\mathrm{\mu,1}}\big{)}}+\frac{\mu_{\mathrm{s,2}}-\mu_{\infty }}{1+\big{(}\iota\omega\tau_{\mathrm{\mu,2}}\big{)}},\]  
where \(\tau_{\mathrm{c}/\mu,1/2}\) are relaxation time constants, the subscript \"\(\infty\)\" refers to a very high frequency value of the relative material property, the subscript \"\(\infty\)\" to a static value of the relative material property, and \(\iota\) denotes the imaginary unit. The waveguide's input and output boundaries are denoted with \(\Gamma_{\mathrm{in}}\) and \(\Gamma_{\mathrm{out}}\), respectively. The remaining waveguide walls are assumed to be perfect electric conductors (PECs) and the corresponding boundary is denoted with \(\Gamma_{\mathrm{PEC}}\). We assume that the input port boundary \(\Gamma_{\mathrm{in}}\) is excited by an incoming plane wave. We further assume that the excitation coincides with the fundamental transverse electric (TE) mode and that all other propagation modes can be neglected, e.g., are quickly attenuated. Under these assumptions, the electric field \(\mathbf{E}\) can be computed by solving the boundary value problem:  
\[\
abla\times\left(\mu^{-1}\
abla\times\mathbf{E}\right)-\omega^{2 }\epsilon\mathbf{E} =0, \text{in }D, \tag{19a}\] \[\mathbf{n}_{\Gamma_{\mathrm{PEC}}}\times\mathbf{E} =0, \text{on }\Gamma_{\mathrm{PEC}},\] (19b) \[\mathbf{n}_{\Gamma_{\mathrm{in}}}\times\left(\
abla\times\mathbf{ E}\right)+\imath k_{\mathrm{z}}\mathbf{n}_{\Gamma_{\mathrm{in}}}\times\left( \mathbf{n}_{\Gamma_{\mathrm{in}}}\times\mathbf{E}\right) =\mathbf{U}^{\mathrm{i}}, \text{on }\Gamma_{\mathrm{in}},\] (19c) \[\mathbf{n}_{\Gamma_{\mathrm{out}}}\times\left(\
abla\times \mathbf{E}\right)+\imath k_{\mathrm{z}}\mathbf{n}_{\Gamma_{\mathrm{out}}} \times\left(\mathbf{n}_{\Gamma_{\mathrm{out}}}\times\mathbf{E}\right) =0, \text{on }\Gamma_{\mathrm{out}}, \tag{19d}\]  
where \(D\) is the computational domain, \(\omega=2\pi f\) the angular frequency, \(f\) the frequency, \(\mathbf{U}^{\mathrm{i}}\) the incoming plane wave, \(\mathbf{n}_{\Gamma_{\mathrm{in}}/\Gamma_{\mathrm{out}}/\Gamma_{\mathrm{PEC}}}\) are outwards-pointing normal vectors, and \(\mathbf{k}=(0,0,k_{z})\) the wavevector. The QoI is the reflection coefficient at the input port \(\Gamma_{\mathrm{in}}\), \(r=\left|\frac{\mathbf{F}_{\Gamma_{\mathrm{in}}}}{\mathbf{F}_{\Gamma_{\mathrm{ in}}}^{\prime}}\right|\in[0,1]\).