Fast recovery of parametric eigenvalues depending on several parameters and location of high order exceptional points

TL;DR

This work tackles the challenge of tracking parametric non-Hermitian eigenvalues and locating high-order exceptional points (EPs) across large parameter spaces. It introduces a Partial Characteristic Polynomial (PCP) built from a subset of eigenvalues and expanded via multivariate Taylor series to yield regular coefficients that can be used with standard root-finding to recover eigenvalues and EP locations. Key contributions include a stable bordered-matrix approach for computing eigenvalue derivatives, hierarchical techniques to efficiently assemble PCP coefficients, and robust strategies for solving the resulting multivariate polynomial systems while filtering spurious roots. The method demonstrates scalability to large sparse matrices from finite-element discretizations and yields accurate EP locations in applications ranging from acoustic waveguides to 3D cavities, offering substantial computational savings over brute-force approaches.

Abstract

A numerical algorithm is proposed to deal with parametric eigenvalue problems involving non-Hermitian matrices and is exploited to find location of defective eigenvalues in the parameter space of non-Hermitian parametric eigenvalue problems. These non-Hermitian degeneracies also called exceptional points (EP) have raised considerable attention in the scientific community as these can have a great impact in a variety of physical problems. The method first requires the computation of high order derivatives of a few selected eigenvalues with respect to each parameter involved. The second step is to recombine these quantities to form new coefficients associated with a partial characteristic polynomial (PCP). By construction, these coefficients are regular functions in a large domain of the parameter space which means that the PCP allows one to recover the selected eigenvalues as well as the localization of high order EPs by simply using standard root-finding algorithms. The versatility of the proposed approach is tested on several applications, from mass-spring systems to guided acoustic waves with absorbing walls and room acoustics. The scalability of the method to large sparse matrices arising from conventional discretization techniques such as the finite element method is demonstrated. The proposed approach can be extended to a large number of applications where EPs play an important role in quantum mechanics, optics and photonics or in mechanical engineering.

Figures (13)

• Figure 1: Toy model with two parameters.
• Figure 2: Taylor coefficients of the eigenvalue $(\hat{\lambda}_k)_{\boldsymbol{\alpha}} = \partial^{\boldsymbol{\alpha}}\lambda_k(\boldsymbol{\nu}_0) / \boldsymbol{\alpha}!$ with $k=0,\, 1,\, 2$. Computed with $\boldsymbol{\nu}_0=(1, 1)$ and $D=7$.
• Figure 3: Taylor coefficients of the PCP coefficients defined in \ref{['eq:ak_Taylor']} computed with $\boldsymbol{\nu}_0=(1, 1)$ and $D=7$.
• Figure 4: Acoustical waveguide lined with 2 admittance boundary condition.
• Figure 5: (a) Error $E$ from Eq. \ref{['eq:error_wg']} with respect to the number $L$ of eigenvalues in the PCP and the modulus of the perturbation parameter $\boldsymbol{\nu} = \boldsymbol{\nu}_0 + \mathrm{e}^{0.3\mathrm{i}}\epsilon 11$. (b) Evolution of the round-off error with respect to the number of eigenvalues in the set when $\epsilon=0$. Values obtained from hierarchical scheme with the initial guess $\boldsymbol{\nu}_0=(4.76715+7.01265\mathrm{i}, 2.470 +2.89872\mathrm{i})$ and combining 2 to 19 eigenvalues with 5 derivatives in all directions.