Universal Methods for Nonlinear Spectral Problems

TL;DR

This work addresses nonlinear spectral problems where the spectral parameter enters nonlinearly and standard discretizations can produce spectral pollution or invisibility. Itdevelops a universally convergent computational framework based on continuity in the gap topology and the SCI hierarchy, leveraging nonlinear injection moduli and two evaluation schemes to achieve monotone convergence for spectra and pseudospectra. The authors establish optimality bounds within the SCI framework and show that Hermiticity does not automatically simplify the nonlinear case, with rigorous results and lower/upper bounds. Computational demonstrations across nonlinear shifts, Klein–Gordon, acoustic-wave boundaries, time-fractional viscoelastic beams, and delayed predator–prey systems illustrate robustness, accuracy, and broad applicability for physics, engineering, and biology, including potential computer-assisted proofs.

Abstract

Nonlinear spectral problems arise across a range of fields, including mechanical vibrations, fluid-solid interactions, and photonic crystals. Discretizing infinite-dimensional nonlinear spectral problems often introduces significant computational challenges, particularly spectral pollution and invisibility, which can distort or obscure the true underlying spectrum. We present the first general, convergent computational method for computing the spectra and pseudospectra of nonlinear spectral problems. Our approach uses new results on nonlinear injection moduli and requires only minimal continuity assumptions: specifically, continuity with respect to the gap metric on operator graphs, making it applicable to a broad class of problems. We use the Solvability Complexity Index (SCI) hierarchy, which has recently been used to resolve the classical linear problem, to systematically classify the computational complexity of nonlinear spectral problems. Our results establish the optimality of the method and reveal that Hermiticity does not necessarily simplify the computational complexity of these nonlinear problems. Comprehensive examples -- including nonlinear shifts, Klein--Gordon equations, wave equations with acoustic boundary conditions, time-fractional beam equations, and biologically inspired delay differential equations -- demonstrate the robustness, accuracy, and broad applicability of our methodology.

Figures (11)

• Figure 1: Pseudosepctra of $40\times 40$ truncation of the nonlinear shift $S-f(z)S^*$ for $f(z)=\sin(4z)(|z|^2+1)$. The color corresponds to a logarithmic grid of $\epsilon$ values. The spectrum of the truncated problem is $\{\pi m/4:m\in\mathbb{Z}\}$, which is completely different to the spectrum ${\mathrm{Sp}}(S-f(z)S^*)=\{z\in\mathbb{C}:|f(z)|=1\}$.
• Figure 3: Output of \ref{['alg1']} for the Klein--Gordon spectral problem. The green and red dots correspond to the eigenvalues of the quadratic eigenvalue problem resulting from truncation. The two eigenvalues shown in red are spurious and are highlighted by the arrows.
• Figure 4: Convergence of local minima of $\gamma_{n}(z,T)$ (essentially an adaptation of \ref{['alg2']}) for three representative eigenvalues in the discrete spectrum. The legend shows the (rounded) approximate location of the eigenvalues. The plateauing of the curves around $10^{-15}$ is due to reaching machine precision.
• Figure 5: Left: Eigenvalues of the quadratic eigenvalue problem after domain truncation followed by FEM discretization. Right: The minimum of the absolute value of the eigenvalues for different discretization sizes $n$.
• Figure 6: Convergence of \ref{['alg1']} (as we move from left to right) for the wave equation example with acoustic boundary condition. In this case, the spectrum is $\{z\in\mathbb{C}:\mathrm{Im}(z)\geq 0\}$.

Theorems & Definitions (50)

• Definition 2.1: Computational problem
• Definition 2.2: General algorithm
• Example 2.1
• Remark 1: Verification
• Definition 2.3
• Remark 2
• Theorem 2.1: Basic properties of generalized convergence kato2013perturbation
• Definition 2.4: Continuous operator pencils
• Remark 3: Domain of the pencil
• Example 2.2: Linear families