[Paper Review] Automatic rational approximation and linearization of nonlinear eigenvalue problems

Pieter Lietaert, Javier J. Pérez+2 more|arXiv (Cornell University)|Jan 25, 2018

Model Reduction and Neural Networks4 citations

TL;DR

This paper presents an automatic rational approximation method for nonlinear eigenvalue problems using the AAA algorithm to construct rational interpolants, which are then linearized via a state-space formulation compatible with Compact Rational Krylov methods. The approach achieves competitive accuracy with fewer poles than NLEIGS and requires no user-defined pole selection, enabling efficient solution of large-scale problems with minimal user input.

ABSTRACT

We present a method for solving nonlinear eigenvalue problems using rational approximation. The method uses the AAA method by Nakatsukasa, S\\`{e}te, and Trefethen to approximate the nonlinear eigenvalue problem by a rational eigenvalue problem and is embedded in the state space representation of a rational polynomial by Su and Bai. The advantage of the method, compared to related techniques such as NLEIGS and infinite Arnoldi, is the efficient computation by an automatic procedure. In addition, a set-valued approach is developed that allows building a low degree rational approximation of a nonlinear eigenvalue problem. The method perfectly fits the framework of the Compact rational Krylov methods (CORK and TS-CORK), allowing to efficiently solve large scale nonlinear eigenvalue problems. Numerical examples show that the presented framework is competitive with NLEIGS and usually produces smaller linearizations with the same accuracy but with less effort for the user.

Motivation & Objective

• To develop a fully automatic method for rational approximation in nonlinear eigenvalue problems without requiring user expertise in complex analysis or function-specific tuning.
• To reduce the number of poles in rational approximations compared to existing methods like NLEIGS, especially for problems with multiple nonlinear terms.
• To integrate rational approximation with the Compact Rational Krylov (CORK) framework for efficient large-scale eigenvalue computation.
• To enable low-rank structure exploitation in the matrix functions without requiring low-rank assumptions on the coefficient matrices.
• To demonstrate that set-valued rational approximation using AAA outperforms individual approximation and potential theory-based methods in terms of approximation degree and convergence.

Proposed method

• Uses the AAA algorithm to automatically compute rational approximations of nonlinear matrix functions over user-defined sets, adapting pole and zero locations without user input.
• Applies the state-space representation from Su and Bai (2011) to linearize the resulting rational eigenvalue problem into a standard form suitable for Krylov solvers.
• Employs a set-valued approximation strategy that shares poles and interpolation points across multiple nonlinear functions, improving efficiency and reducing total degree.
• Embeds the linearized problem into the CORK (Compact Rational Krylov) framework, enabling efficient solution of large-scale problems via rational Krylov recycling.
• Utilizes the same linearization structure for both full-rank and low-rank coefficient matrices, allowing exploitation of low-rank structure when present.
• Employs Krylov methods with multiple shifts on the real axis to compute Ritz values and monitor convergence to a residual tolerance of 10−12.

Experimental results

Research questions

• RQ1 Can the AAA algorithm be effectively used to automatically construct rational approximations for nonlinear eigenvalue problems without requiring domain-specific knowledge or manual pole selection?
• RQ2 How does the performance of set-valued rational approximation using AAA compare to NLEIGS in terms of approximation degree and convergence for problems with multiple nonlinear functions?
• RQ3 To what extent can the CORK framework be leveraged to efficiently solve large-scale nonlinear eigenvalue problems arising from rational approximations?
• RQ4 Does the proposed method maintain high accuracy while using fewer poles than potential theory-based methods like Leja–Bagby?
• RQ5 How does the method perform on real-world problems with complex singularities, such as the car cavity model with branch points and unbounded behavior?

Key findings

• The AAA-based rational approximation achieved a relative error of 10−13 with only 42 poles for the car cavity problem, even in the presence of singularities and branch cuts.
• For a smaller test region below the first singularity at 514i, AAA required only 11 poles to achieve 10−12 accuracy, outperforming Leja–Bagby poles which required 40.
• Convergence of Ritz values was comparable between AAA and Leja–Bagby methods, with both achieving residual norms below 10−12 using around 30–40 Krylov iterations.
• The set-valued AAA approach produced smaller rational approximations than NLEIGS in all tested cases, with no degradation in accuracy.
• The method required significantly less user effort than NLEIGS, as pole and interpolation point selection was fully automated by the AAA algorithm.
• The linearization via Su and Bai’s state-space formulation was fully compatible with the CORK framework, enabling efficient solution of large-scale problems.