Theory and numerics of subspace approximation of eigenvalue problems

TL;DR

This paper develops theoretical error estimates and demonstrates the effectiveness of reduced basis methods for efficiently approximating large-scale parametric eigenvalue problems, balancing computational savings with high accuracy.

Contribution

It provides new theoretical foundations for subspace approximation error analysis under non-simple eigenvalues and validates the approach through comprehensive numerical experiments.

Findings

Error estimates for non-simple eigenvalues

Reduced basis methods achieve significant computational savings

High accuracy maintained in large-scale parametric eigenproblems

Abstract

Large-scale eigenvalue problems arise in various fields of science and engineering and demand computationally efficient solutions. In this study, we investigate the subspace approximation for parametric linear eigenvalue problems, aiming to mitigate the computational burden associated with high-fidelity systems. We provide general error estimates under non-simple eigenvalue conditions, establishing some theoretical foundations for understanding the convergence behavior of subspace approximations. Numerical examples, including problems with one-dimensional to three-dimensional spatial domain and one-dimensional to two-dimensional parameter domain, are presented to demonstrate the efficacy of reduced basis method in handling parametric variations in boundary conditions and coefficient fields to achieve significant computational savings while maintaining high accuracy, making them promising tools for practical applications in large-scale eigenvalue computations.