A Fast-Convergence Resolution of the Stochastic Eigenproblem Using Halley's Method and the Spectral-Chaos Approach

Abstract: Solving stochastic eigenvalue problems has long been essential for informed decision-making, advancing scientific knowledge, and ensuring the reliability of engineering designs and applications. This paper underscores the need to continue enhancing existing numerical methods for solving the stochastic eigenproblem in order to improve convergence rates, computational efficiency, and robustness. Specifically, we propose a novel spectral-chaos method for solving the stochastic (linear) eigenvalue problem, employing Halley's method as the root-finding algorithm to leverage its cubic convergence properties. Our method achieves maximal convergence in solving stochastic eigenvalue problems since its rate cannot be further improved using a higher-order Householder method due to the quadratic nature of the resulting system of equations. Additionally, due to the complexity of the resulting system of equations, a tensorial approach was developed to tackle the challenges associated with the dimensional multiplicity of the stochastic eigenvalue problem, without which the solution would have been intractable. The method is derived rigorously, with a detailed error analysis that highlights the benefit of using our approach when the eigenvector components are nearly known, the computational cost of the method is also rigorously presented, and an illustrative example is provided to demonstrate the implementation of the method. Subsequently, a case study is demoed to analyze the results and validate the advantages of using Halley's method over Newton's method and Monte Carlo simulations.