Numerical estimation of the eigenvalue problem with stochastic density function based on quasi-Monte Carlo method

The expectation value of the least eigenvalue of the Helmholtz equation with stochastic density function is investigated. The density function is assumed to admit series expansion, where the stochastic variables and the physical variables are separated. By truncating the density function to a finite dimension, the finite element (FE) method is applied to solve the eigenvalue problem. The finite-dimensional integral is approximated by quasi-Monte Carlo (QMC) method. The uniform bounds of the FE errors of the least eigenvalue and the least eigenfunction are proved. Detailed study of the bounds of the derivatives of the least eigenvalue and the least eigenfunction with respect to the stochastic variables are provided. The total truncation-FE-QMC error is derived. The numerical example is conducted to verify the estimation of the total truncation-FE-QMC error.