The bounds of the mixed derivatives of the least eigenvalue and eigenfunction of the eigenvalue problem with stochastic density function

The eigenvalue problem with stochastic density function is investigated. The density function is assumed to admit series expansion. By truncating the density function series to a finite dimension, the finite element (FE) method is applied to solve the eigenvalue problem. The finite-dimensional integral is approximated by the quasi-Monte Carlo (QMC) method. The bounds of the mixed derivatives of the least eigenvalue and the eigenfunction with respect to the stochastic parameter are estimated by induction. Then, the total truncation-FE-QMC error is obtained based on the bounds of the mixed derivatives of the least eigenvalue and eigenfunction for the eigenvalue problem. Numerical results validate the estimation of the total truncation-FE-QMC error.