What question did this study set out to answer?

To estimate the least eigenvalue of the Helmholtz equation with a stochastic density function using numerical methods.

March 26, 2026

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

Puntos clave

To estimate the least eigenvalue of the Helmholtz equation with a stochastic density function using numerical methods.
Investigated the least eigenvalue with a series expansion of density function.
Applied finite element method to solve the eigenvalue problem.
Utilized quasi-Monte Carlo method for integral approximation.
Proved uniform bounds of finite element errors for eigenvalues and eigenfunctions.
Derived total truncation-FE-QMC error and verified with a numerical example.
Established uniform bounds for the least eigenvalue and eigenfunction.
Provided detailed bounds regarding their derivatives with respect to stochastic variables.
Verified estimation accuracy through numerical example.

Resumen

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.

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